{"group":{"id":1,"name":"Community","lockable":false,"created_at":"2012-01-18T18:02:15.000Z","updated_at":"2026-04-06T14:01:22.000Z","description":"Problems submitted by members of the MATLAB Central community.","is_default":true,"created_by":161519,"badge_id":null,"featured":false,"trending":false,"solution_count_in_trending_period":0,"trending_last_calculated":"2026-04-06T00:00:00.000Z","image_id":null,"published":true,"community_created":false,"status_id":2,"is_default_group_for_player":false,"deleted_by":null,"deleted_at":null,"restored_by":null,"restored_at":null,"description_opc":null,"description_html":null,"published_at":null},"problems":[{"id":2736,"title":"Pernicious Anniversary Problem","description":"Since Cody is 5 years old, it's pernicious. A \u003chttp://rosettacode.org/wiki/Pernicious_numbers Pernicious number\u003e is an integer whose population count is a prime. Check if the given number is pernicious.","description_html":"\u003cp\u003eSince Cody is 5 years old, it's pernicious. A \u003ca href = \"http://rosettacode.org/wiki/Pernicious_numbers\"\u003ePernicious number\u003c/a\u003e is an integer whose population count is a prime. Check if the given number is pernicious.\u003c/p\u003e","function_template":"function y = isPernicious(x)\r\n  y = false;\r\nend","test_suite":"%%\r\nx = 5;\r\ny_correct = true;\r\nassert(isequal(isPernicious(x),y_correct))\r\n%%\r\nx = 1;\r\ny_correct = false;\r\nassert(isequal(isPernicious(x),y_correct))\r\n%%\r\nx = 2^randi(16);\r\ny_correct = false;\r\nassert(isequal(isPernicious(x),y_correct))\r\n%%\r\nx = 17;\r\ny_correct = true;\r\nassert(isequal(isPernicious(x),y_correct))\r\n%%\r\nx = 18;\r\ny_correct = true;\r\nassert(isequal(isPernicious(x),y_correct))\r\n%%\r\nx = 61;\r\ny_correct = true;\r\nassert(isequal(isPernicious(x),y_correct))\r\n%%\r\nx = 6;\r\ny_correct = true;\r\nassert(isequal(isPernicious(x),y_correct))\r\n%%\r\nx = 2115;\r\ny_correct = false;\r\nassert(isequal(isPernicious(x),y_correct))\r\n%%\r\nx = 2114;\r\ny_correct = true;\r\nassert(isequal(isPernicious(x),y_correct))\r\n%%\r\nx = 2017;\r\ny_correct = true;\r\nassert(isequal(isPernicious(x),y_correct))\r\n\r\n\r\n\r\n\r\n","published":true,"deleted":false,"likes_count":13,"comments_count":1,"created_by":14358,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":838,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":34,"created_at":"2014-12-08T08:48:45.000Z","updated_at":"2026-04-10T14:31:08.000Z","published_at":"2017-10-16T01:45:06.000Z","restored_at":"2017-10-25T14:37:50.000Z","restored_by":null,"spam":false,"simulink":false,"admin_reviewed":true,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eSince Cody is 5 years old, it's pernicious. A\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://rosettacode.org/wiki/Pernicious_numbers\\\"\u003e\u003cw:r\u003e\u003cw:t\u003ePernicious number\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e is an integer whose population count is a prime. Check if the given number is pernicious.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":248,"title":"Twins in a Window","description":"\u003chttp://en.wikipedia.org/wiki/Twin_primes Twin primes\u003e are prime numbers that differ by 2, such as [11,13] or [41,43]. Write a function that returns a sorted list of the twin primes between lowVal and highVal (all the primes must be greater than lowVal and less than highVal). The primes should be arranged in an n-by-2 matrix as shown in the examples below.\r\n\r\nExamples:\r\n\r\n Input  lowVal = 10, highVal = 50\r\n Output p is [ 11    13\r\n               17    19\r\n               29    31\r\n               41    43 ]\r\n\r\n Input  lowVal = 1000, highVal = 1050\r\n Output p is [ 1019  1021\r\n               1031  1033 ]\r\n","description_html":"\u003cp\u003e\u003ca href=\"http://en.wikipedia.org/wiki/Twin_primes\"\u003eTwin primes\u003c/a\u003e are prime numbers that differ by 2, such as [11,13] or [41,43]. Write a function that returns a sorted list of the twin primes between lowVal and highVal (all the primes must be greater than lowVal and less than highVal). The primes should be arranged in an n-by-2 matrix as shown in the examples below.\u003c/p\u003e\u003cp\u003eExamples:\u003c/p\u003e\u003cpre\u003e Input  lowVal = 10, highVal = 50\r\n Output p is [ 11    13\r\n               17    19\r\n               29    31\r\n               41    43 ]\u003c/pre\u003e\u003cpre\u003e Input  lowVal = 1000, highVal = 1050\r\n Output p is [ 1019  1021\r\n               1031  1033 ]\u003c/pre\u003e","function_template":"function p = window_twins(lowVal,highVal)\r\n  p = 0;\r\nend","test_suite":"%%\r\nlowVal = 10;\r\nhighVal = 50;\r\np = [11    13\r\n     17    19\r\n     29    31\r\n     41    43];\r\nassert(isequal(window_twins(lowVal,highVal),p))\r\n\r\n%%\r\nlowVal = 1000;\r\nhighVal = 1050;\r\np = [1019  1021\r\n     1031  1033];\r\nassert(isequal(window_twins(lowVal,highVal),p))\r\n\r\n%%\r\nlowVal = 3120;\r\nhighVal = 3300;\r\np = [ 3167  3169\r\n      3251  3253\r\n      3257  3259];\r\nassert(isequal(window_twins(lowVal,highVal),p))\r\n","published":true,"deleted":false,"likes_count":2,"comments_count":0,"created_by":7,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":162,"test_suite_updated_at":"2012-02-03T17:12:57.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2012-02-03T16:30:52.000Z","updated_at":"2026-04-10T14:29:19.000Z","published_at":"2012-02-03T17:12:56.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"http://en.wikipedia.org/wiki/Twin_primes\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eTwin primes\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e are prime numbers that differ by 2, such as [11,13] or [41,43]. Write a function that returns a sorted list of the twin primes between lowVal and highVal (all the primes must be greater than lowVal and less than highVal). The primes should be arranged in an n-by-2 matrix as shown in the examples below.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eExamples:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[ Input  lowVal = 10, highVal = 50\\n Output p is [ 11    13\\n               17    19\\n               29    31\\n               41    43 ]\\n\\n Input  lowVal = 1000, highVal = 1050\\n Output p is [ 1019  1021\\n               1031  1033 ]]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":44230,"title":"I'm going to enjoy watching you calculate, Mr Anderson","description":"Smith numbers are numbers such that if you add up all of the digits in the number, that sum equals the sum of all of the digits in all of their factors. This term was coined by Albert Wilansky, when he noticed the defining property in the phone number of his brother-in-law Harold Smith: 493-7775.\r\n4+9+3+7+7+7+5=42\r\nThe prime factors of 4937775 are 3, 5, 5, and 65837. 3+5+5+(6+5+8+3+7) = 42\r\nSince all prime numbers obviously meet this criteria, Smith numbers are defined as composite numbers. Write a MATLAB function that will tell you if a given number is a Smith number.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 174px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 87px; transform-origin: 407px 87px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 375.5px 8px; transform-origin: 375.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eSmith numbers are numbers such that if you add up all of the digits in the number, that sum equals the sum of all of the digits in all of their factors. This term was coined by Albert Wilansky, when he noticed the defining property in the phone number of his brother-in-law Harold Smith: 493-7775.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 64px 8px; transform-origin: 64px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e4+9+3+7+7+7+5=42\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 251px 8px; transform-origin: 251px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe prime factors of 4937775 are 3, 5, 5, and 65837. 3+5+5+(6+5+8+3+7) = 42\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 380.5px 8px; transform-origin: 380.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eSince all prime numbers obviously meet this criteria, Smith numbers are defined as composite numbers. Write a MATLAB function that will tell you if a given number is a Smith number.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = smith(x)\r\n  y = x;\r\nend","test_suite":"%%\r\nassert(isequal(smith(4937775),1))\r\n%%\r\nassert(isequal(smith(1164),0))\r\n%%\r\nassert(isequal(smith(19683),1))\r\n%%\r\nassert(isequal(smith(11),0))  % Remember - Smith numbers are composite\r\n%%\r\nassert(isequal(smith(11^2),1))\r\n%%\r\nassert(isequal(smith(345741),1))\r\n%%\r\nassert(isequal(smith(19876),0))\r\n%%\r\nassert(isequal(smith(314159),0))\r\n%%\r\nassert(isequal(smith(612985),1))\r\n%%\r\nassert(isequal(smith(12379887),1))\r\n%%\r\nassert(isequal(smith(23456789),0))\r\n%%\r\nassert(isequal(smith(13),0))\r\n%%\r\nassert(isequal(smith(23),0))\r\n%%\r\ny=primes(randi(1e5));\r\nassert(isequal(smith(y(end)),0))\r\n","published":true,"deleted":false,"likes_count":2,"comments_count":1,"created_by":1615,"edited_by":223089,"edited_at":"2023-01-07T08:26:27.000Z","deleted_by":null,"deleted_at":null,"solvers_count":49,"test_suite_updated_at":"2023-01-07T08:26:27.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2017-06-01T19:16:25.000Z","updated_at":"2026-03-16T15:29:16.000Z","published_at":"2017-06-01T19:16:25.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eSmith numbers are numbers such that if you add up all of the digits in the number, that sum equals the sum of all of the digits in all of their factors. This term was coined by Albert Wilansky, when he noticed the defining property in the phone number of his brother-in-law Harold Smith: 493-7775.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e4+9+3+7+7+7+5=42\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe prime factors of 4937775 are 3, 5, 5, and 65837. 3+5+5+(6+5+8+3+7) = 42\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eSince all prime numbers obviously meet this criteria, Smith numbers are defined as composite numbers. Write a MATLAB function that will tell you if a given number is a Smith number.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":44384,"title":"Find the nearest prime number","description":"Happy 5th birthday, Cody! Since 5 is a prime number, let's have some fun looking for other prime numbers.\r\n\r\nGiven a positive integer x, find the nearest prime number. Keep in mind that the nearest prime may be less than x.\r\n\r\n*Examples*\r\n\r\n  nearestprime(5) = 5\r\n  nearestprime(36) = 37\r\n  nearestprime(200) = 199\r\n\r\nNOTE: You may ignore cases in which two prime numbers are equally close to x. (e.g., x=12 --\u003e 11 and 13 are both primes). ","description_html":"\u003cp\u003eHappy 5th birthday, Cody! Since 5 is a prime number, let's have some fun looking for other prime numbers.\u003c/p\u003e\u003cp\u003eGiven a positive integer x, find the nearest prime number. Keep in mind that the nearest prime may be less than x.\u003c/p\u003e\u003cp\u003e\u003cb\u003eExamples\u003c/b\u003e\u003c/p\u003e\u003cpre class=\"language-matlab\"\u003enearestprime(5) = 5\r\nnearestprime(36) = 37\r\nnearestprime(200) = 199\r\n\u003c/pre\u003e\u003cp\u003eNOTE: You may ignore cases in which two prime numbers are equally close to x. (e.g., x=12 --\u0026gt; 11 and 13 are both primes).\u003c/p\u003e","function_template":"function y = nearestprime(x)\r\n  y = x;\r\nend","test_suite":"%%\r\nx = 0;\r\ny_correct = 2;\r\nassert(isequal(nearestprime(x),y_correct))\r\n\r\n%%\r\nx = 5;\r\ny_correct = 5;\r\nassert(isequal(nearestprime(x),y_correct))\r\n\r\n%%\r\nx = 100;\r\ny_correct = 101;\r\nassert(isequal(nearestprime(x),y_correct))\r\n\r\n%%\r\nx = 500;\r\ny_correct = 499;\r\nassert(isequal(nearestprime(x),y_correct))\r\n\r\n%%\r\nx = 911;\r\ny_correct = 911;\r\nassert(isequal(nearestprime(x),y_correct))\r\n\r\n%%\r\nx = 2500;\r\ny_correct = 2503;\r\nassert(isequal(nearestprime(x),y_correct))\r\n\r\n%%\r\nx = 8000;\r\ny_correct = 7993;\r\nassert(isequal(nearestprime(x),y_correct))\r\n\r\n%%\r\nx = 100000;\r\ny_correct = 100003;\r\nassert(isequal(nearestprime(x),y_correct))\r\n\r\n%%\r\nx = 1300000;\r\ny_correct = 1299989;\r\nassert(isequal(nearestprime(x),y_correct))\r\n\r\n%%\r\nx = 179424710;\r\ny_correct = 179424719;\r\nassert(isequal(nearestprime(x),y_correct))\r\n","published":true,"deleted":false,"likes_count":9,"comments_count":1,"created_by":4793,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":664,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":34,"created_at":"2017-10-13T19:42:15.000Z","updated_at":"2026-04-07T15:16:58.000Z","published_at":"2017-10-16T01:45:10.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eHappy 5th birthday, Cody! Since 5 is a prime number, let's have some fun looking for other prime numbers.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eGiven a positive integer x, find the nearest prime number. Keep in mind that the nearest prime may be less than x.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eExamples\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[nearestprime(5) = 5\\nnearestprime(36) = 37\\nnearestprime(200) = 199]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eNOTE: You may ignore cases in which two prime numbers are equally close to x. (e.g., x=12 --\u0026gt; 11 and 13 are both primes).\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":3016,"title":"Twin Primes","description":"Twin primes are pairs of primes that are immediately next to each other (difference of two). The lesser of twin primes are 3, 5, 11, 17, 29, ... ( \u003chttp://oeis.org/A001359 ref.\u003e ). The greater of twin primes are 5, 7, 13, 19, 31, ... ( \u003chttp://oeis.org/A006512 ref.\u003e ). Therefore, the first five twin primes are [3,5] [5,7] [11,13] [17,19] [29,31].\r\n\r\nFor a given index range n, return the twin primes corresponding to that range as a two-row column array.","description_html":"\u003cp\u003eTwin primes are pairs of primes that are immediately next to each other (difference of two). The lesser of twin primes are 3, 5, 11, 17, 29, ... ( \u003ca href = \"http://oeis.org/A001359\"\u003eref.\u003c/a\u003e ). The greater of twin primes are 5, 7, 13, 19, 31, ... ( \u003ca href = \"http://oeis.org/A006512\"\u003eref.\u003c/a\u003e ). Therefore, the first five twin primes are [3,5] [5,7] [11,13] [17,19] [29,31].\u003c/p\u003e\u003cp\u003eFor a given index range n, return the twin primes corresponding to that range as a two-row column array.\u003c/p\u003e","function_template":"function [twins] = twin_primes(n)\r\n\r\ntwins = n;\r\n\r\nend","test_suite":"%%\r\nn = 1:5;\r\ntwins_corr = [3, 5, 11, 17, 29; 5, 7, 13, 19, 31];\r\nassert(isequal(twin_primes(n),twins_corr))\r\n\r\n%%\r\nn = 1:10;\r\ntwins_corr = [3, 5, 11, 17, 29, 41, 59, 71, 101, 107; 5, 7, 13, 19, 31, 43, 61, 73, 103, 109];\r\nassert(isequal(twin_primes(n),twins_corr))\r\n\r\n%%\r\nn = 1:25;\r\ntwins_corr = [3, 5, 11, 17, 29, 41, 59, 71, 101, 107, 137, 149, 179, 191, 197, 227, 239, 269, 281, 311, 347, 419, 431, 461, 521; 5, 7, 13, 19, 31, 43, 61, 73, 103, 109, 139, 151, 181, 193, 199, 229, 241, 271, 283, 313, 349, 421, 433, 463, 523];\r\nassert(isequal(twin_primes(n),twins_corr))\r\n\r\n%%\r\nn = 1:51;\r\ntwins_corr = [3, 5, 11, 17, 29, 41, 59, 71, 101, 107, 137, 149, 179, 191, 197, 227, 239, 269, 281, 311, 347, 419, 431, 461, 521, 569, 599, 617, 641, 659, 809, 821, 827, 857, 881, 1019, 1031, 1049, 1061, 1091, 1151, 1229, 1277, 1289, 1301, 1319, 1427, 1451, 1481, 1487, 1607; 5, 7, 13, 19, 31, 43, 61, 73, 103, 109, 139, 151, 181, 193, 199, 229, 241, 271, 283, 313, 349, 421, 433, 463, 523, 571, 601, 619, 643, 661, 811, 823, 829, 859, 883, 1021, 1033, 1051, 1063, 1093, 1153, 1231, 1279, 1291, 1303, 1321, 1429, 1453, 1483, 1489, 1609];\r\nassert(isequal(twin_primes(n),twins_corr))\r\n\r\n%%\r\nn = 10:29;\r\ntwins_corr = [107, 137, 149, 179, 191, 197, 227, 239, 269, 281, 311, 347, 419, 431, 461, 521, 569, 599, 617, 641; 109, 139, 151, 181, 193, 199, 229, 241, 271, 283, 313, 349, 421, 433, 463, 523, 571, 601, 619, 643];\r\nassert(isequal(twin_primes(n),twins_corr))\r\n\r\n%%\r\nn = 2:8;\r\ntwins_corr = [5, 11, 17, 29, 41, 59, 71; 7, 13, 19, 31, 43, 61, 73];\r\nassert(isequal(twin_primes(n),twins_corr))\r\n\r\n%%\r\nn = 35:42;\r\ntwins_corr = [881, 1019, 1031, 1049, 1061, 1091, 1151, 1229; 883, 1021, 1033, 1051, 1063, 1093, 1153, 1231];\r\nassert(isequal(twin_primes(n),twins_corr))\r\n\r\n%%\r\nn = 34:47;\r\ntwins_corr = [857, 881, 1019, 1031, 1049, 1061, 1091, 1151, 1229, 1277, 1289, 1301, 1319, 1427; 859, 883, 1021, 1033, 1051, 1063, 1093, 1153, 1231, 1279, 1291, 1303, 1321, 1429];\r\nassert(isequal(twin_primes(n),twins_corr))\r\n\r\n%%\r\nn = 9:-1:4;\r\ntwins_corr = [101, 71, 59, 41, 29, 17; 103, 73, 61, 43, 31, 19];\r\nassert(isequal(twin_primes(n),twins_corr))\r\n","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":26769,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":98,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":30,"created_at":"2015-02-14T03:03:50.000Z","updated_at":"2026-03-16T14:18:09.000Z","published_at":"2015-02-14T03:03:50.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eTwin primes are pairs of primes that are immediately next to each other (difference of two). The lesser of twin primes are 3, 5, 11, 17, 29, ... (\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://oeis.org/A001359\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eref.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e ). The greater of twin primes are 5, 7, 13, 19, 31, ... (\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://oeis.org/A006512\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eref.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e ). Therefore, the first five twin primes are [3,5] [5,7] [11,13] [17,19] [29,31].\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor a given index range n, return the twin primes corresponding to that range as a two-row column array.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":44385,"title":"Extra safe primes","description":"Did you know that the number 5 is the first safe prime? A safe prime is a prime number that can be expressed as 2p+1, where p is also a prime.\r\n\r\nTo celebrate Cody's Five-Year Anniversary, write a function to determine if a positive integer n is a safe prime in which the prime p (such that n=2p+1) is *also a safe prime*.\r\n\r\n*Examples*\r\n\r\n  isextrasafe(5) = false % because 5=2*2+1 and 2 is not a safe prime\r\n  isextrasafe(23) = true % because 23=2*11+1 and 11 is also a safe prime (11=2*5+1)\r\n","description_html":"\u003cp\u003eDid you know that the number 5 is the first safe prime? A safe prime is a prime number that can be expressed as 2p+1, where p is also a prime.\u003c/p\u003e\u003cp\u003eTo celebrate Cody's Five-Year Anniversary, write a function to determine if a positive integer n is a safe prime in which the prime p (such that n=2p+1) is \u003cb\u003ealso a safe prime\u003c/b\u003e.\u003c/p\u003e\u003cp\u003e\u003cb\u003eExamples\u003c/b\u003e\u003c/p\u003e\u003cpre class=\"language-matlab\"\u003eisextrasafe(5) = false % because 5=2*2+1 and 2 is not a safe prime\r\nisextrasafe(23) = true % because 23=2*11+1 and 11 is also a safe prime (11=2*5+1)\r\n\u003c/pre\u003e","function_template":"function tf = isextrasafe(x)\r\n    tf = false;\r\nend","test_suite":"%%\r\nx = 0;\r\nassert(isequal(isextrasafe(x),false))\r\n\r\n%%\r\nx = 5;\r\nassert(isequal(isextrasafe(x),false))\r\n\r\n%%\r\nx = 7;\r\nassert(isequal(isextrasafe(x),false))\r\n\r\n%%\r\nx = 11;\r\nassert(isequal(isextrasafe(x),true))\r\n\r\n%%\r\nx = 15;\r\nassert(isequal(isextrasafe(x),false))\r\n\r\n%%\r\nx = 23;\r\nassert(isequal(isextrasafe(x),true))\r\n\r\n%%\r\nx = 71;\r\nassert(isequal(isextrasafe(x),false))\r\n\r\n%%\r\nx = 719;\r\nassert(isequal(isextrasafe(x),true))\r\n\r\n%%\r\nx = 2039;\r\nassert(isequal(isextrasafe(x),true))\r\n\r\n%%\r\nx = 2040;\r\nassert(isequal(isextrasafe(x),false))\r\n\r\n%%\r\nx = 5807;\r\nassert(isequal(isextrasafe(x),true))","published":true,"deleted":false,"likes_count":13,"comments_count":4,"created_by":4793,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":757,"test_suite_updated_at":"2017-10-19T17:09:19.000Z","rescore_all_solutions":true,"group_id":34,"created_at":"2017-10-13T20:02:13.000Z","updated_at":"2026-04-10T14:37:08.000Z","published_at":"2017-10-16T01:45:10.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eDid you know that the number 5 is the first safe prime? A safe prime is a prime number that can be expressed as 2p+1, where p is also a prime.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eTo celebrate Cody's Five-Year Anniversary, write a function to determine if a positive integer n is a safe prime in which the prime p (such that n=2p+1) is\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ealso a safe prime\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eExamples\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[isextrasafe(5) = false % because 5=2*2+1 and 2 is not a safe prime\\nisextrasafe(23) = true % because 23=2*11+1 and 11 is also a safe prime (11=2*5+1)]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":1204,"title":"Prime Time","description":"All you need to do here is submit your solution a prime number of seconds after the top of the hour.  Any hour at all...  Easy, right?","description_html":"\u003cp\u003eAll you need to do here is submit your solution a prime number of seconds after the top of the hour.  Any hour at all...  Easy, right?\u003c/p\u003e","function_template":"function y = prime_time(x)\r\n  y = x;\r\nend","test_suite":"%%\r\nfiletext = fileread('prime_time.m')\r\nassert(isempty(strfind(filetext, 'fopen')))\r\n%%\r\nx=123\r\ny = prime_time(x)\r\na=clock\r\nb=floor(polyval(a(5:6),60))\r\nif isprime(b)\r\ny_correct= y\r\nelse \r\ny_correct = NaN; \r\nend\r\nassert(isequal(y,y_correct),sprintf('%s%g%s','Time is ',datestr(now,13),', or ',b,' seconds after the hour.'))","published":true,"deleted":false,"likes_count":2,"comments_count":0,"created_by":1615,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":63,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2013-01-14T18:34:19.000Z","updated_at":"2025-11-22T17:34:33.000Z","published_at":"2013-01-14T18:34:19.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eAll you need to do here is submit your solution a prime number of seconds after the top of the hour. Any hour at all... Easy, right?\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":896,"title":"Sophie Germain prime","description":"In number theory, a prime number p is a *Sophie Germain prime* if 2p + 1 is also prime. For example, 23 is a Sophie Germain prime because it is a prime and 2 × 23 + 1 = 47, and 47 is also a prime number. These numbers are named after French mathematician Marie-Sophie Germain.\r\n\r\nSee \u003chttp://en.wikipedia.org/wiki/Sophie_Germain_prime Sophie Germain prime\u003e article on Wikipedia.\r\n\r\n\r\nIn this Problem , the input is a number and you must return true or false if this number is a Sophie Germain prime.","description_html":"\u003cp\u003eIn number theory, a prime number p is a \u003cb\u003eSophie Germain prime\u003c/b\u003e if 2p + 1 is also prime. For example, 23 is a Sophie Germain prime because it is a prime and 2 × 23 + 1 = 47, and 47 is also a prime number. These numbers are named after French mathematician Marie-Sophie Germain.\u003c/p\u003e\u003cp\u003eSee \u003ca href=\"http://en.wikipedia.org/wiki/Sophie_Germain_prime\"\u003eSophie Germain prime\u003c/a\u003e article on Wikipedia.\u003c/p\u003e\u003cp\u003eIn this Problem , the input is a number and you must return true or false if this number is a Sophie Germain prime.\u003c/p\u003e","function_template":"function tf = your_fcn_name(x)\r\n  tf = true;\r\nend","test_suite":"%%\r\np = 233;\r\ny_correct = true;\r\nassert(isequal(your_fcn_name(p),y_correct))\r\n\r\n%%\r\np = 23;\r\ny_correct14 = true;\r\nassert(isequal(your_fcn_name(p),y_correct14))\r\n\r\n%%\r\np = 22;\r\ny_correct14 = false;\r\nassert(isequal(your_fcn_name(p),y_correct14))\r\n\r\n%% \r\np = 1 % p must also be a prime number !!\r\ny_correct1t = false;\r\nassert(isequal(your_fcn_name(p),y_correct1t))\r\n\r\n%% \r\np = 14 % p must also be a prime number !!\r\ncorrect1t = false;\r\nassert(isequal(your_fcn_name(p),correct1t))\r\n\r\n%% \r\np = 29 \r\ncorrect1tp = true;\r\nassert(isequal(your_fcn_name(p),correct1tp))\r\n","published":true,"deleted":false,"likes_count":6,"comments_count":2,"created_by":639,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":1066,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":44,"created_at":"2012-08-10T13:04:11.000Z","updated_at":"2026-04-09T08:16:22.000Z","published_at":"2012-08-10T13:04:11.000Z","restored_at":"2018-10-10T14:57:27.000Z","restored_by":null,"spam":false,"simulink":false,"admin_reviewed":true,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIn number theory, a prime number p is a\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eSophie Germain prime\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e if 2p + 1 is also prime. For example, 23 is a Sophie Germain prime because it is a prime and 2 × 23 + 1 = 47, and 47 is also a prime number. These numbers are named after French mathematician Marie-Sophie Germain.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eSee\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://en.wikipedia.org/wiki/Sophie_Germain_prime\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eSophie Germain prime\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e article on Wikipedia.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIn this Problem , the input is a number and you must return true or false if this number is a Sophie Germain prime.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":2337,"title":"Sum of big primes without primes","description":"Inspired by Project Euler n°10 (I am quite obviously a fan).\r\nWith problem n°250 by Doug, you can find some global methods to compute the sum of all the primes below the input n.\r\nFor example, the sum of the primes below 10 is 2 + 3 + 5 + 7 = 17.\r\nBut how to proceed (in time) with big number and WITHOUT the primes function ?\r\nHINTS: sum(primes(n)) is possible here but why miss the wonderfull Sieve of Eratosthenes ?\r\nhttp://en.wikipedia.org/wiki/Sieve_of_Eratosthenes","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 171px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 85.5px; transform-origin: 407px 85.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 183px 8px; transform-origin: 183px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eInspired by Project Euler n°10 (I am quite obviously a fan).\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 376px 8px; transform-origin: 376px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWith problem n°250 by Doug, you can find some global methods to compute the sum of all the primes below the input n.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 208px 8px; transform-origin: 208px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eFor example, the sum of the primes below 10 is 2 + 3 + 5 + 7 = 17.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 255.5px 8px; transform-origin: 255.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eBut how to proceed (in time) with big number and WITHOUT the primes function ?\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 288.5px 8px; transform-origin: 288.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eHINTS: sum(primes(n)) is possible here but why miss the wonderfull Sieve of Eratosthenes ?\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"/#null\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003ehttp://en.wikipedia.org/wiki/Sieve_of_Eratosthenes\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = big_euler10(n)\r\n  y = x;\r\nend","test_suite":"%%\r\nfiletext = fileread('big_euler10.m');\r\nillegal = contains(filetext, 'assignin') || contains(filetext, 'regexp') || ...\r\n          contains(filetext, 'primes'); \r\nassert(~illegal)\r\n\r\n%%\r\nx = 1;\r\ny_correct = 0;\r\nassert(isequal(big_euler10(x),y_correct))\r\n%%\r\nx = 10;\r\ny_correct = 17;\r\nassert(isequal(big_euler10(x),y_correct))\r\n%%\r\nx = 100;\r\ny_correct = 1060;\r\nassert(isequal(big_euler10(x),y_correct))\r\n%%\r\nx = 1000;\r\ny_correct = 76127;\r\nassert(isequal(big_euler10(x),y_correct))\r\n%%\r\nx = 10000;\r\ny_correct = 5736396;\r\nassert(isequal(big_euler10(x),y_correct))\r\n%%\r\nx = 100000;\r\ny_correct = 454396537;\r\nassert(isequal(big_euler10(x),y_correct))\r\n%%\r\nx = 1000000;\r\ny_correct = 37550402023;\r\nassert(isequal(big_euler10(x),y_correct))\r\n%%\r\nx = 1000000-100;\r\ny_correct = 37542402433;\r\nassert(isequal(big_euler10(x),y_correct))\r\n%%\r\nx = 2000000-1000;\r\ny_correct = 142781862782;\r\nassert(isequal(big_euler10(x),y_correct))\r\n%% Solution of Project Euler 10 with n=2000000\r\n\r\n","published":true,"deleted":false,"likes_count":3,"comments_count":3,"created_by":5390,"edited_by":223089,"edited_at":"2023-06-05T10:25:19.000Z","deleted_by":null,"deleted_at":null,"solvers_count":239,"test_suite_updated_at":"2023-06-05T10:25:19.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2014-05-27T21:25:58.000Z","updated_at":"2026-03-29T22:02:38.000Z","published_at":"2014-05-27T21:51:18.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eInspired by Project Euler n°10 (I am quite obviously a fan).\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWith problem n°250 by Doug, you can find some global methods to compute the sum of all the primes below the input n.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor example, the sum of the primes below 10 is 2 + 3 + 5 + 7 = 17.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eBut how to proceed (in time) with big number and WITHOUT the primes function ?\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eHINTS: sum(primes(n)) is possible here but why miss the wonderfull Sieve of Eratosthenes ?\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"\\\"\u003e\u003cw:r\u003e\u003cw:t\u003ehttp://en.wikipedia.org/wiki/Sieve_of_Eratosthenes\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":44403,"title":"Goldbach's marginal conjecture - Write integer as sum of three primes","description":"Goldbach's strong conjecture states that every even integer greater than 2 can be expressed as the sum of two primes. For example: 4 = 2+2, 6 = 3+3, 8 = 3+5, 10 = 3+7 = 5+5, 12 = 5+7 etc.\r\n\r\nAs a corrollary, Goldbach's weak conjecture states that every odd integer greater than 7 can be expressed as the sum of three odd primes. For example: 9 = 3+3+3, 11 = 3+3+5, 13 = 3+3+7 = 3+5+5, 15 = 3+5+7 = 5+5+5 etc.\r\n\r\nA third conjecture was written by Goldbach in the margin of a letter, and (in its modern version) states that \r\n\r\n\" _Every integer greater than 5 can be expressed as the sum of three primes._ \"\r\n\r\nExamples:\r\n\r\n*  6 = 2 + 2 + 2\r\n*  7 = 2 + 2 + 3\r\n*  8 = 2 + 3 + 3 \r\n*  9 = 2 + 2 + 5 = 3 + 3 + 3 \r\n* 10 = 2 + 3 + 5\r\n* 11 = 2 + 2 + 7 = 3 + 3 + 5\r\n* 12 = 2 + 3 + 7 = 2 + 5 + 5\r\n* 13 = 3 + 3 + 7 = 3 + 5 + 5\r\n* 14 = 2 + 5 + 7\r\n* 15 = 2 + 2 + 11 = 3 + 5 + 7 = 5 + 5 + 5\r\n\r\nYour task is to write a function which takes a positive integer _n_ as input, and which returns a 1-by-3 vector _y_, which contains three numbers that are primes and whose sum equals _n_. If there exist multiple solutions for _y_, then any one of those solutions will suffice. However, _y_ must be in sorted order. You can assume that _n_ will be an integer greater than 5.\r\n\r\n","description_html":"\u003cp\u003eGoldbach's strong conjecture states that every even integer greater than 2 can be expressed as the sum of two primes. For example: 4 = 2+2, 6 = 3+3, 8 = 3+5, 10 = 3+7 = 5+5, 12 = 5+7 etc.\u003c/p\u003e\u003cp\u003eAs a corrollary, Goldbach's weak conjecture states that every odd integer greater than 7 can be expressed as the sum of three odd primes. For example: 9 = 3+3+3, 11 = 3+3+5, 13 = 3+3+7 = 3+5+5, 15 = 3+5+7 = 5+5+5 etc.\u003c/p\u003e\u003cp\u003eA third conjecture was written by Goldbach in the margin of a letter, and (in its modern version) states that\u003c/p\u003e\u003cp\u003e\" \u003ci\u003eEvery integer greater than 5 can be expressed as the sum of three primes.\u003c/i\u003e \"\u003c/p\u003e\u003cp\u003eExamples:\u003c/p\u003e\u003cul\u003e\u003cli\u003e6 = 2 + 2 + 2\u003c/li\u003e\u003cli\u003e7 = 2 + 2 + 3\u003c/li\u003e\u003cli\u003e8 = 2 + 3 + 3\u003c/li\u003e\u003cli\u003e9 = 2 + 2 + 5 = 3 + 3 + 3\u003c/li\u003e\u003cli\u003e10 = 2 + 3 + 5\u003c/li\u003e\u003cli\u003e11 = 2 + 2 + 7 = 3 + 3 + 5\u003c/li\u003e\u003cli\u003e12 = 2 + 3 + 7 = 2 + 5 + 5\u003c/li\u003e\u003cli\u003e13 = 3 + 3 + 7 = 3 + 5 + 5\u003c/li\u003e\u003cli\u003e14 = 2 + 5 + 7\u003c/li\u003e\u003cli\u003e15 = 2 + 2 + 11 = 3 + 5 + 7 = 5 + 5 + 5\u003c/li\u003e\u003c/ul\u003e\u003cp\u003eYour task is to write a function which takes a positive integer \u003ci\u003en\u003c/i\u003e as input, and which returns a 1-by-3 vector \u003ci\u003ey\u003c/i\u003e, which contains three numbers that are primes and whose sum equals \u003ci\u003en\u003c/i\u003e. If there exist multiple solutions for \u003ci\u003ey\u003c/i\u003e, then any one of those solutions will suffice. However, \u003ci\u003ey\u003c/i\u003e must be in sorted order. You can assume that \u003ci\u003en\u003c/i\u003e will be an integer greater than 5.\u003c/p\u003e","function_template":"function y = goldbach3(n)\r\n  y = [n,n,n];\r\nend","test_suite":"%%\r\nn = 6;\r\ny = goldbach3(n);\r\ny_correct = [2,2,2];\r\nassert(isequal(y,y_correct))\r\n\r\n%%\r\nn = 7;\r\ny = goldbach3(n);\r\ny_correct = [2,2,3];\r\nassert(isequal(y,y_correct))\r\n\r\n%%\r\nn = 8;\r\ny = goldbach3(n);\r\ny_correct = [2,3,3];\r\nassert(isequal(y,y_correct))\r\n\r\n%%\r\nn = 9;\r\ny = goldbach3(n);\r\ny_correct1 = [2,2,5];\r\ny_correct2 = [3,3,3];\r\nassert(isequal(y,y_correct1)|isequal(y,y_correct2))\r\n\r\n%%\r\nn = 10;\r\ny = goldbach3(n);\r\ny_correct = [2,3,5];\r\nassert(isequal(y,y_correct))\r\n\r\n%%\r\nn = 11;\r\ny = goldbach3(n);\r\ny_correct1 = [2,2,7];\r\ny_correct2 = [3,3,5];\r\nassert(isequal(y,y_correct1)|isequal(y,y_correct2))\r\n\r\n%%\r\nn = 12;\r\ny = goldbach3(n);\r\ny_correct1 = [2,3,7];\r\ny_correct2 = [2,5,5];\r\nassert(isequal(y,y_correct1)|isequal(y,y_correct2))\r\n\r\n%%\r\nn = 13;\r\ny = goldbach3(n);\r\ny_correct1 = [3,3,7];\r\ny_correct2 = [3,5,5];\r\nassert(isequal(y,y_correct1)|isequal(y,y_correct2))\r\n\r\n%%\r\nn = 14;\r\ny = goldbach3(n);\r\ny_correct = [2,5,7];\r\nassert(isequal(y,y_correct))\r\n\r\n%%\r\nn = 15;\r\ny = goldbach3(n);\r\ny_correct1 = [2,2,11];\r\ny_correct2 = [3,5,7];\r\ny_correct3 = [5,5,5];\r\nassert(isequal(y,y_correct1)|isequal(y,y_correct2)|isequal(y,y_correct3))\r\n\r\n%%\r\nn = 101;\r\ny = goldbach3(n);\r\nassert(isequal(y,sort(y)))\r\nassert(all(isprime(y)))\r\nassert(sum(y)==n)\r\n\r\n%%\r\nn = 102;\r\ny = goldbach3(n);\r\nassert(isequal(y,sort(y)))\r\nassert(all(isprime(y)))\r\nassert(sum(y)==n)\r\n\r\n%% \r\nfor n = 250:300\r\n    y = goldbach3(n);\r\n    assert(isequal(y,sort(y)));\r\n    assert(all(isprime(y)));\r\n    assert(sum(y)==n);\r\nend\r\n\r\n%%\r\nn = randi(2000)+5; % generate a random integer greater than 5 and smaller than 2006\r\ny = goldbach3(n);\r\nassert(isequal(y,sort(y)))\r\nassert(all(isprime(y)))\r\nassert(sum(y)==n)\r\n\r\n%% \r\nvalid = zeros(1,50);\r\nfor k = 1:50\r\n    n = randi(1000)+5; % generate a random integer greater than 5 and smaller than 1006\r\n    yk = goldbach3(n);\r\n    valid(k) = (isequal(yk,sort(yk)) \u0026 all(isprime(yk)) \u0026 sum(yk)==n);\r\nend\r\nassert(all(valid));\r\n","published":true,"deleted":false,"likes_count":3,"comments_count":2,"created_by":108199,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":75,"test_suite_updated_at":"2017-11-18T23:12:48.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2017-11-14T00:05:38.000Z","updated_at":"2026-03-16T15:38:02.000Z","published_at":"2017-11-14T01:21:48.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eGoldbach's strong conjecture states that every even integer greater than 2 can be expressed as the sum of two primes. For example: 4 = 2+2, 6 = 3+3, 8 = 3+5, 10 = 3+7 = 5+5, 12 = 5+7 etc.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eAs a corrollary, Goldbach's weak conjecture states that every odd integer greater than 7 can be expressed as the sum of three odd primes. For example: 9 = 3+3+3, 11 = 3+3+5, 13 = 3+3+7 = 3+5+5, 15 = 3+5+7 = 5+5+5 etc.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eA third conjecture was written by Goldbach in the margin of a letter, and (in its modern version) states that\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\"\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eEvery integer greater than 5 can be expressed as the sum of three primes.\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \\\"\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eExamples:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e6 = 2 + 2 + 2\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e7 = 2 + 2 + 3\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e8 = 2 + 3 + 3\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e9 = 2 + 2 + 5 = 3 + 3 + 3\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e10 = 2 + 3 + 5\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e11 = 2 + 2 + 7 = 3 + 3 + 5\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e12 = 2 + 3 + 7 = 2 + 5 + 5\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e13 = 3 + 3 + 7 = 3 + 5 + 5\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e14 = 2 + 5 + 7\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e15 = 2 + 2 + 11 = 3 + 5 + 7 = 5 + 5 + 5\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eYour task is to write a function which takes a positive integer\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e as input, and which returns a 1-by-3 vector\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ey\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, which contains three numbers that are primes and whose sum equals\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e. If there exist multiple solutions for\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ey\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, then any one of those solutions will suffice. However,\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ey\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e must be in sorted order. You can assume that\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e will be an integer greater than 5.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":2910,"title":"Mersenne Primes vs. All Primes","description":"A Mersenne prime (M) is a prime number of the form M = 2^p - 1, where p is another prime number. \u003chttps://www.mathworks.com/matlabcentral/cody/problems/525-mersenne-primes Problem 525\u003e asks the user to determine if a number is a Mersenne prime. In this problem, you are tasked with returning the number of primes numbers below the input number, n, that are Mersenne primes and the fraction of all primes below that input number that the Mersenne primes represent.\r\n\r\nFor example, for n = 100, there are 25 primes numbers: 2, 3, 5, 7, ..., 89, 97. As far as Mersenne primes go, there are only three that are less than 100: 2^2 - 1 = 3, 2^3 - 1 = 7, and 2^5 - 1 = 31. The corresponding fraction would be 3/25.","description_html":"\u003cp\u003eA Mersenne prime (M) is a prime number of the form M = 2^p - 1, where p is another prime number. \u003ca href = \"https://www.mathworks.com/matlabcentral/cody/problems/525-mersenne-primes\"\u003eProblem 525\u003c/a\u003e asks the user to determine if a number is a Mersenne prime. In this problem, you are tasked with returning the number of primes numbers below the input number, n, that are Mersenne primes and the fraction of all primes below that input number that the Mersenne primes represent.\u003c/p\u003e\u003cp\u003eFor example, for n = 100, there are 25 primes numbers: 2, 3, 5, 7, ..., 89, 97. As far as Mersenne primes go, there are only three that are less than 100: 2^2 - 1 = 3, 2^3 - 1 = 7, and 2^5 - 1 = 31. The corresponding fraction would be 3/25.\u003c/p\u003e","function_template":"function [y,f] = Mersenne_prime_comp(n)\r\n y = 1;\r\n f = 0;\r\nend","test_suite":"%%\r\nn = 1e2;\r\ny_correct = 3;\r\nf_correct = 3/25;\r\n[y,f] = Mersenne_prime_comp(n);\r\nassert(isequal(y,y_correct))\r\nassert(isequal(f,f_correct))\r\n\r\n%%\r\nn = 1e3;\r\ny_correct = 4;\r\nf_correct = 0.023809523809524;\r\n[y,f] = Mersenne_prime_comp(n);\r\nassert(isequal(y,y_correct))\r\nassert(abs(f-f_correct)\u003c(10*eps))\r\n\r\n%%\r\nn = 1e4;\r\ny_correct = 5;\r\nf_correct = 0.004068348250610;\r\n[y,f] = Mersenne_prime_comp(n);\r\nassert(isequal(y,y_correct))\r\nassert(abs(f-f_correct)\u003c(10*eps))\r\n\r\n%%\r\nn = 1e5;\r\ny_correct = 5;\r\nf_correct = 5.212677231025855e-04;\r\n[y,f] = Mersenne_prime_comp(n);\r\nassert(isequal(y,y_correct))\r\nassert(abs(f-f_correct)\u003c(10*eps))\r\n\r\n%%\r\nn = 1e6;\r\ny_correct = 7;\r\nf_correct = 8.917424647761727e-05;\r\n[y,f] = Mersenne_prime_comp(n);\r\nassert(isequal(y,y_correct))\r\nassert(abs(f-f_correct)\u003c(10*eps))\r\n","published":true,"deleted":false,"likes_count":8,"comments_count":2,"created_by":26769,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":846,"test_suite_updated_at":"2015-02-01T04:14:08.000Z","rescore_all_solutions":false,"group_id":29,"created_at":"2015-02-01T03:56:07.000Z","updated_at":"2026-04-01T10:02:01.000Z","published_at":"2015-02-01T04:14:08.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eA Mersenne prime (M) is a prime number of the form M = 2^p - 1, where p is another prime number.\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/525-mersenne-primes\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eProblem 525\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e asks the user to determine if a number is a Mersenne prime. In this problem, you are tasked with returning the number of primes numbers below the input number, n, that are Mersenne primes and the fraction of all primes below that input number that the Mersenne primes represent.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor example, for n = 100, there are 25 primes numbers: 2, 3, 5, 7, ..., 89, 97. As far as Mersenne primes go, there are only three that are less than 100: 2^2 - 1 = 3, 2^3 - 1 = 7, and 2^5 - 1 = 31. The corresponding fraction would be 3/25.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":2595,"title":"Polite numbers. Politeness.","description":"A polite number is an integer that sums of two or more consecutive positive integers.\r\nPoliteness of a positive integer is a number of nontrivial ways to write n as a sum of two or more consecutive positive integers.\r\n\r\nFor example _9 = 4+5 = 2+3+4_  and politeness of 9 is 2.\r\n\r\nGiven _N_ return politeness of _N_.\r\n\r\nSee also \u003chttp://www.mathworks.co.uk/matlabcentral/cody/problems/2593 2593\u003e","description_html":"\u003cp\u003eA polite number is an integer that sums of two or more consecutive positive integers.\r\nPoliteness of a positive integer is a number of nontrivial ways to write n as a sum of two or more consecutive positive integers.\u003c/p\u003e\u003cp\u003eFor example \u003ci\u003e9 = 4+5 = 2+3+4\u003c/i\u003e  and politeness of 9 is 2.\u003c/p\u003e\u003cp\u003eGiven \u003ci\u003eN\u003c/i\u003e return politeness of \u003ci\u003eN\u003c/i\u003e.\u003c/p\u003e\u003cp\u003eSee also \u003ca href = \"http://www.mathworks.co.uk/matlabcentral/cody/problems/2593\"\u003e2593\u003c/a\u003e\u003c/p\u003e","function_template":"function P = politeness(N)\r\n  P=N;\r\nend","test_suite":"%%\r\nx = 1;\r\ny_correct = 0;\r\nassert(isequal(politeness(x),y_correct))\r\n%%\r\nx = 2;\r\ny_correct = 0;\r\nassert(isequal(politeness(x),y_correct))\r\n%%\r\nx = 3;\r\ny_correct = 1;\r\nassert(isequal(politeness(x),y_correct))\r\n%%\r\nx = 7;\r\ny_correct = 1;\r\nassert(isequal(politeness(x),y_correct))\r\n%%\r\nx = 9;\r\ny_correct = 2;\r\nassert(isequal(politeness(x),y_correct))\r\n%%\r\nx = 15;\r\ny_correct = 3;\r\nassert(isequal(politeness(x),y_correct))\r\n%%\r\nx = 18;\r\ny_correct = 2;\r\nassert(isequal(politeness(x),y_correct))\r\n%%\r\nx = 21;\r\ny_correct = 3;\r\nassert(isequal(politeness(x),y_correct))\r\n%%\r\nx = 1024;\r\ny_correct = 0;\r\nassert(isequal(politeness(x),y_correct))\r\n%%\r\nx = 1025;\r\ny_correct = 5;\r\nassert(isequal(politeness(x),y_correct))\r\n%%\r\nx = 25215;\r\ny_correct = 11;\r\nassert(isequal(politeness(x),y_correct))\r\n%%\r\nx = 62;\r\ny_correct = 1;\r\nassert(isequal(politeness(x),y_correct))\r\n%%\r\nx = 63;\r\ny_correct = 5;\r\nassert(isequal(politeness(x),y_correct))\r\n%%\r\nx = 65;\r\ny_correct = 3;\r\nassert(isequal(politeness(x),y_correct))\r\n%%\r\n% anti-lookup \u0026 clue\r\nnums=primes(200);\r\npattern=[1 nums([false ~randi([0 25],1,45)])];\r\nx=prod(pattern)*2^randi([0 5]);\r\ny_correct=2^numel(pattern)/2-1;\r\nassert(isequal(politeness(x),y_correct))\r\n%%\r\nfor k=randi(2e4,1,20)\r\n  assert(isequal(politeness(k*(k-1))+1,(politeness(k)+1)*(politeness(k-1)+1)))\r\nend","published":true,"deleted":false,"likes_count":8,"comments_count":6,"created_by":14358,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":186,"test_suite_updated_at":"2014-09-17T15:38:21.000Z","rescore_all_solutions":false,"group_id":8,"created_at":"2014-09-16T22:47:12.000Z","updated_at":"2026-02-16T10:30:04.000Z","published_at":"2014-09-17T10:56:54.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eA polite number is an integer that sums of two or more consecutive positive integers. Politeness of a positive integer is a number of nontrivial ways to write n as a sum of two or more consecutive positive integers.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor example\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e9 = 4+5 = 2+3+4\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e and politeness of 9 is 2.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eGiven\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eN\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e return politeness of\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eN\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eSee also\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://www.mathworks.co.uk/matlabcentral/cody/problems/2593\\\"\u003e\u003cw:r\u003e\u003cw:t\u003e2593\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":1672,"title":"Leftovers?  Again?!","description":"I am thinking of a positive number X.  To determine what number I am thinking of, I will give you two 1xN vectors.  The first vector (V1) is several numbers, none of which will share a factor.  The second vector (V2) is the remainder of X when divided by each of the numbers in V1. Calculate what the lowest possible value of X can be given these criteria.  For example:\r\n\r\nV1 = [2 3] ; V2 = [1 2]\r\n\r\nwould give an X value of 5.  There are an infinite number of other values of X that would satisfy V1 and V2, but I want the lowest one.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: normal; text-decoration: none; white-space: normal; \"\u003e\u003cdiv style=\"display: block; min-width: 0px; padding-top: 0px; transform-origin: 332px 82.5px; vertical-align: baseline; perspective-origin: 332px 82.5px; \"\u003e\u003cdiv style=\"font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-bottom: 9px; margin-left: 4px; margin-right: 10px; margin-top: 2px; text-align: left; transform-origin: 309px 42px; white-space: pre-wrap; perspective-origin: 309px 42px; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"display: inline; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; transform-origin: 0px 0px; perspective-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eI am thinking of a positive number X. To determine what number I am thinking of, I will give you two 1xN vectors. The first vector (V1) is several numbers, none of which will share a factor. The second vector (V2) is the remainder of X when divided by each of the numbers in V1. Calculate what the lowest possible value of X can be given these criteria. For example:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-bottom: 9px; margin-left: 4px; margin-right: 10px; margin-top: 2px; text-align: left; transform-origin: 309px 10.5px; white-space: pre-wrap; perspective-origin: 309px 10.5px; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"display: inline; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; transform-origin: 0px 0px; perspective-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eV1 = [2 3] ; V2 = [1 2]\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-bottom: 9px; margin-left: 4px; margin-right: 10px; margin-top: 2px; text-align: left; transform-origin: 309px 21px; white-space: pre-wrap; perspective-origin: 309px 21px; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"display: inline; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; transform-origin: 0px 0px; perspective-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003ewould give an X value of 5. There are an infinite number of other values of X that would satisfy V1 and V2, but I want the lowest one.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function X = leftovers(n,a)\r\n X = pi;\r\nend","test_suite":"%!cp leftovers.m safe\r\n%!rm *.*\r\n%!mv safe leftovers.m\r\n%!rm @*\r\n\r\n% Clean user's function from some known jailbreaking mechanisms\r\nfid = fopen('leftovers.m');\r\nst = regexprep(char(fread(fid)'), '!', 'error(''No shell commands!''); %');\r\nst = regexprep(st, 'feval', 'error(''No fancy functions!''); %');\r\nst = regexprep(st, 'str2func', 'error(''No fancy functions!''); %');\r\nst = regexprep(st, 'regex', 'error(''No fancy functions!''); %');\r\nst = regexprep(st, 'system', 'error(''No shell commands!''); %');\r\nst = regexprep(st, 'dos', 'error(''No shell commands!''); %');\r\nst = regexprep(st, 'unix', 'error(''No shell commands!''); %');\r\nst = regexprep(st, 'perl', 'error(''No external languages commands!''); %');\r\nst = regexprep(st, 'java', 'error(''No external languages commands!''); %');\r\nst = regexprep(st, 'assert', 'error(''No overwriting!''); %');\r\nfclose(fid)\r\n\r\nfid = fopen('leftovers.m' , 'w');\r\nfwrite(fid,st);\r\nfclose(fid)\r\n%%\r\nV1 = [2 3] ; V2 = [1 2];; y_correct = 5; assert(isequal(leftovers(V1,V2),y_correct))\r\n%%\r\nV1=[3 5 7] ; V2=[1 2 3]; y_correct = 52; assert(isequal(leftovers(V1,V2),y_correct))\r\n%%\r\nV1=[3 4 5] ; V2=[2 3 1]; y_correct = 11; assert(isequal(leftovers(V1,V2),y_correct))\r\n%%\r\nV1=[4 9 25] ; V2=[3 2 7]; y_correct = 407; assert(isequal(leftovers(V1,V2),y_correct))\r\n%%\r\nV1=[9 10 77] ; V2=[1 2 69]; y_correct = 6922; assert(isequal(leftovers(V1,V2),y_correct))\r\n%%\r\na=primes(30);\r\nb=ceil(8*rand()+2)\r\nV1=a(1:b);V2=1:b;\r\ny_correct=[23 53 1523 29243 299513 4383593 188677703 5765999453];\r\nassert(isequal(leftovers(V1,V2),y_correct(b-2)))\r\n%%\r\nV1=[leftovers([6 35],[3 9]) leftovers([3 5 7],[1 3 1])];\r\nassert(isequal(leftovers(V1,V1-8),379))\r\n%%\r\n% Discourage the for x=1:inf loops\r\nV1=[74 93 145 161 209 221]; V2=[66 85 137 153 201 213];\r\ny_correct=7420738134802;\r\nassert(isequal(leftovers(V1,V2),y_correct))\r\n%%\r\n% Discourage the for x=1:inf loops\r\nV1=[17 82 111 155 203 247 253] ; V2=[11 50 68 95 124 150 154];\r\ny_correct=59652745309190;\r\nassert(isequal(leftovers(V1,V2),y_correct))\r\n","published":true,"deleted":false,"likes_count":4,"comments_count":10,"created_by":1615,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":32,"test_suite_updated_at":"2020-09-29T13:24:49.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2013-06-21T18:45:36.000Z","updated_at":"2025-11-22T17:35:45.000Z","published_at":"2013-06-21T18:45:36.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eI am thinking of a positive number X. To determine what number I am thinking of, I will give you two 1xN vectors. The first vector (V1) is several numbers, none of which will share a factor. The second vector (V2) is the remainder of X when divided by each of the numbers in V1. Calculate what the lowest possible value of X can be given these criteria. For example:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eV1 = [2 3] ; V2 = [1 2]\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewould give an X value of 5. There are an infinite number of other values of X that would satisfy V1 and V2, but I want the lowest one.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":60,"title":"The Goldbach Conjecture","description":"The Goldbach conjecture asserts that every even integer greater than 2 can be expressed as the sum of two primes.\r\nGiven the even integer n, return primes p1 and p2 that satisfy the condition n = p1 + p2. Note that the primes are not always unique. The test is not sensitive to order or uniqueness. You just need to meet the appropriate conditions.\r\nExample:\r\n Input  n = 286\r\n Output (any of the following is acceptable) \r\n        [  3 283]\r\n        [283   3]\r\n        [  5 281]\r\n        [107 179]\r\n        [137 149]","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 255.033px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 127.517px; transform-origin: 407px 127.517px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 12.5px 8px; transform-origin: 12.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"/#null\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eGoldbach conjecture\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 287px 8px; transform-origin: 287px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e asserts that every even integer greater than 2 can be expressed as the sum of two primes.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 368.5px 8px; transform-origin: 368.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eGiven the even integer n, return primes p1 and p2 that satisfy the condition n = p1 + p2. Note that the primes are not always unique. The test is not sensitive to order or uniqueness. You just need to meet the appropriate conditions.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 28.5px 8px; transform-origin: 28.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eExample:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgb(247, 247, 247); block-size: 143.033px; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-end-end-radius: 4px; border-end-start-radius: 4px; border-start-end-radius: 4px; border-start-start-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; margin-block-end: 10px; margin-block-start: 10px; margin-bottom: 10px; margin-inline-end: 3px; margin-inline-start: 3px; margin-left: 3px; margin-right: 3px; margin-top: 10px; perspective-origin: 404px 71.5167px; transform-origin: 404px 71.5167px; margin-left: 3px; margin-top: 10px; margin-bottom: 10px; margin-right: 3px; \"\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 60px 8.5px; tab-size: 4; transform-origin: 60px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; perspective-origin: 32px 8.5px; transform-origin: 32px 8.5px; \"\u003e Input  \u003c/span\u003e\u003cspan style=\"border-block-end-color: rgb(170, 4, 249); border-block-start-color: rgb(170, 4, 249); border-bottom-color: rgb(170, 4, 249); border-inline-end-color: rgb(170, 4, 249); border-inline-start-color: rgb(170, 4, 249); border-left-color: rgb(170, 4, 249); border-right-color: rgb(170, 4, 249); border-top-color: rgb(170, 4, 249); caret-color: rgb(170, 4, 249); color: rgb(170, 4, 249); column-rule-color: rgb(170, 4, 249); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(170, 4, 249); perspective-origin: 28px 8.5px; text-decoration-color: rgb(170, 4, 249); text-emphasis-color: rgb(170, 4, 249); transform-origin: 28px 8.5px; \"\u003en = 286\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 180px 8.5px; tab-size: 4; transform-origin: 180px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e Output (any of the following is acceptable) \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 68px 8.5px; tab-size: 4; transform-origin: 68px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e        [  3 283]\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 68px 8.5px; tab-size: 4; transform-origin: 68px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e        [283   3]\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 68px 8.5px; tab-size: 4; transform-origin: 68px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e        [  5 281]\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 68px 8.5px; tab-size: 4; transform-origin: 68px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e        [107 179]\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 68px 8.5px; tab-size: 4; transform-origin: 68px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e        [137 149]\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function [p1,p2] = goldbach(n)\r\n  p1 = n;\r\n  p2 = n;\r\nend","test_suite":"%%\r\nfiletext = fileread('goldbach.m');\r\nillegal = contains(filetext, 'assignin') || contains(filetext, 'regexp'); \r\nassert(~illegal)\r\n\r\n%%\r\nnList = 28:6:76;\r\nfor i = 1:length(nList)\r\n  n = nList(i);\r\n  [p1,p2] = goldbach(n)\r\n  assert(isprime(p1) \u0026\u0026 isprime(p2) \u0026\u0026 (p1+p2==n));\r\nend\r\n\r\n%%\r\nnList = [18 20 22 100 102 114 1000 2000 36 3600];\r\nfor i = 1:length(nList)\r\n  n = nList(i);\r\n  [p1,p2] = goldbach(n)\r\n  assert(isprime(p1) \u0026\u0026 isprime(p2) \u0026\u0026 (p1+p2==n));\r\nend","published":true,"deleted":false,"likes_count":60,"comments_count":17,"created_by":1,"edited_by":223089,"edited_at":"2023-06-05T15:48:22.000Z","deleted_by":null,"deleted_at":null,"solvers_count":5973,"test_suite_updated_at":"2023-06-05T15:48:22.000Z","rescore_all_solutions":false,"group_id":2,"created_at":"2012-01-18T01:00:25.000Z","updated_at":"2026-04-09T08:17:56.000Z","published_at":"2012-01-18T01:00:25.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eGoldbach conjecture\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e asserts that every even integer greater than 2 can be expressed as the sum of two primes.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eGiven the even integer n, return primes p1 and p2 that satisfy the condition n = p1 + p2. Note that the primes are not always unique. The test is not sensitive to order or uniqueness. You just need to meet the appropriate conditions.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eExample:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[ Input  n = 286\\n Output (any of the following is acceptable) \\n        [  3 283]\\n        [283   3]\\n        [  5 281]\\n        [107 179]\\n        [137 149]]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":64,"title":"The Goldbach Conjecture, Part 2","description":"The \u003chttp://en.wikipedia.org/wiki/Goldbach's_conjecture Goldbach\nconjecture\u003e asserts that every even integer greater than 2 can be\nexpressed as the sum of two primes.\n \nGiven the even integer n, return c, the number of different ways two primes can be added to result in n. Only count a pair once; the order is unimportant.\n\nExample:\n\n Input  n = 10\n Output c is 2\n\nbecause of the prime pairs [3 7] and [5 5].\n\n Input  n = 50\n Output c is 4\n\nbecause of [3 47], [7 43], [13 37], and [19 31].\n","description_html":"\u003cp\u003eThe \u003ca href=\"http://en.wikipedia.org/wiki/Goldbach's_conjecture\"\u003eGoldbach\nconjecture\u003c/a\u003e asserts that every even integer greater than 2 can be\nexpressed as the sum of two primes.\u003c/p\u003e\u003cp\u003eGiven the even integer n, return c, the number of different ways two primes can be added to result in n. Only count a pair once; the order is unimportant.\u003c/p\u003e\u003cp\u003eExample:\u003c/p\u003e\u003cpre\u003e Input  n = 10\n Output c is 2\u003c/pre\u003e\u003cp\u003ebecause of the prime pairs [3 7] and [5 5].\u003c/p\u003e\u003cpre\u003e Input  n = 50\n Output c is 4\u003c/pre\u003e\u003cp\u003ebecause of [3 47], [7 43], [13 37], and [19 31].\u003c/p\u003e","function_template":"function c = goldbach2(n)\n  c = 1;\nend","test_suite":"%%\nn = 6;\nc_correct = 1;\nassert(isequal(goldbach2(n),c_correct))\n\n%%\nn = 10;\nc_correct = 2;\nassert(isequal(goldbach2(n),c_correct))\n\n%%\nn = 50;\nc_correct = 4;\nassert(isequal(goldbach2(n),c_correct))\n\n%%\nn = 480;\nc_correct = 29;\nassert(isequal(goldbach2(n),c_correct))","published":true,"deleted":false,"likes_count":18,"comments_count":4,"created_by":1,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":2413,"test_suite_updated_at":"2012-01-18T01:00:26.000Z","rescore_all_solutions":false,"group_id":2,"created_at":"2012-01-18T01:00:26.000Z","updated_at":"2026-02-04T16:46:15.000Z","published_at":"2012-01-18T01:00:26.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://en.wikipedia.org/wiki/Goldbach's_conjecture\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eGoldbach conjecture\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e asserts that every even integer greater than 2 can be expressed as the sum of two primes.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eGiven the even integer n, return c, the number of different ways two primes can be added to result in n. Only count a pair once; the order is unimportant.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eExample:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[ Input  n = 10\\n Output c is 2]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ebecause of the prime pairs [3 7] and [5 5].\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[ Input  n = 50\\n Output c is 4]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ebecause of [3 47], [7 43], [13 37], and [19 31].\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":83,"title":"Prime factor digits","description":"Consider the following number system. Calculate the prime factorization for each number n, then represent the prime factors in a vector like so:\r\n       13 11 7 5 3 2\r\n      ---------------\r\n    2:             1\r\n    3:           1 0\r\n    4:             2\r\n    5:         1 0 0\r\n    6:           1 1\r\n   12:           1 2\r\n   14:       1 0 0 1\r\n   18:           2 1\r\n   26:   1 0 0 0 0 1\r\n   60:         1 1 2\r\nEach \"place\" in the number system represents a prime number. Given n, return the vector p.\r\nAs shown above, if n = 26, then p = [1 0 0 0 0 1].\r\nThe input n is always an integer greater than 1. Suppress any leading zeros. The length of the vector is determined by the largest prime factor.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 409.2px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 204.6px; transform-origin: 407px 204.6px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 368.5px 8px; transform-origin: 368.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eConsider the following number system. Calculate the prime factorization for each number n, then represent the prime factors in a vector like so:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgb(247, 247, 247); block-size: 245.2px; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-end-end-radius: 4px; border-end-start-radius: 4px; border-start-end-radius: 4px; border-start-start-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; margin-block-end: 10px; margin-block-start: 10px; margin-bottom: 10px; margin-inline-end: 3px; margin-inline-start: 3px; margin-left: 3px; margin-right: 3px; margin-top: 10px; perspective-origin: 404px 122.6px; transform-origin: 404px 122.6px; margin-left: 3px; margin-top: 10px; margin-bottom: 10px; margin-right: 3px; \"\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 80px 8.5px; transform-origin: 80px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e       13 11 7 5 3 2\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 84px 8.5px; transform-origin: 84px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e      ---------------\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 80px 8.5px; transform-origin: 80px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e    2:             1\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 80px 8.5px; transform-origin: 80px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e    3:           1 0\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 80px 8.5px; transform-origin: 80px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e    4:             2\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 80px 8.5px; transform-origin: 80px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e    5:         1 0 0\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 80px 8.5px; transform-origin: 80px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e    6:           1 1\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 80px 8.5px; transform-origin: 80px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e   12:           1 2\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 80px 8.5px; transform-origin: 80px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e   14:       1 0 0 1\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 80px 8.5px; transform-origin: 80px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e   18:           2 1\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 80px 8.5px; transform-origin: 80px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e   26:   1 0 0 0 0 1\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 80px 8.5px; transform-origin: 80px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e   60:         1 1 2\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 10px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 10px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 10px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 289.5px 8px; transform-origin: 289.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eEach \"place\" in the number system represents a prime number. Given n, return the vector p.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 153px 8px; transform-origin: 153px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eAs shown above, if n = 26, then p = [1 0 0 0 0 1].\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 384px 8px; transform-origin: 384px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe input n is always an integer greater than 1. Suppress any leading zeros. The length of the vector is determined by the largest prime factor.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function p = factor_digits(n)\r\n  p = 0;\r\nend","test_suite":"%%\r\nn = 26;\r\np_correct = [1 0 0 0 0 1];\r\nassert(isequal(factor_digits(n),p_correct))\r\n\r\n%%\r\nn = 168;\r\np_correct = [1 0 1 3];\r\nassert(isequal(factor_digits(n),p_correct))\r\n\r\n%%\r\nn = 2;\r\np_correct = 1;\r\nassert(isequal(factor_digits(n),p_correct))\r\n\r\n%%\r\nn = 1444;\r\np_correct = 2*[1 0 0 0 0 0 0 1];\r\nassert(isequal(factor_digits(n),p_correct))\r\n\r\n%%\r\nn = 47;\r\np_correct = [1 0 0 0 0 0 0 0 0 0 0 0 0 0 0];\r\nassert(isequal(factor_digits(n),p_correct))\r\n\r\n%%\r\nn = 999;\r\np_correct = [1 0 0 0 0 0 0 0 0 0 3 0];\r\nassert(isequal(factor_digits(n),p_correct))\r\n\r\n%%\r\nn = 20;\r\np_correct = [1 0 2];\r\nassert(isequal(factor_digits(n),p_correct))","published":true,"deleted":false,"likes_count":28,"comments_count":6,"created_by":1,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":2540,"test_suite_updated_at":"2021-08-08T11:30:25.000Z","rescore_all_solutions":false,"group_id":2,"created_at":"2012-01-18T01:00:29.000Z","updated_at":"2026-04-10T14:34:09.000Z","published_at":"2012-01-18T01:00:29.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eConsider the following number system. Calculate the prime factorization for each number n, then represent the prime factors in a vector like so:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[       13 11 7 5 3 2\\n      ---------------\\n    2:             1\\n    3:           1 0\\n    4:             2\\n    5:         1 0 0\\n    6:           1 1\\n   12:           1 2\\n   14:       1 0 0 1\\n   18:           2 1\\n   26:   1 0 0 0 0 1\\n   60:         1 1 2]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eEach \\\"place\\\" in the number system represents a prime number. Given n, return the vector p.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eAs shown above, if n = 26, then p = [1 0 0 0 0 1].\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe input n is always an integer greater than 1. Suppress any leading zeros. The length of the vector is determined by the largest prime factor.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":1095,"title":"Circular Primes (based on Project Euler, problem 35)","description":"The number, 197, is called a circular prime because all rotations of the digits: 197, 971, and 719, are themselves prime.\r\n\r\nThere are thirteen such primes below 100: 2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 79, and 97.\r\n\r\nGiven a number x, write a MATLAB script that will tell you the number of circular primes less than or equal to x as well as a sorted list of what the circular prime numbers are.","description_html":"\u003cp\u003eThe number, 197, is called a circular prime because all rotations of the digits: 197, 971, and 719, are themselves prime.\u003c/p\u003e\u003cp\u003eThere are thirteen such primes below 100: 2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 79, and 97.\u003c/p\u003e\u003cp\u003eGiven a number x, write a MATLAB script that will tell you the number of circular primes less than or equal to x as well as a sorted list of what the circular prime numbers are.\u003c/p\u003e","function_template":"function [how_many what_numbers]=circular_prime(x)\r\n    how_many=3;\r\n    what_numbers=[2 3 5];\r\nend","test_suite":"%%\r\n[y numbers]=circular_prime(197)\r\nassert(isequal(y,16)\u0026\u0026isequal(numbers,[2 3 5 7 11 13 17 31 37 71 73 79 97 113 131 197]))\r\n%%\r\n[y numbers]=circular_prime(100)\r\nassert(isequal(y,13)\u0026\u0026isequal(numbers,[2 3 5 7 11 13 17 31 37 71 73 79 97]))\r\n%%\r\n[y numbers]=circular_prime(250)\r\nassert(isequal(y,17)\u0026\u0026isequal(numbers,[2 3 5 7 11 13 17 31 37 71 73 79 97 113 131 197 199]))\r\n%%\r\n[y numbers]=circular_prime(2000)\r\nassert(isequal(y,27)\u0026\u0026isequal(numbers,[2 3 5 7 11 13 17 31 37 71 73 79 97 113 131 197 199 311 337 373 719 733 919 971 991 1193 1931]))\r\n%%\r\n[y numbers]=circular_prime(10000)\r\nassert(isequal(y,33)\u0026\u0026isequal(numbers,[2 3 5 7 11 13 17 31 37 71 73 79 97 113 131 197 199 311 337 373 719 733 919 971 991 1193 1931 3119 3779 7793 7937 9311 9377]))\r\n%%\r\n[y numbers]=circular_prime(54321)\r\nassert(isequal(y,38)\u0026\u0026isequal(numbers,[2 3 5 7 11 13 17 31 37 71 73 79 97 113 131 197 199 311 337 373 719 733 919 971 991 1193 1931 3119 3779 7793 7937 9311 9377 11939 19391 19937 37199 39119]))\r\n","published":true,"deleted":false,"likes_count":10,"comments_count":6,"created_by":1615,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":652,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2012-12-05T18:02:09.000Z","updated_at":"2026-04-09T08:20:38.000Z","published_at":"2012-12-05T18:02:09.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe number, 197, is called a circular prime because all rotations of the digits: 197, 971, and 719, are themselves prime.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThere are thirteen such primes below 100: 2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 79, and 97.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eGiven a number x, write a MATLAB script that will tell you the number of circular primes less than or equal to x as well as a sorted list of what the circular prime numbers are.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":2342,"title":"Numbers spiral diagonals (Part 2)","description":"Inspired by Project Euler n°28 and 58.\r\nA n x n spiral matrix is obtained by starting with the number 1 and moving to the right in a clockwise direction.\r\nFor example with n=5, the spiral matrix is :\r\n                       21 22 23 24 25\r\n                       20  7  8  9 10\r\n                       19  6  1  2 11\r\n                       18  5  4  3 12\r\n                       17 16 15 14 13\r\nThe sum of the numbers on the diagonals is 101 (See problem 2340) and you have 5 primes (3, 5, 7, 13, 17) out of the 9 numbers lying along both diagonals. So the prime ratio is 5/9 ≈ 55%.\r\nWith a 7x7 spiral matrix, the ratio is 62% (8 primes out of the 13 diagonal numbers).\r\nWhat is the side length (always odd and greater than 1) of the square spiral for which the ratio of primes along both diagonals FIRST falls below p% ? (0\u003cp\u003c1)","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 326.167px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 163.083px; transform-origin: 407px 163.083px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 120px 8px; transform-origin: 120px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eInspired by Project Euler n°28 and 58.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 341px 8px; transform-origin: 341px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eA n x n spiral matrix is obtained by starting with the number 1 and moving to the right in a clockwise direction.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 131.5px 8px; transform-origin: 131.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eFor example with n=5, the spiral matrix is :\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgb(247, 247, 247); block-size: 102.167px; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-end-end-radius: 4px; border-end-start-radius: 4px; border-start-end-radius: 4px; border-start-start-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; margin-block-end: 10px; margin-block-start: 10px; margin-bottom: 10px; margin-inline-end: 3px; margin-inline-start: 3px; margin-left: 3px; margin-right: 3px; margin-top: 10px; perspective-origin: 404px 51.0833px; transform-origin: 404px 51.0833px; margin-left: 3px; margin-top: 10px; margin-bottom: 10px; margin-right: 3px; \"\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 148px 8.5px; tab-size: 4; transform-origin: 148px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e                       21 22 23 24 25\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 148px 8.5px; tab-size: 4; transform-origin: 148px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e                       20  7  8  9 10\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 148px 8.5px; tab-size: 4; transform-origin: 148px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e                       19  6  1  2 11\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 148px 8.5px; tab-size: 4; transform-origin: 148px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e                       18  5  4  3 12\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 148px 8.5px; tab-size: 4; transform-origin: 148px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e                       17 16 15 14 13\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 10px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 10px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 10px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 382.5px 8px; transform-origin: 382.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe sum of the numbers on the diagonals is 101 (See problem 2340) and you have 5 primes (3, 5, 7, 13, 17) out of the 9 numbers lying along both diagonals. So the prime ratio is 5/9 ≈ 55%.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 262px 8px; transform-origin: 262px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWith a 7x7 spiral matrix, the ratio is 62% (8 primes out of the 13 diagonal numbers).\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 364.5px 8px; transform-origin: 364.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"perspective-origin: 74px 8px; transform-origin: 74px 8px; \"\u003eWhat is the side length \u003c/span\u003e\u003cspan style=\"border-block-end-style: solid; border-block-end-width: 1px; border-bottom-style: solid; border-bottom-width: 1px; perspective-origin: 2.5px 8.5px; transform-origin: 2.5px 8.5px; \"\u003e(\u003c/span\u003e\u003cspan style=\"perspective-origin: 96px 8px; transform-origin: 96px 8px; \"\u003ealways odd and greater than 1\u003c/span\u003e\u003cspan style=\"border-block-end-style: solid; border-block-end-width: 1px; border-bottom-style: solid; border-bottom-width: 1px; perspective-origin: 2.5px 8.5px; transform-origin: 2.5px 8.5px; \"\u003e)\u003c/span\u003e\u003cspan style=\"perspective-origin: 189.5px 8px; transform-origin: 189.5px 8px; \"\u003e of the square spiral for which the ratio of primes along both diagonals FIRST falls below p% ? (0\u0026lt;p\u0026lt;1)\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function res=spiral_ratio(pourcentage)\r\nres=pourcentage*2;\r\nend","test_suite":"%%\r\nx = 0.8;\r\ny_correct = 3;\r\nassert(isequal(spiral_ratio(x),y_correct))\r\n%%\r\nx = 0.5;\r\ny_correct = 11;\r\nassert(isequal(spiral_ratio(x),y_correct))\r\n%%\r\nx = 0.4;\r\ny_correct = 31;\r\nassert(isequal(spiral_ratio(x),y_correct))\r\n%%\r\nx = 0.3;\r\ny_correct = 49;\r\nassert(isequal(spiral_ratio(x),y_correct))\r\n%%\r\nx = 0.25;\r\ny_correct = 99;\r\nassert(isequal(spiral_ratio(x),y_correct))\r\n%%\r\nx = 0.2;\r\ny_correct = 309;\r\nassert(isequal(spiral_ratio(x),y_correct))\r\n%%\r\nx = 0.15;\r\ny_correct = 981;\r\nassert(isequal(spiral_ratio(x),y_correct))\r\n%%\r\nx = 0.14;\r\ny_correct = 1883;\r\nassert(isequal(spiral_ratio(x),y_correct))\r\n%%\r\nx = 0.13;\r\ny_correct = 3593;\r\nassert(isequal(spiral_ratio(x),y_correct))\r\n%%\r\nx = 0.12;\r\ny_correct = 6523;\r\nassert(isequal(spiral_ratio(x),y_correct))\r\n%%\r\nx = 0.11;\r\ny_correct = 12201;\r\nassert(isequal(spiral_ratio(x),y_correct))","published":true,"deleted":false,"likes_count":7,"comments_count":5,"created_by":5390,"edited_by":223089,"edited_at":"2022-09-26T17:42:20.000Z","deleted_by":null,"deleted_at":null,"solvers_count":196,"test_suite_updated_at":"2022-07-09T19:28:50.000Z","rescore_all_solutions":false,"group_id":31,"created_at":"2014-05-31T18:36:25.000Z","updated_at":"2026-03-04T11:06:42.000Z","published_at":"2014-05-31T18:53:35.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eInspired by Project Euler n°28 and 58.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eA n x n spiral matrix is obtained by starting with the number 1 and moving to the right in a clockwise direction.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor example with n=5, the spiral matrix is :\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[                       21 22 23 24 25\\n                       20  7  8  9 10\\n                       19  6  1  2 11\\n                       18  5  4  3 12\\n                       17 16 15 14 13]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe sum of the numbers on the diagonals is 101 (See problem 2340) and you have 5 primes (3, 5, 7, 13, 17) out of the 9 numbers lying along both diagonals. So the prime ratio is 5/9 ≈ 55%.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWith a 7x7 spiral matrix, the ratio is 62% (8 primes out of the 13 diagonal numbers).\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWhat is the side length (always odd and greater than 1) of the square spiral for which the ratio of primes along both diagonals FIRST falls below p% ? (0\u0026lt;p\u0026lt;1)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":1394,"title":"Prime Ladders","description":"A \u003chttp://en.wikipedia.org/wiki/Word_ladder word ladder\u003e transforms one word to another by means of single-letter mutations. So COLD can become WARM like so (there are often multiple solutions):\r\n\r\n COLD\r\n CORD\r\n CARD\r\n WARD\r\n WARM\r\n\r\nA number ladder does much the same thing, changing one digit at a time. A *prime ladder* is a number ladder with the additional constraint that each element is a prime number. Here is a prime ladder that connects 757 and 139\r\n\r\n 757 \r\n 157\r\n 137\r\n 139\r\n\r\nGiven two numbers p1 and p2, construct a prime ladder column vector in which p1 is the first element, p2 is the last element, and each successive row differs by exactly one digit from the preceding element. \r\n\r\nTo restate the above example, consider\r\n\r\n p1 = 757\r\n p2 = 139\r\n\r\nfor which an acceptable answer is\r\n\r\n ladder = [757; 157; 137; 139]\r\n\r\nYou can assume that p1 and p2 contain the same number of digits. I am not looking for a unique answer. I will only check that the conditions of a prime ladder are met.\r\n\r\n","description_html":"\u003cp\u003eA \u003ca href = \"http://en.wikipedia.org/wiki/Word_ladder\"\u003eword ladder\u003c/a\u003e transforms one word to another by means of single-letter mutations. So COLD can become WARM like so (there are often multiple solutions):\u003c/p\u003e\u003cpre\u003e COLD\r\n CORD\r\n CARD\r\n WARD\r\n WARM\u003c/pre\u003e\u003cp\u003eA number ladder does much the same thing, changing one digit at a time. A \u003cb\u003eprime ladder\u003c/b\u003e is a number ladder with the additional constraint that each element is a prime number. Here is a prime ladder that connects 757 and 139\u003c/p\u003e\u003cpre\u003e 757 \r\n 157\r\n 137\r\n 139\u003c/pre\u003e\u003cp\u003eGiven two numbers p1 and p2, construct a prime ladder column vector in which p1 is the first element, p2 is the last element, and each successive row differs by exactly one digit from the preceding element.\u003c/p\u003e\u003cp\u003eTo restate the above example, consider\u003c/p\u003e\u003cpre\u003e p1 = 757\r\n p2 = 139\u003c/pre\u003e\u003cp\u003efor which an acceptable answer is\u003c/p\u003e\u003cpre\u003e ladder = [757; 157; 137; 139]\u003c/pre\u003e\u003cp\u003eYou can assume that p1 and p2 contain the same number of digits. I am not looking for a unique answer. I will only check that the conditions of a prime ladder are met.\u003c/p\u003e","function_template":"function ladder = prime_ladder(p1,p2)\r\n  ladder = 0;\r\nend","test_suite":"%%\r\n\r\np1 = 13;\r\np2 = 29;\r\nladder = prime_ladder(p1,p2);\r\n\r\nassert(all(isprime(ladder)))\r\nassert(iscolumn(ladder))\r\nassert(ladder(1)==p1)\r\nassert(ladder(end)==p2)\r\nassert(all(sum(diff(num2str(ladder))~=0,2)==1))\r\n\r\n%%\r\n\r\np1 = 389;\r\np2 = 269;\r\nladder = prime_ladder(p1,p2);\r\n\r\nassert(all(isprime(ladder)))\r\nassert(iscolumn(ladder))\r\nassert(ladder(1)==p1)\r\nassert(ladder(end)==p2)\r\nassert(all(sum(diff(num2str(ladder))~=0,2)==1))\r\n\r\n%%\r\n\r\np1 = 761;\r\np2 = 397;\r\nladder = prime_ladder(p1,p2);\r\n\r\nassert(all(isprime(ladder)))\r\nassert(iscolumn(ladder))\r\nassert(ladder(1)==p1)\r\nassert(ladder(end)==p2)\r\nassert(all(sum(diff(num2str(ladder))~=0,2)==1))\r\n\r\n%%\r\n\r\np1 = 983;\r\np2 = 239;\r\nladder = prime_ladder(p1,p2);\r\n\r\nassert(all(isprime(ladder)))\r\nassert(iscolumn(ladder))\r\nassert(ladder(1)==p1)\r\nassert(ladder(end)==p2)\r\nassert(all(sum(diff(num2str(ladder))~=0,2)==1))\r\n\r\n%%\r\n\r\np1 = 271;\r\np2 = 439;\r\nladder = prime_ladder(p1,p2);\r\n\r\nassert(all(isprime(ladder)))\r\nassert(iscolumn(ladder))\r\nassert(ladder(1)==p1)\r\nassert(ladder(end)==p2)\r\nassert(all(sum(diff(num2str(ladder))~=0,2)==1))\r\n\r\n%%\r\n\r\np1 = 877;\r\np2 = 733;\r\nladder = prime_ladder(p1,p2);\r\n\r\nassert(all(isprime(ladder)))\r\nassert(iscolumn(ladder))\r\nassert(ladder(1)==p1)\r\nassert(ladder(end)==p2)\r\nassert(all(sum(diff(num2str(ladder))~=0,2)==1))\r\n\r\n%%\r\n\r\np1 = 2267;\r\np2 = 1153;\r\nladder = prime_ladder(p1,p2);\r\n\r\nassert(all(isprime(ladder)))\r\nassert(iscolumn(ladder))\r\nassert(ladder(1)==p1)\r\nassert(ladder(end)==p2)\r\nassert(all(sum(diff(num2str(ladder))~=0,2)==1))\r\n","published":true,"deleted":false,"likes_count":10,"comments_count":3,"created_by":7,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":55,"test_suite_updated_at":"2013-03-27T21:24:26.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2013-03-26T22:51:16.000Z","updated_at":"2026-01-03T14:28:57.000Z","published_at":"2013-03-27T15:28:59.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eA\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://en.wikipedia.org/wiki/Word_ladder\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eword ladder\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e transforms one word to another by means of single-letter mutations. So COLD can become WARM like so (there are often multiple solutions):\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[ COLD\\n CORD\\n CARD\\n WARD\\n WARM]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eA number ladder does much the same thing, changing one digit at a time. A\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eprime ladder\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e is a number ladder with the additional constraint that each element is a prime number. Here is a prime ladder that connects 757 and 139\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[ 757 \\n 157\\n 137\\n 139]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eGiven two numbers p1 and p2, construct a prime ladder column vector in which p1 is the first element, p2 is the last element, and each successive row differs by exactly one digit from the preceding element.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eTo restate the above example, consider\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[ p1 = 757\\n p2 = 139]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003efor which an acceptable answer is\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[ ladder = [757; 157; 137; 139]]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eYou can assume that p1 and p2 contain the same number of digits. I am not looking for a unique answer. I will only check that the conditions of a prime ladder are met.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":44377,"title":"Five steps to enlightenment","description":"This problem asks you to identify valid variations of the famous \u003chttps://en.wikipedia.org/wiki/Sum_and_Product_Puzzle sum and product puzzle\u003e.\r\n\r\n\u003c\u003chttp://www.alfnie.com/software/printnumbers.jpg\u003e\u003e\r\n\r\nThe original _sum and product_ puzzle goes somewhat like this:\r\n\r\nScott and Priscilla are asked to guess two numbers X and Y, *ranging between 2 and 100* and with X\u003cY. Scott is told their sum S (=X+Y) and Priscilla is told their product P (=X*Y). After this, they have the following conversation:\r\n\r\n  Scott:      I don't know X and Y                                                (Step 1)\r\n  Priscilla:  Neither do I                                                        (Step 2)\r\n  Scott:      Before our conversation I already knew that you didn't know         (Step 3)\r\n  Priscilla:  But now I do know X and Y                                           (Step 4)\r\n  Scott:      And so do I!                                                        (Step 5)\r\n\r\nThe original puzzle asks you to deduce the value of X and Y from the above information\r\n\r\nTo solve this puzzle you may assume that: a) Scott and Priscilla are perfect mathematicians/logicians and always speak the truth; b) they are aware of each other circumstances (Scott knows that Priscilla has been told the product of X and Y, and Priscilla knows that Scott has been told the sum of X and Y before the beginning of their conversation); and c) the third sentence refers to Scott and Priscilla state before the beginning of the conversation (i.e. Scott already knew -at the outset of this conversation- that Priscilla did not know the solution -at the time-)\r\n\r\nBriefly, the deduction for this seemingly impossible puzzle starts with all possible pairs of values of X and Y, and uses  the information in the sentences above to sequentially reduce the range of possible (X,Y) pairs to those that would be consistent with each sentence (see figure below; blue dots represent (X,Y) pairs still possible after each of the sentences/steps above; plot uses matrix convention with origin at the upper-left corner, X in vertical axis, and Y in horizontal axis; upper triangular segment represents all possible X\u003cY pairs before the start of S\u0026P conversation). \r\n\r\n\u003c\u003chttp://www.alfnie.com/software/printnumbers_b.jpg\u003e\u003e\r\n\r\nFor example, in *Step 1* after S says _\"I don't know X and Y\"_ we learn that the pair (X,Y) = (2,3) (and a few others) are no longer possible, since otherwise S would have known the solution as soon as he was told their sum (X+Y=5), which takes a unique value among all possible (X,Y) pairs. Similarly, in *Step 2*, after P says _\"Neither do I\"_, the pair (X,Y) = (5,7) (and many others) are no longer possible, since otherwise P would have known the solution as soon as she was told their product X*Y=35, which turns out to be unique among all remaining X,Y pairs. In *Step 3*, S asserting _\"I knew that you didn't know\"_ tells us that, knowing only X+Y, he was able to determine beforehand that it would be _impossible_ for P to know both X and Y from their product X*Y alone, which, again, rules out a considerable number of (X,Y) pairs (e.g. all solutions with X+Y=12 can be ruled out, because if S was told that X+Y=12 he could not have possibly dismissed beforehand the possibility that perhaps X=5 and Y=7 which would have allowed P to know both X and Y as soon as she was told their product X*Y=35). In *Step 4*, P suddenly become aware of the solution after S revelation informs us that, unlike in Step 2, she is now (after step-3 crop in possible X,Y pairs) able to uniquely determine the solution (X,Y) from their product (X*Y). This allows us to rule out many (X,Y) pairs among the remaining possible values, such as (X,Y)=(2,15) or (5,6), because P would not have been able to uniquely identify the solution at this point if she was told that the product X*Y was 30. Last, in *Step 5*, S suddenly becoming aware of the solution after P revelation, again allows us to rule out any remaining (X,Y) pair where knowing the sum S would still not suffice to uniquely identify the solution.\r\n\r\nThis puzzle is very neat because, somewhat surprisingly, after sequentially reducing the range of possible (X,Y) pairs from the five sentences/steps above, only one possible pair remains. The solution X=4 and Y=13, which Priscilla learns after the third sentence, Scott learns after the fourth sentence, and you, the reader, learn after the fifth sentence. \r\n\r\nKey to the existence of a unique solution to this puzzle is the initial range of possible (X,Y) pairs that we are told to consider. If, for example, instead of considering all X,Y values between 2 and 100, we were told to consider all X,Y values between 1 and 100, the puzzle would not be solvable, as in this case there will be multiple possible solutions that would all be consistent with the five sentences above (see figure below; at Step 5 there still exist 6 different possible solutions to this puzzle). \r\n\r\n\u003c\u003chttp://www.alfnie.com/software/printnumbers_c.jpg\u003e\u003e\r\n\r\nSimilarly, if we were told to consider all X,Y values between 2 and 50, the puzzle would again not be solvable, as in this case there would be _no solution_ consistent with all five sentences (perhaps surprisingly, since the X=4 Y=13 solution above is in fact within the stated range). \r\n\r\n\u003c\u003chttp://www.alfnie.com/software/printnumbers_d.jpg\u003e\u003e\r\n\r\nOn the other hand, if we were told to consider X and Y values ranging between 1 and 24, for example, the puzzle would again become solvable, now with a new unique solution X=1 and Y=6. \r\n\r\n\u003c\u003chttp://www.alfnie.com/software/printnumbers_e.jpg\u003e\u003e\r\n\r\nIn this problem you are tasked to create a function that would determine whether a particular variation of this puzzle would work or not. Specifically, given an initial set of possible (X,Y) pairs (entered as a Nx2 matrix and representing the full set of possible X,Y pairs that Scott and Priscilla are told to consider), you should determine whether it is possible to solve this puzzle (i.e. whether one, and only one, (X,Y) pair is consistent with the five sequential sentences above). Your function should simply return 1 (or true) if the puzzle is solvable, and 0 (or false) otherwise. \r\n\r\nGood luck!","description_html":"\u003cp\u003eThis problem asks you to identify valid variations of the famous \u003ca href = \"https://en.wikipedia.org/wiki/Sum_and_Product_Puzzle\"\u003esum and product puzzle\u003c/a\u003e.\u003c/p\u003e\u003cimg src = \"http://www.alfnie.com/software/printnumbers.jpg\"\u003e\u003cp\u003eThe original \u003ci\u003esum and product\u003c/i\u003e puzzle goes somewhat like this:\u003c/p\u003e\u003cp\u003eScott and Priscilla are asked to guess two numbers X and Y, \u003cb\u003eranging between 2 and 100\u003c/b\u003e and with X\u0026lt;Y. Scott is told their sum S (=X+Y) and Priscilla is told their product P (=X*Y). After this, they have the following conversation:\u003c/p\u003e\u003cpre class=\"language-matlab\"\u003eScott:      I don't know X and Y                                                (Step 1)\r\nPriscilla:  Neither do I                                                        (Step 2)\r\nScott:      Before our conversation I already knew that you didn't know         (Step 3)\r\nPriscilla:  But now I do know X and Y                                           (Step 4)\r\nScott:      And so do I!                                                        (Step 5)\r\n\u003c/pre\u003e\u003cp\u003eThe original puzzle asks you to deduce the value of X and Y from the above information\u003c/p\u003e\u003cp\u003eTo solve this puzzle you may assume that: a) Scott and Priscilla are perfect mathematicians/logicians and always speak the truth; b) they are aware of each other circumstances (Scott knows that Priscilla has been told the product of X and Y, and Priscilla knows that Scott has been told the sum of X and Y before the beginning of their conversation); and c) the third sentence refers to Scott and Priscilla state before the beginning of the conversation (i.e. Scott already knew -at the outset of this conversation- that Priscilla did not know the solution -at the time-)\u003c/p\u003e\u003cp\u003eBriefly, the deduction for this seemingly impossible puzzle starts with all possible pairs of values of X and Y, and uses  the information in the sentences above to sequentially reduce the range of possible (X,Y) pairs to those that would be consistent with each sentence (see figure below; blue dots represent (X,Y) pairs still possible after each of the sentences/steps above; plot uses matrix convention with origin at the upper-left corner, X in vertical axis, and Y in horizontal axis; upper triangular segment represents all possible X\u0026lt;Y pairs before the start of S\u0026P conversation).\u003c/p\u003e\u003cimg src = \"http://www.alfnie.com/software/printnumbers_b.jpg\"\u003e\u003cp\u003eFor example, in \u003cb\u003eStep 1\u003c/b\u003e after S says \u003ci\u003e\"I don't know X and Y\"\u003c/i\u003e we learn that the pair (X,Y) = (2,3) (and a few others) are no longer possible, since otherwise S would have known the solution as soon as he was told their sum (X+Y=5), which takes a unique value among all possible (X,Y) pairs. Similarly, in \u003cb\u003eStep 2\u003c/b\u003e, after P says \u003ci\u003e\"Neither do I\"\u003c/i\u003e, the pair (X,Y) = (5,7) (and many others) are no longer possible, since otherwise P would have known the solution as soon as she was told their product X*Y=35, which turns out to be unique among all remaining X,Y pairs. In \u003cb\u003eStep 3\u003c/b\u003e, S asserting \u003ci\u003e\"I knew that you didn't know\"\u003c/i\u003e tells us that, knowing only X+Y, he was able to determine beforehand that it would be \u003ci\u003eimpossible\u003c/i\u003e for P to know both X and Y from their product X*Y alone, which, again, rules out a considerable number of (X,Y) pairs (e.g. all solutions with X+Y=12 can be ruled out, because if S was told that X+Y=12 he could not have possibly dismissed beforehand the possibility that perhaps X=5 and Y=7 which would have allowed P to know both X and Y as soon as she was told their product X*Y=35). In \u003cb\u003eStep 4\u003c/b\u003e, P suddenly become aware of the solution after S revelation informs us that, unlike in Step 2, she is now (after step-3 crop in possible X,Y pairs) able to uniquely determine the solution (X,Y) from their product (X*Y). This allows us to rule out many (X,Y) pairs among the remaining possible values, such as (X,Y)=(2,15) or (5,6), because P would not have been able to uniquely identify the solution at this point if she was told that the product X*Y was 30. Last, in \u003cb\u003eStep 5\u003c/b\u003e, S suddenly becoming aware of the solution after P revelation, again allows us to rule out any remaining (X,Y) pair where knowing the sum S would still not suffice to uniquely identify the solution.\u003c/p\u003e\u003cp\u003eThis puzzle is very neat because, somewhat surprisingly, after sequentially reducing the range of possible (X,Y) pairs from the five sentences/steps above, only one possible pair remains. The solution X=4 and Y=13, which Priscilla learns after the third sentence, Scott learns after the fourth sentence, and you, the reader, learn after the fifth sentence.\u003c/p\u003e\u003cp\u003eKey to the existence of a unique solution to this puzzle is the initial range of possible (X,Y) pairs that we are told to consider. If, for example, instead of considering all X,Y values between 2 and 100, we were told to consider all X,Y values between 1 and 100, the puzzle would not be solvable, as in this case there will be multiple possible solutions that would all be consistent with the five sentences above (see figure below; at Step 5 there still exist 6 different possible solutions to this puzzle).\u003c/p\u003e\u003cimg src = \"http://www.alfnie.com/software/printnumbers_c.jpg\"\u003e\u003cp\u003eSimilarly, if we were told to consider all X,Y values between 2 and 50, the puzzle would again not be solvable, as in this case there would be \u003ci\u003eno solution\u003c/i\u003e consistent with all five sentences (perhaps surprisingly, since the X=4 Y=13 solution above is in fact within the stated range).\u003c/p\u003e\u003cimg src = \"http://www.alfnie.com/software/printnumbers_d.jpg\"\u003e\u003cp\u003eOn the other hand, if we were told to consider X and Y values ranging between 1 and 24, for example, the puzzle would again become solvable, now with a new unique solution X=1 and Y=6.\u003c/p\u003e\u003cimg src = \"http://www.alfnie.com/software/printnumbers_e.jpg\"\u003e\u003cp\u003eIn this problem you are tasked to create a function that would determine whether a particular variation of this puzzle would work or not. Specifically, given an initial set of possible (X,Y) pairs (entered as a Nx2 matrix and representing the full set of possible X,Y pairs that Scott and Priscilla are told to consider), you should determine whether it is possible to solve this puzzle (i.e. whether one, and only one, (X,Y) pair is consistent with the five sequential sentences above). Your function should simply return 1 (or true) if the puzzle is solvable, and 0 (or false) otherwise.\u003c/p\u003e\u003cp\u003eGood luck!\u003c/p\u003e","function_template":"function valid = fivesteps(XY)\r\n  % XY is a Nx2 matrix, where each row represents a possible X,Y combination\r\n  % valid is 1 if the puzzle is solvable or 0 otherwise\r\n  valid = true;\r\nend","test_suite":"%%\r\nassessFunctionAbsence({'regexp','regexpi','regexprep','str2num'},'FileName','fivesteps.m')\r\nassert(isempty(regexp(fileread('fivesteps.m'),'assert')));\r\n[~,~]=system('rm freepass*');\r\n%%\r\n%lines=textread('fivesteps.m','%s'); \r\n%id=str2num(regexp(lines{end},'\\d+','match','once'));\r\n%assert(~ismember(id,[3430216]),'Please submit a valid non-cheating solution and ask the problem author to manually evaluate it'); % [3931805,3397427,3430216]\r\n%%\r\n% X,Y X\u003cY between 2 and 100\r\n[x,y]=find(triu(ones(100),1));\r\nz=[x y];\r\nvalid=all(z\u003e1,2);\r\nassert(fivesteps(z(valid,:))==true);\r\n%%\r\n% X,Y X\u003cY between 2 and 60\r\n[x,y]=find(triu(ones(60),1));\r\nz=[x y];\r\nvalid=all(z\u003e1,2);\r\nassert(fivesteps(z(valid,:))==false);\r\n%%\r\n% X,Y X\u003cY between 2 and 200\r\n[x,y]=find(triu(ones(200),1));\r\nz=[x y];\r\nvalid=all(z\u003e1,2);\r\nassert(fivesteps(z(valid,:))==true);\r\n%%\r\n% X,Y X\u003cY between 2 and 1680\r\n[x,y]=find(triu(ones(1680),1));\r\nz=[x y];\r\nvalid=all(z\u003e1,2);\r\nassert(fivesteps(z(valid,:))==true);\r\n%%\r\n% X,Y X\u003cY between 2 and 1700\r\n[x,y]=find(triu(ones(1700),1));\r\nz=[x y];\r\nvalid=all(z\u003e1,2);\r\nassert(fivesteps(z(valid,:))==false);\r\n%%\r\n% X,Y X\u003cY between 1 and 20\r\n[x,y]=find(triu(ones(20),1));\r\nz=[x y];\r\nvalid=all(z\u003e=1,2);\r\nassert(fivesteps(z(valid,:))==false);\r\n%%\r\n% X,Y X\u003cY between 1 and 30\r\n[x,y]=find(triu(ones(30),1));\r\nz=[x y];\r\nvalid=all(z\u003e=1,2);\r\nassert(fivesteps(z(valid,:))==true);\r\n%%\r\n% X,Y X\u003cY between 1 and 40\r\n[x,y]=find(triu(ones(40),1));\r\nz=[x y];\r\nvalid=all(z\u003e=1,2);\r\nassert(fivesteps(z(valid,:))==false);\r\n%%\r\n% X,Y X\u003cY between 3 and 5000\r\n[x,y]=find(triu(ones(3000),1));\r\nz=[x y];\r\nvalid=all(z\u003e=3,2);\r\nassert(fivesteps(z(valid,:))==true);\r\n%%\r\n% X,Y X\u003cY between 3 and 100\r\n[x,y]=find(triu(ones(100),1));\r\nz=[x y];\r\nvalid=all(z\u003e=3,2);\r\nassert(fivesteps(z(valid,:))==false);\r\n%%\r\n% X,Y X\u003cY even between 2 and 40\r\n[x,y]=meshgrid(2:2:40);\r\nz=[x(:) y(:)];\r\nvalid=y\u003ex;\r\nassert(fivesteps(z(valid,:))==false);\r\n%%\r\n% X,Y X\u003cY odd between 1 and 1000\r\n[x,y]=meshgrid(1:2:1000);\r\nz=[x(:) y(:)];\r\nvalid=y\u003ex;\r\nassert(fivesteps(z(valid,:))==true);\r\n%%\r\n% X,Y X\u003cY non-primes between 1 and 50\r\n[x,y]=meshgrid(setdiff(1:50,primes(50)));\r\nz=[x(:) y(:)];\r\nvalid=y\u003ex;\r\nassert(fivesteps(z(valid,:))==true);\r\n%%\r\n% X,Y X\u003cY primes between 1 and 50\r\n[x,y]=meshgrid(primes(50));\r\nz=[x(:) y(:)];\r\nvalid=y\u003ex;\r\nassert(fivesteps(z(valid,:))==false);\r\n%%\r\n% X,Y X\u003cY between 3 and 100\r\n[x,y]=find(triu(ones(randi([10,100])),1));\r\nz=[x y];\r\nvalid=all(z\u003e2,2);\r\nassert(fivesteps(z(valid,:))==false);\r\n%%\r\n% X,Y X\u003cY between 1 and 100\r\n[x,y]=find(triu(ones(randi([40,100])),1));\r\nz=[x y];\r\nvalid=all(z\u003e=1,2);\r\nassert(fivesteps(z(valid,:))==false);\r\n%%\r\n% X,Y X\u003cY between 2 and 100\r\n[x,y]=find(triu(ones(randi([62,100])),1));\r\nz=[x y];\r\nvalid=all(z\u003e1,2);\r\nassert(fivesteps(z(valid,:))==true);\r\n%%\r\n% X,Y X\u003cY between 2 and 100\r\n[x,y]=find(triu(ones(randi([10,61])),1));\r\nz=[x y];\r\nvalid=all(z\u003e1,2);\r\nassert(fivesteps(z(valid,:))==false); \r\n%%\r\n% X,Y X\u003cY in 3 5 12 18 20 28 30\r\n% Possible solutions after step 1: [3,20] [3,30] [5,18] [5,28] [18,30] [20,28] \r\n% Possible solutions after step 2: [3,30] [5,18] \r\n% Possible solutions after step 3: [5,18] \r\n% Possible solutions after step 4: [5,18] \r\n% Possible solutions after step 5: [5,18] \r\n[x,y]=meshgrid([3 5 12 18 20 28 30]);\r\nz=[x(:) y(:)];\r\nvalid=y\u003ex;\r\nassert(fivesteps(z(valid,:))==true);\r\n%%\r\n% X,Y X\u003cY in 3 5 12 15 28 30\r\n% Possible solutions after step 1: [3,30] [5,28] \r\n% Possible solutions after step 2: \r\n[x,y]=meshgrid([3 5 12 15 28 30]);\r\nz=[x(:) y(:)];\r\nvalid=y\u003ex;\r\nassert(fivesteps(z(valid,:))==false);\r\n%%\r\n% X,Y X\u003cY in 2 4 6 8 12 16 18\r\n% Possible solutions after step 1: [2,8] [2,12] [2,16] [2,18] [4,6] [4,16] [4,18] [6,8] [6,12] [6,16] [6,18] [8,12] [8,16] \r\n% Possible solutions after step 2: [2,12] [4,6] [4,18] [6,12] [6,16] [8,12] \r\n% Possible solutions after step 3: [2,12] [4,18] [6,12] [6,16] \r\n% Possible solutions after step 4: [2,12] [6,16] \r\n% Possible solutions after step 5: [2,12] [6,16] \r\n[x,y]=meshgrid([2 4 6 8 12 16 18]);\r\nz=[x(:) y(:)];\r\nvalid=y\u003ex;\r\nassert(fivesteps(z(valid,:))==false);\r\n%%\r\n% X,Y X\u003cY in 4 6 8 12 16 18\r\n% Possible solutions after step 1: [4,16] [4,18] [6,16] [6,18] [8,12] [8,16] \r\n% Possible solutions after step 2: [6,16] [8,12] \r\n% Possible solutions after step 3: [6,16] \r\n% Possible solutions after step 4: [6,16] \r\n% Possible solutions after step 5: [6,16] \r\n[x,y]=meshgrid([4 6 8 12 16 18]);\r\nz=[x(:) y(:)];\r\nvalid=y\u003ex;\r\nassert(fivesteps(z(valid,:))==true);\r\n%%\r\n% X,Y X\u003cY in 2 3 10 14 15 20 21\r\n% Possible solutions after step 1: [2,15] [2,21] [3,14] [3,20] [3,21] [10,14] [14,21] [15,20] \r\n% Possible solutions after step 2: [2,21] [3,14] \r\n% Possible solutions after step 3: [3,14] \r\n% Possible solutions after step 4: [3,14] \r\n% Possible solutions after step 5: [3,14] \r\n[x,y]=meshgrid([2 3 10 14 15 20 21]);\r\nz=[x(:) y(:)];\r\nvalid=y\u003ex;\r\nassert(fivesteps(z(valid,:))==true);\r\n%%\r\n% X,Y X\u003cY in 2 3 5 8 10 14 15 20 21\r\n% Possible solutions after step 1: [2,15] [2,20] [2,21] [3,10] [3,14] [3,15] [3,20] [3,21] [5,8] [5,20] [8,10] [8,14] [8,15] [8,21] [10,14] [10,15] [14,15] [14,21] [15,20] \r\n% Possible solutions after step 2: [2,15] [2,20] [2,21] [3,10] [3,14] [5,8] \r\n% Possible solutions after step 3: [2,15] [3,10] [3,14] [5,8] \r\n% Possible solutions after step 4: [3,14] [5,8] \r\n% Possible solutions after step 5: [3,14] [5,8] \r\n[x,y]=meshgrid([2 3 5 8 10 14 15 20 21]);\r\nz=[x(:) y(:)];\r\nvalid=y\u003ex;\r\nassert(fivesteps(z(valid,:))==false);\r\n%%\r\n% a few random cases to discourage look-up table solutions\r\n    data={[2 5 6 7 9 10 12 16 23 25 26 27 28 29 31],[49 94 97 109 112 113 154 157 158 161 184 187 188 191 193 199 202 203 206 208 209 214 217 218 221 223 224 225];...\r\n        [2 4 6 9 10 11 12 13 14 20 22 23 25 26 28],[49 64 79 81 109 111 113 124 125 126 128 129 154 156 158 159 161 169 171 173 174 176 177 184 186 188 189 191 192 193 199 201 203 204 206 207 208 209 214 216 218 219 221 222 223 224 225];...\r\n        [2 3 4 6 7 9 10 11 12 14 19 20 21 24 25 28 31],[19 35 87 89 108 109 121 124 126 127 142 144 145 155 156 171 174 175 177 180 206 215 223 230 240 241 245 247 253 257 262 263 264 270 271 274 278 279 280 281 285 288 289];...\r\n        [2 4 5 6 8 9 10 11 15 16 18 19 20 21 23 24 25 26 27 28 29],[45 66 67 87 109 111 150 152 153 171 174 177 214 219 221 234 235 237 240 242 243 256 258 261 263 264 265 276 277 279 282 284 285 286 287 297 298 300 303 305 306 307 308 309 319 321 324 326 327 328 329 330 331 339 340 342 345 347 348 349 350 351 352 353 360 361 363 366 368 369 370 371 372 373 374 375 381 382 384 387 389 390 391 392 393 394 395 396 397 402 403 405 408 410 411 412 413 414 415 416 417 418 419 423 424 426 429 431 432 433 434 435 436 437 438 439 440 441];...\r\n        [2 4 5 6 8 10 11 12 14 16 17 25 27 28 29 30 31],[1 54 89 109 125 137 140 144 145 154 156 171 177 179 181 188 194 196 198 199 205 208 211 213 216 217 222 225 228 229 230 232 233 234 235 239 245 247 248 250 251 252 253 256 262 264 266 267 268 269 270 271 273 279 281 283 284 285 286 287 288 289];...\r\n        [2 5 6 7 10 14 15 16 17 20 21 23 24 25 29 30],[120 152 154 168 170 171 184 185 186 187 188 200 202 203 204 205 216 218 219 220 221 222 232 234 235 236 237 238 239 248 250 251 252 253 254 255 256];...\r\n        [4 5 7 8 9 11 12 15 16 17 20 21 24 27 28 30],[18 68 82 85 86 98 100 102 103 116 118 119 120 129 130 136 146 150 151 152 153 162 166 168 169 171 194 196 198 199 202 203 205 212 226 230 231 232 233 235 237 239 244 249 250];...\r\n        [2 3 4 6 8 9 11 12 13 15 17 19 20 21 22 23 25 28 29 31],[22 43 102 122 123 127 147 148 162 163 168 169 182 187 188 190 202 203 207 208 209 210 211 222 223 227 228 229 230 231 232 242 243 247 248 249 250 251 252 253 262 263 267 268 269 270 271 272 273 274 282 283 287 288 289 290 291 292 293 294 295 302 303 307 308 309 310 311 312 313 314 315 316 322 323 327 328 329 330 331 332 333 334 335 336 337 342 343 347 348 349 350 351 352 353 354 355 356 357 358 362 363 367 368 369 370 371 372 373 374 375 376 377 378 379 382 383 387 388 389 390 391 392 393 394 395 396 397 398 399 400];...\r\n        [3 4 5 6 8 9 10 11 13 14 18 19 21 22 23 25 26 28 29 30 31],[111 150 153 154 155 174 176 177 195 197 198 199 216 218 219 220 221 237 239 240 242 243 258 260 261 262 263 264 265 279 281 282 283 284 285 286 287 300 302 303 304 305 306 307 308 309 321 323 324 325 326 327 328 329 330 331 342 344 345 346 347 348 349 350 351 352 353 363 365 366 367 368 369 370 371 372 373 374 375 384 386 387 388 389 390 391 392 393 394 395 396 397 405 407 408 409 410 411 412 413 414 415 416 417 418 419 426 428 429 430 431 432 433 434 435 436 437 438 439 440 441];...\r\n        [2 4 5 6 8 10 11 13 14 15 16 17 20 22 28 31],[51 52 83 100 103 132 135 136 137 148 151 153 154 168 183 185 187 188 196 199 200 201 202 204 205 212 215 217 218 220 221 222 228 231 232 233 234 236 237 238 239 244 247 249 250 252 253 254 255 256];...\r\n        [2 4 5 8 10 11 12 15 20 21 22 26 28 29 30 31],[1 65 82 103 119 120 129 145 151 152 154 177 183 184 186 188 193 199 200 204 205 209 215 216 218 220 221 222 225 231 232 234 236 237 238 239 241 247 248 250 252 253 254 255 256];...\r\n        [2 4 5 7 8 12 13 17 18 19 22 26 28 29 30],[49 64 65 109 110 113 124 125 128 129 139 140 143 144 145 154 155 158 159 160 161 184 185 188 189 191 193 199 200 203 204 205 206 208 209];...\r\n        [2 3 6 8 9 12 15 17 18 19 21 22 23 24 25 27 28 29 30],[81 96 119 121 137 138 140 141 157 159 160 161 176 180 195 197 198 199 201 214 216 217 218 220 221 233 235 237 239 240 241 254 255 256 258 259 260 261 271 273 274 275 277 278 279 280 281 290 292 293 294 296 297 298 299 300 301 309 311 312 313 315 316 317 318 319 320 321 328 330 331 332 334 335 336 337 338 339 340 341 347 349 350 351 352 353 354 355 356 357 358 359 360 361];...\r\n        [6 7 9 11 17 18 21 22 23 27 28 29],[53 101 105 125 129 131 137 141 143 144];...\r\n        [2 3 4 6 8 9 11 12 15 16 18 19 20 22 24 25 27 28 29 30],[22 81 83 102 103 121 122 127 147 148 161 164 166 167 168 188 189 190 202 203 208 210 211 222 231 232 242 247 248 250 251 252 253 262 263 267 268 269 270 271 272 273 274 285 287 288 290 291 292 293 294 295 310 314 315 316 322 323 327 328 330 331 332 333 334 335 336 337 347 348 350 351 352 353 354 355 356 357 358 362 367 368 370 371 372 373 374 375 376 377 378 379 382 387 388 390 391 392 393 394 395 396 397 398 399 400];...\r\n        [3 4 5 6 10 11 13 17 20 21 25 26 30],[57 70 71 96 97 99 161 162 164 169];...\r\n        [2 3 4 10 12 15 16 18 19 20 21 22 25 26 28 30 31],[1 52 55 120 123 127 140 144 145 157 161 162 163 171 174 178 179 180 181 188 191 195 196 197 198 199 205 208 212 213 214 215 216 217 222 225 229 230 231 232 233 234 235 239 242 246 247 248 249 250 251 252 253 256 259 263 264 266 267 268 269 270 271 273 276 280 281 282 283 284 285 286 287 288 289];...\r\n        [2 4 5 7 10 15 16 18 20 24 25 27 28 29 30 31],[17 18 82 86 98 102 103 114 118 119 120 146 150 151 152 154 178 182 183 186 187 188 209 210 214 215 216 218 220 222 226 230 231 232 234 236 238 239 242 246 247 248 250 252 254 255 256];...\r\n        [4 6 7 10 11 12 18 20 21 22 24 28],[26 27 40 51 52 88 97 99 101 103 105 112 113 117 123 124 131 133 134 136 139 141 143];...\r\n        [2 3 4 6 8 10 12 14 15 17 18 19 20 21 22 24 26 28 30],[20 41 79 80 117 118 119 121 137 138 141 156 157 160 174 178 181 193 197 198 200 201 212 213 214 220 221 229 231 232 233 235 238 239 240 241 251 252 255 269 272 274 276 277 279 281 288 292 293 295 296 297 298 300 301 307 311 312 314 315 316 317 319 320 321 326 330 331 333 334 335 336 338 339 340 341 345 349 350 352 353 354 355 357 358 359 360 361];...\r\n        [2 3 4 6 8 10 12 13 15 16 18 19 20 22 25 26 29],[18 35 37 70 72 88 91 105 106 107 121 124 139 140 141 144 145 160 161 163 173 176 179 180 181 190 195 197 207 208 209 211 214 215 216 217 224 227 230 231 232 234 235 241 247 248 249 251 252 253 258 259 260 275 278 281 282 283 285 286 287 289];...\r\n        [3 4 5 8 9 12 14 21 23 24 28 30],[1 49 53 97 101 105 121 125 129 131];...\r\n        [2 3 6 7 8 9 12 13 15 16 18 19 22 23 24 26 28],[52 55 89 103 120 124 140 145 157 159 174 179 181 191 196 198 199 208 213 215 216 217 230 232 235 242 245 247 249 250 251 252 253 259 261 276 278];...\r\n        [2 4 6 8 9 10 11 12 21 22 24 25 26 27 28 29 31],[1 19 37 55 73 86 87 89 90 104 121 123 124 125 127 137 139 140 141 144 145 155 159 172 174 178 179 181 188 189 190 192 194 195 198 199 205 206 207 208 212 215 216 217 222 224 226 227 239 241 242 243 246 247 249 250 251 253 256 257 258 259 260 263 264 266 267 268 270 271 273 274 275 276 277 280 281 283 284 285 287 288 289]};\r\n    for ndata=randi(size(data,1),1,20)\r\n        a=data{ndata,1};\r\n        N=numel(a);\r\n        A=zeros(N);\r\n        A(data{ndata,2})=1;A=A|A';\r\n        if rand\u003c.5, idx=find(A); else idx=find(~A); end\r\n        [b,c]=ind2sub(size(A),idx(randi(numel(idx))));\r\n        d=a;d([b c])=[];\r\n        [x,y]=meshgrid(d);\r\n        z=[x(:) y(:)];\r\n        valid=y\u003ex;\r\n        assert(fivesteps(z(valid,:))==A(b,c),'failed on d = %s (correct output = %s)',mat2str(d),mat2str(A(b,c)));\r\n    end\r\n","published":true,"deleted":false,"likes_count":6,"comments_count":36,"created_by":43,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":42,"test_suite_updated_at":"2018-01-02T21:35:48.000Z","rescore_all_solutions":true,"group_id":35,"created_at":"2017-10-11T19:04:33.000Z","updated_at":"2026-01-17T20:25:15.000Z","published_at":"2017-10-16T01:51:01.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/image\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/media/image1.JPEG\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/image\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/media/image2.JPEG\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/image\",\"targetMode\":\"\",\"relationshipId\":\"rId3\",\"target\":\"/media/image3.JPEG\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/image\",\"targetMode\":\"\",\"relationshipId\":\"rId4\",\"target\":\"/media/image4.JPEG\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/image\",\"targetMode\":\"\",\"relationshipId\":\"rId5\",\"target\":\"/media/image5.JPEG\"}],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThis problem asks you to identify valid variations of the famous\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://en.wikipedia.org/wiki/Sum_and_Product_Puzzle\\\"\u003e\u003cw:r\u003e\u003cw:t\u003esum and product puzzle\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe original\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003esum and product\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e puzzle goes somewhat like this:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eScott and Priscilla are asked to guess two numbers X and Y,\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eranging between 2 and 100\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e and with X\u0026lt;Y. Scott is told their sum S (=X+Y) and Priscilla is told their product P (=X*Y). After this, they have the following conversation:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[Scott:      I don't know X and Y                                                (Step 1)\\nPriscilla:  Neither do I                                                        (Step 2)\\nScott:      Before our conversation I already knew that you didn't know         (Step 3)\\nPriscilla:  But now I do know X and Y                                           (Step 4)\\nScott:      And so do I!                                                        (Step 5)]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe original puzzle asks you to deduce the value of X and Y from the above information\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eTo solve this puzzle you may assume that: a) Scott and Priscilla are perfect mathematicians/logicians and always speak the truth; b) they are aware of each other circumstances (Scott knows that Priscilla has been told the product of X and Y, and Priscilla knows that Scott has been told the sum of X and Y before the beginning of their conversation); and c) the third sentence refers to Scott and Priscilla state before the beginning of the conversation (i.e. Scott already knew -at the outset of this conversation- that Priscilla did not know the solution -at the time-)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eBriefly, the deduction for this seemingly impossible puzzle starts with all possible pairs of values of X and Y, and uses the information in the sentences above to sequentially reduce the range of possible (X,Y) pairs to those that would be consistent with each sentence (see figure below; blue dots represent (X,Y) pairs still possible after each of the sentences/steps above; plot uses matrix convention with origin at the upper-left corner, X in vertical axis, and Y in horizontal axis; upper triangular segment represents all possible X\u0026lt;Y pairs before the start of S\u0026amp;P conversation).\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId2\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor example, in\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eStep 1\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e after S says\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\\\"I don't know X and Y\\\"\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e we learn that the pair (X,Y) = (2,3) (and a few others) are no longer possible, since otherwise S would have known the solution as soon as he was told their sum (X+Y=5), which takes a unique value among all possible (X,Y) pairs. Similarly, in\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eStep 2\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, after P says\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\\\"Neither do I\\\"\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, the pair (X,Y) = (5,7) (and many others) are no longer possible, since otherwise P would have known the solution as soon as she was told their product X*Y=35, which turns out to be unique among all remaining X,Y pairs. In\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eStep 3\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, S asserting\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\\\"I knew that you didn't know\\\"\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e tells us that, knowing only X+Y, he was able to determine beforehand that it would be\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eimpossible\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e for P to know both X and Y from their product X*Y alone, which, again, rules out a considerable number of (X,Y) pairs (e.g. all solutions with X+Y=12 can be ruled out, because if S was told that X+Y=12 he could not have possibly dismissed beforehand the possibility that perhaps X=5 and Y=7 which would have allowed P to know both X and Y as soon as she was told their product X*Y=35). In\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eStep 4\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, P suddenly become aware of the solution after S revelation informs us that, unlike in Step 2, she is now (after step-3 crop in possible X,Y pairs) able to uniquely determine the solution (X,Y) from their product (X*Y). This allows us to rule out many (X,Y) pairs among the remaining possible values, such as (X,Y)=(2,15) or (5,6), because P would not have been able to uniquely identify the solution at this point if she was told that the product X*Y was 30. Last, in\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eStep 5\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, S suddenly becoming aware of the solution after P revelation, again allows us to rule out any remaining (X,Y) pair where knowing the sum S would still not suffice to uniquely identify the solution.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThis puzzle is very neat because, somewhat surprisingly, after sequentially reducing the range of possible (X,Y) pairs from the five sentences/steps above, only one possible pair remains. The solution X=4 and Y=13, which Priscilla learns after the third sentence, Scott learns after the fourth sentence, and you, the reader, learn after the fifth sentence.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eKey to the existence of a unique solution to this puzzle is the initial range of possible (X,Y) pairs that we are told to consider. If, for example, instead of considering all X,Y values between 2 and 100, we were told to consider all X,Y values between 1 and 100, the puzzle would not be solvable, as in this case there will be multiple possible solutions that would all be consistent with the five sentences above (see figure below; at Step 5 there still exist 6 different possible solutions to this puzzle).\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId3\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eSimilarly, if we were told to consider all X,Y values between 2 and 50, the puzzle would again not be solvable, as in this case there would be\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eno solution\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e consistent with all five sentences (perhaps surprisingly, since the X=4 Y=13 solution above is in fact within the stated range).\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId4\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eOn the other hand, if we were told to consider X and Y values ranging between 1 and 24, for example, the puzzle would again become solvable, now with a new unique solution X=1 and Y=6.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId5\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIn this problem you are tasked to create a function that would determine whether a particular variation of this puzzle would work or not. Specifically, given an initial set of possible (X,Y) pairs (entered as a Nx2 matrix and representing the full set of possible X,Y pairs that Scott and Priscilla are told to consider), you should determine whether it is possible to solve this puzzle (i.e. whether one, and only one, (X,Y) pair is consistent with the five sequential sentences above). Your function should simply return 1 (or true) if the puzzle is solvable, and 0 (or false) otherwise.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eGood luck!\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray 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FABRRRQAUUUUAFFFFABRRRQAUUUUAZ2qxh4IyY5W2yA7oj80f+0B3+nvVCDT5rgXbxs8YmKLunUkyBc5JHBwc4xxwK36KpS0MZUVKXMznbq2unna38t2WeSOOWRVO0pglvpwCvX+IUmuI8rzW7R3iJ5GIfsyFllJz8rYHHQdcDBrWubsQS7WkSJMD5mQkEknj9KYbwuySRTwtC7hV2nO7nB5/Os/rKi/QuEVBWKGp2l5dtYA2kUyIhaWNpNiB8AD1Pdu3as68OnSaR5jW6xywF0W3ZyAr7sHBH3ucfnW9qdp5i/a4pJ47iCNtpgxlx124IIOcDtXFPqTSWdtp1vdWs88zw3EckI3fOxO5X3NhmHXqufRRVNKSOjmbhZHUaVD/AGa9vbl5P3ysSnG0PnJIx29OemK1L9ituFUDzHYIhx0J7/zP4Vzvh+8vb2S2W5eEeW0qzKkeMyLt568cNyPXNa0d7NJdKsyI0SsSG8sg9OvPTB4/Gi3K9TnUnF3mYF/u07WdQ2xOnnJHHAdmfMOzBC+p5znPGKyrFbVNK0y1gtRFqltcoJZnjwqFWyzGTpgrkYz3HFdtqd1ZLFE11GPKZynnMgIhODycjj0z71m6Fas6XywyrLD5wKXBGRLxz0xnHTjjj2ra14N9TeLTRt2KReUZElSdyxLSLg8+g9PpU7QRsSWUFj/ERzUoqKdpUt5Hhj82UKSqbtu49hntWEVZWM2Z+pGUSBYWcsI2c/Ptx0wffHp0pLi3N5fWrfaJVHltJGVwCp+UZ6dwx65qrHfS3U95BLBZ38lrHuBi+ULJkjyzuJweOv6UQaw51J7aaSzkZIt/nRKQsYJA2nk9eDn2ocWtTNq+5ffSUMDKs0onLmQTkjcHxtz6dOMeleS/F+TWNC0jRNG0G2luxcSNJdEW3mhtgQKCMEAHP/jo5r2eGVJkDI6t6lTkVJTK5UtDF0rw/baFoi2GnW8Ebn55XVfLEsh+87bfX0/DpXhXx70yPQtY0DVrNlgv5jIxkj4cmPy9rlsctknn6V77ruuWHhvRbrVtTm8q1t13M2MknoAB3JOAK8b8MX1/8UPE8/iDU9DhTRkzZQySOxIRjgxqM7SSCdzY9vaneyKvZWPZdLt1Nvb3hDiWSLcQ+MjcFJHA7YqXULVrqNAn3kcPgsVz7ZH1qLWI3bTpiszIqocqoHzVojoKSk7jUrMwHsLqCSGQqXRWDybZR1GcfexnjAyT26VpR34kdESCTLKGP3RtBzjPPt2rA168e41dtO8y7EUUKSNFaKS8m5jnOOwC+3LVVuZILvSRdNetI1tHvNwjGF3Vcnp3bjpxzmifM7M2eqXMdrUYhRZPMA569Tj8q5/SryS3trWJppJJrht7LLNuaJSMjr1xwOPc1vS3McIO4nIGcAE1Mmluc+7sT1h63p9lNNbXF1b3cqpICxt2f5SAcMVXk9eo5/CrP2gEsnn5Z3UYU9B3AqxFIkU0kTSccFQzc9Oev+eaUZ32LcLHIRaZJbzJPOt+tnJ5qptjeR1XeGUFOSN2M9O3PWtrS9Ku/wCxoreecxpIuZIinzKD1XOePy9a3nQOu0jjNKqhVAHQVPso7siyK9yqrEMOItvRuw4rhryBobG4vtLt2gt4rJ0E4XD3GSDvJA9AW3Nzzn69pqlwltYuXRXD/IFZcqc+vtXL21ppkFjqAsoIROZ0jR0iClt2z5SeMKTnI9M1boueprCF1ct6RbCw1tbRIrXEkCTbrVGQAfMBuGTzz97jP4V1dYGgWkdje30QtYbZ22SGO3HybcEAjgc5DdhW/TUOTQmSsxaKKKZIUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRUcsnlRM/GFGTuOBQBWvY7kwSm3IZyuFRh3+uaqz6XxbiNVkEUisu4fMo/iIP61oJMDbLM5VAVDHLcD8aoXOphZv3c8axBQfMEZkBznuDx0rGdOG7Ik0ldmmQCMH07Vz914cshaTPe3d1MQEKzS7GaIIdwwNuDz1yCW75rVa5k+1wIFQwyAndnk8Z6VLeRiazmjZC4ZCNoOCfpWnNdaAp3TsZ2jW9pYoLVI5VmG6TdMqhmyeSNvAHQYGMYHFTQTG4uLhFVDJEAAc/KMk8Dj259xVOG3uZ7lpYZZ9ywlBJOm3BJHAGB0x1x3pbCaS31KSOQRBSqRjZnHBI6n34qYzlcyvKbVzUV1t1xMQHY54JbPv0p73MMYBZxyMjAzx61BIHa+kAkCfuxg4yepqNW820byXihn8sZZkyoHPbI4rUam1ojRUhl3Agg9DVK5jvpoJo4ZfIc5EcmAdp5wcfl+tc9p+uam14ttPFbw2ot1lSZYXfcpcqpPzfLwPetfTrrUZdUubeeW1lgt1Cu8ULIRIcHbyzZwpyf8AeHvScTeLuipf2uomO6v5xaQTLaNAgjdmDFiDuYlRwMcDB6mudsd8U8t0/lxr9rSJ2W7MhHAILKV+YZ5JJB49BXoorJ1SO1iPmLZRS3kwMaN5Slj8pzkntirU0lZjXpcztBuSdUuPJknltJIlKyzOT5kg+8wB6A5H+Faq6kxMbtAwhkYKr5556ZFZmnWwtL2AwaZBakxsCFwGYArycCrE9zFctDLGzC6V1HkEk455yPpnmh2epwzrybbvY8/+POka/r2jaPp+i6dc3kRuHmnEK52lVAXP/fTflXdaFZWNloOkaPZK0cVvDCy4TjgBufc9SfU+9cF8YNT8e2OraanhGPVWt2t2M/2K0My7t3GSFODivQNDt1TSNIu75mgvGtYWlikAQ+Z5QU5BG4EZIxUM7VqjT1VkGmzCRioZdowMkk8AAdyTxVGTXJItEh1A2Mjs4+ZEYEJzjk/4CrWpWsVy9uZLx7Zo2LIVZRkkY6MD6/rVSPRrhPDR04ziSYj7zngc57D+lUuU0io21M3XYNWs9RutV04bo2t9sgO0YCjIK9c8knGB0xWTPo8MtpY2UV/FcXN3IJHkyWAGWZvoDnp359676eN5LeSOKTy5CpCvtztPY4rGtNIuTeRSzxWtuIQxH2bOZHIxuPAxwTxz1o3RUZLlszDa1WDUba2+2Tm886Mywtnyz0GV4/yK6y4TDHYJfO2gIRnBPvjj86zhp2qXC2ttePbmK3lSQ3CsS8m05Hy44J7nNb4HFY1KXNuzOa13M+a3ka5UtGrKz8kMem0+3FTvBFHGSIlc5/jOck+5zVqqWoSmOIAqrRkEyFl3AKOvGaUIKm2yHKyHwXZmMYMe3ehfr0IIBH61ZyPWs6JkeKUXEEXkxLxhOB1yvvjA6f0qvIbKy07z7hUTI8woF4JxwMDrRKbvaKFGVzWmz5D7QWO04AOCa5+2jtLXT7mO98tI5SpAkBh3vtHADHOflHI70tj4itItNm82KeOW3fa0O0FssTgLg4xwe/aq13q1vNfPKHjt5Y4nglivB0BwcjGQT/s/xfhXTTUuXVWNoxki9aS2tjOqi4QPM4XDzBnk6Y+8SSBnt05rerh7mKe20o2xliF1cafDGY5s+ZvCkAJ/eOf++Tz3ruB0pTXUU+4tFFFQQFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQBWvBObdvs/wDrMjH0zz+mazJba5nsbgXjygLFkYcKCcHI4PPbrW3Ve4IdHhYcOu04PPNTOqoR1LjLSxj6lK8Wkad5cImcvEFiLY3nHT0qPTWmvrm+TCWwlCSbANwKMCAwPHJweo9KvXGnm5WBDvCQcxMj4ZWHGfQ8ZqOwSx0q5+ztM5u58A7yTwCdq+i9envWaTm/Ih+9oaRtR5kDA4EWRj14xVg1Un1C3t5NkkmGxkgAnA9TjoPerQORkdK25bBycouKoHSrb7Us53F0AVcgcAHOOmav9qz7TUxdEkCEIBuOJcsB7jFNRbJlJJ2Zckhjlx5iK2OmRms6+jgeUrOqumAoj3ENg9SAP88VJHqLP5LPAVjnH7ohsk8EjI7cCq95dXUTyEyrGRjyo+D5n9azq3SsZyqwir2uUoLOO41EXxSWZWtxF5EcmNu1yRuBIGOentWpo0HkWcpJZmluJZmZlAzuckdCeAMDr2rOknmSOWDz2iCkIQFyBk5JLduDW40qW6IuPlxgY7Cppt21CNZTu9iYDH4VmalHFLdW6tNIkoVyiIdu/pnmtSq91aQ3kPlzorL1GVBwfxrSV7aHRB2dzL07UXRGilYShZiiOz8svHTj5up/KtvA9BVEaVAvCM6jHIAAH8uPwqwLu3OcTxHnbw46+lZ0udX5xTcW7k9ZGpXUcc0kUUDy3DxjeQMhF55/nx3rW71lXsVpdTywHZHchA3mNGpJHPTP0raKTepVO19SpeXtuLNEtCZJZIgOUJJQA8+1ammXS3diki8Y+Uqeq47HNYsUMcsNg8Yg80K6LHIDtcA9QefTP41o6Ir7bt2YMHnJDDoeADj8QR+FXKkldo2qRio6GvRUJlEcbSTFY0UZJZuAPUmmXF9a2qhp7iKMEEruYDI9vWszluNe+iSRlKuVU7WcD5VPuaimmhju3kKyu0YBYquQnB/xzXN21+puWuhdOspjlZUxwPnBC++eatQ3aXmsARyCOSeFlfKn5BleB/tYHX/AVnzXHys6RW3KWU5BGQaY0MUxAkjVwORvGcH8aljjSOMIo+UAAfSnBQOlLkdw3Wpyeq3jWVzfwJc29iqRK0EPkqftRIPHPXnjAqbU7HUNTs5oSFhh+yB2D7TulIPy5P3QMDn39qW+tJ4LvVJJdMW9iulBSUugCAIBtbccgAgnjPWqfm3L2ltDdW11cWEFmPMCFR5rrwxfcc4GBgd/wrp91WaGoXV0iro+nX13o9zftBI89wsBhzIqtlCfnGOB14HcD3roPD1jc2gvXu45Fkmm375WUsRtA52k9wTjtnFXtNUm2Eu10VwNsbNnYvar1Eqrlcr2jcbC0UUVmQFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAJWPdR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Anniversary Problem","description":"Since Cody is 5 years old, it's pernicious. A \u003chttp://rosettacode.org/wiki/Pernicious_numbers Pernicious number\u003e is an integer whose population count is a prime. Check if the given number is pernicious.","description_html":"\u003cp\u003eSince Cody is 5 years old, it's pernicious. A \u003ca href = \"http://rosettacode.org/wiki/Pernicious_numbers\"\u003ePernicious number\u003c/a\u003e is an integer whose population count is a prime. Check if the given number is pernicious.\u003c/p\u003e","function_template":"function y = isPernicious(x)\r\n  y = false;\r\nend","test_suite":"%%\r\nx = 5;\r\ny_correct = true;\r\nassert(isequal(isPernicious(x),y_correct))\r\n%%\r\nx = 1;\r\ny_correct = false;\r\nassert(isequal(isPernicious(x),y_correct))\r\n%%\r\nx = 2^randi(16);\r\ny_correct = false;\r\nassert(isequal(isPernicious(x),y_correct))\r\n%%\r\nx = 17;\r\ny_correct = true;\r\nassert(isequal(isPernicious(x),y_correct))\r\n%%\r\nx = 18;\r\ny_correct = true;\r\nassert(isequal(isPernicious(x),y_correct))\r\n%%\r\nx = 61;\r\ny_correct = true;\r\nassert(isequal(isPernicious(x),y_correct))\r\n%%\r\nx = 6;\r\ny_correct = true;\r\nassert(isequal(isPernicious(x),y_correct))\r\n%%\r\nx = 2115;\r\ny_correct = false;\r\nassert(isequal(isPernicious(x),y_correct))\r\n%%\r\nx = 2114;\r\ny_correct = true;\r\nassert(isequal(isPernicious(x),y_correct))\r\n%%\r\nx = 2017;\r\ny_correct = true;\r\nassert(isequal(isPernicious(x),y_correct))\r\n\r\n\r\n\r\n\r\n","published":true,"deleted":false,"likes_count":13,"comments_count":1,"created_by":14358,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":838,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":34,"created_at":"2014-12-08T08:48:45.000Z","updated_at":"2026-04-10T14:31:08.000Z","published_at":"2017-10-16T01:45:06.000Z","restored_at":"2017-10-25T14:37:50.000Z","restored_by":null,"spam":false,"simulink":false,"admin_reviewed":true,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eSince Cody is 5 years old, it's pernicious. A\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://rosettacode.org/wiki/Pernicious_numbers\\\"\u003e\u003cw:r\u003e\u003cw:t\u003ePernicious number\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e is an integer whose population count is a prime. Check if the given number is pernicious.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":248,"title":"Twins in a Window","description":"\u003chttp://en.wikipedia.org/wiki/Twin_primes Twin primes\u003e are prime numbers that differ by 2, such as [11,13] or [41,43]. Write a function that returns a sorted list of the twin primes between lowVal and highVal (all the primes must be greater than lowVal and less than highVal). The primes should be arranged in an n-by-2 matrix as shown in the examples below.\r\n\r\nExamples:\r\n\r\n Input  lowVal = 10, highVal = 50\r\n Output p is [ 11    13\r\n               17    19\r\n               29    31\r\n               41    43 ]\r\n\r\n Input  lowVal = 1000, highVal = 1050\r\n Output p is [ 1019  1021\r\n               1031  1033 ]\r\n","description_html":"\u003cp\u003e\u003ca href=\"http://en.wikipedia.org/wiki/Twin_primes\"\u003eTwin primes\u003c/a\u003e are prime numbers that differ by 2, such as [11,13] or [41,43]. Write a function that returns a sorted list of the twin primes between lowVal and highVal (all the primes must be greater than lowVal and less than highVal). The primes should be arranged in an n-by-2 matrix as shown in the examples below.\u003c/p\u003e\u003cp\u003eExamples:\u003c/p\u003e\u003cpre\u003e Input  lowVal = 10, highVal = 50\r\n Output p is [ 11    13\r\n               17    19\r\n               29    31\r\n               41    43 ]\u003c/pre\u003e\u003cpre\u003e Input  lowVal = 1000, highVal = 1050\r\n Output p is [ 1019  1021\r\n               1031  1033 ]\u003c/pre\u003e","function_template":"function p = window_twins(lowVal,highVal)\r\n  p = 0;\r\nend","test_suite":"%%\r\nlowVal = 10;\r\nhighVal = 50;\r\np = [11    13\r\n     17    19\r\n     29    31\r\n     41    43];\r\nassert(isequal(window_twins(lowVal,highVal),p))\r\n\r\n%%\r\nlowVal = 1000;\r\nhighVal = 1050;\r\np = [1019  1021\r\n     1031  1033];\r\nassert(isequal(window_twins(lowVal,highVal),p))\r\n\r\n%%\r\nlowVal = 3120;\r\nhighVal = 3300;\r\np = [ 3167  3169\r\n      3251  3253\r\n      3257  3259];\r\nassert(isequal(window_twins(lowVal,highVal),p))\r\n","published":true,"deleted":false,"likes_count":2,"comments_count":0,"created_by":7,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":162,"test_suite_updated_at":"2012-02-03T17:12:57.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2012-02-03T16:30:52.000Z","updated_at":"2026-04-10T14:29:19.000Z","published_at":"2012-02-03T17:12:56.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"http://en.wikipedia.org/wiki/Twin_primes\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eTwin primes\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e are prime numbers that differ by 2, such as [11,13] or [41,43]. Write a function that returns a sorted list of the twin primes between lowVal and highVal (all the primes must be greater than lowVal and less than highVal). The primes should be arranged in an n-by-2 matrix as shown in the examples below.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eExamples:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[ Input  lowVal = 10, highVal = 50\\n Output p is [ 11    13\\n               17    19\\n               29    31\\n               41    43 ]\\n\\n Input  lowVal = 1000, highVal = 1050\\n Output p is [ 1019  1021\\n               1031  1033 ]]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":44230,"title":"I'm going to enjoy watching you calculate, Mr Anderson","description":"Smith numbers are numbers such that if you add up all of the digits in the number, that sum equals the sum of all of the digits in all of their factors. This term was coined by Albert Wilansky, when he noticed the defining property in the phone number of his brother-in-law Harold Smith: 493-7775.\r\n4+9+3+7+7+7+5=42\r\nThe prime factors of 4937775 are 3, 5, 5, and 65837. 3+5+5+(6+5+8+3+7) = 42\r\nSince all prime numbers obviously meet this criteria, Smith numbers are defined as composite numbers. Write a MATLAB function that will tell you if a given number is a Smith number.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 174px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 87px; transform-origin: 407px 87px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 375.5px 8px; transform-origin: 375.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eSmith numbers are numbers such that if you add up all of the digits in the number, that sum equals the sum of all of the digits in all of their factors. This term was coined by Albert Wilansky, when he noticed the defining property in the phone number of his brother-in-law Harold Smith: 493-7775.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 64px 8px; transform-origin: 64px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e4+9+3+7+7+7+5=42\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 251px 8px; transform-origin: 251px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe prime factors of 4937775 are 3, 5, 5, and 65837. 3+5+5+(6+5+8+3+7) = 42\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 380.5px 8px; transform-origin: 380.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eSince all prime numbers obviously meet this criteria, Smith numbers are defined as composite numbers. Write a MATLAB function that will tell you if a given number is a Smith number.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = smith(x)\r\n  y = x;\r\nend","test_suite":"%%\r\nassert(isequal(smith(4937775),1))\r\n%%\r\nassert(isequal(smith(1164),0))\r\n%%\r\nassert(isequal(smith(19683),1))\r\n%%\r\nassert(isequal(smith(11),0))  % Remember - Smith numbers are composite\r\n%%\r\nassert(isequal(smith(11^2),1))\r\n%%\r\nassert(isequal(smith(345741),1))\r\n%%\r\nassert(isequal(smith(19876),0))\r\n%%\r\nassert(isequal(smith(314159),0))\r\n%%\r\nassert(isequal(smith(612985),1))\r\n%%\r\nassert(isequal(smith(12379887),1))\r\n%%\r\nassert(isequal(smith(23456789),0))\r\n%%\r\nassert(isequal(smith(13),0))\r\n%%\r\nassert(isequal(smith(23),0))\r\n%%\r\ny=primes(randi(1e5));\r\nassert(isequal(smith(y(end)),0))\r\n","published":true,"deleted":false,"likes_count":2,"comments_count":1,"created_by":1615,"edited_by":223089,"edited_at":"2023-01-07T08:26:27.000Z","deleted_by":null,"deleted_at":null,"solvers_count":49,"test_suite_updated_at":"2023-01-07T08:26:27.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2017-06-01T19:16:25.000Z","updated_at":"2026-03-16T15:29:16.000Z","published_at":"2017-06-01T19:16:25.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eSmith numbers are numbers such that if you add up all of the digits in the number, that sum equals the sum of all of the digits in all of their factors. This term was coined by Albert Wilansky, when he noticed the defining property in the phone number of his brother-in-law Harold Smith: 493-7775.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e4+9+3+7+7+7+5=42\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe prime factors of 4937775 are 3, 5, 5, and 65837. 3+5+5+(6+5+8+3+7) = 42\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eSince all prime numbers obviously meet this criteria, Smith numbers are defined as composite numbers. Write a MATLAB function that will tell you if a given number is a Smith number.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":44384,"title":"Find the nearest prime number","description":"Happy 5th birthday, Cody! Since 5 is a prime number, let's have some fun looking for other prime numbers.\r\n\r\nGiven a positive integer x, find the nearest prime number. Keep in mind that the nearest prime may be less than x.\r\n\r\n*Examples*\r\n\r\n  nearestprime(5) = 5\r\n  nearestprime(36) = 37\r\n  nearestprime(200) = 199\r\n\r\nNOTE: You may ignore cases in which two prime numbers are equally close to x. (e.g., x=12 --\u003e 11 and 13 are both primes). ","description_html":"\u003cp\u003eHappy 5th birthday, Cody! Since 5 is a prime number, let's have some fun looking for other prime numbers.\u003c/p\u003e\u003cp\u003eGiven a positive integer x, find the nearest prime number. Keep in mind that the nearest prime may be less than x.\u003c/p\u003e\u003cp\u003e\u003cb\u003eExamples\u003c/b\u003e\u003c/p\u003e\u003cpre class=\"language-matlab\"\u003enearestprime(5) = 5\r\nnearestprime(36) = 37\r\nnearestprime(200) = 199\r\n\u003c/pre\u003e\u003cp\u003eNOTE: You may ignore cases in which two prime numbers are equally close to x. (e.g., x=12 --\u0026gt; 11 and 13 are both primes).\u003c/p\u003e","function_template":"function y = nearestprime(x)\r\n  y = x;\r\nend","test_suite":"%%\r\nx = 0;\r\ny_correct = 2;\r\nassert(isequal(nearestprime(x),y_correct))\r\n\r\n%%\r\nx = 5;\r\ny_correct = 5;\r\nassert(isequal(nearestprime(x),y_correct))\r\n\r\n%%\r\nx = 100;\r\ny_correct = 101;\r\nassert(isequal(nearestprime(x),y_correct))\r\n\r\n%%\r\nx = 500;\r\ny_correct = 499;\r\nassert(isequal(nearestprime(x),y_correct))\r\n\r\n%%\r\nx = 911;\r\ny_correct = 911;\r\nassert(isequal(nearestprime(x),y_correct))\r\n\r\n%%\r\nx = 2500;\r\ny_correct = 2503;\r\nassert(isequal(nearestprime(x),y_correct))\r\n\r\n%%\r\nx = 8000;\r\ny_correct = 7993;\r\nassert(isequal(nearestprime(x),y_correct))\r\n\r\n%%\r\nx = 100000;\r\ny_correct = 100003;\r\nassert(isequal(nearestprime(x),y_correct))\r\n\r\n%%\r\nx = 1300000;\r\ny_correct = 1299989;\r\nassert(isequal(nearestprime(x),y_correct))\r\n\r\n%%\r\nx = 179424710;\r\ny_correct = 179424719;\r\nassert(isequal(nearestprime(x),y_correct))\r\n","published":true,"deleted":false,"likes_count":9,"comments_count":1,"created_by":4793,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":664,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":34,"created_at":"2017-10-13T19:42:15.000Z","updated_at":"2026-04-07T15:16:58.000Z","published_at":"2017-10-16T01:45:10.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eHappy 5th birthday, Cody! Since 5 is a prime number, let's have some fun looking for other prime numbers.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eGiven a positive integer x, find the nearest prime number. Keep in mind that the nearest prime may be less than x.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eExamples\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[nearestprime(5) = 5\\nnearestprime(36) = 37\\nnearestprime(200) = 199]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eNOTE: You may ignore cases in which two prime numbers are equally close to x. (e.g., x=12 --\u0026gt; 11 and 13 are both primes).\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":3016,"title":"Twin Primes","description":"Twin primes are pairs of primes that are immediately next to each other (difference of two). The lesser of twin primes are 3, 5, 11, 17, 29, ... ( \u003chttp://oeis.org/A001359 ref.\u003e ). The greater of twin primes are 5, 7, 13, 19, 31, ... ( \u003chttp://oeis.org/A006512 ref.\u003e ). Therefore, the first five twin primes are [3,5] [5,7] [11,13] [17,19] [29,31].\r\n\r\nFor a given index range n, return the twin primes corresponding to that range as a two-row column array.","description_html":"\u003cp\u003eTwin primes are pairs of primes that are immediately next to each other (difference of two). The lesser of twin primes are 3, 5, 11, 17, 29, ... ( \u003ca href = \"http://oeis.org/A001359\"\u003eref.\u003c/a\u003e ). The greater of twin primes are 5, 7, 13, 19, 31, ... ( \u003ca href = \"http://oeis.org/A006512\"\u003eref.\u003c/a\u003e ). Therefore, the first five twin primes are [3,5] [5,7] [11,13] [17,19] [29,31].\u003c/p\u003e\u003cp\u003eFor a given index range n, return the twin primes corresponding to that range as a two-row column array.\u003c/p\u003e","function_template":"function [twins] = twin_primes(n)\r\n\r\ntwins = n;\r\n\r\nend","test_suite":"%%\r\nn = 1:5;\r\ntwins_corr = [3, 5, 11, 17, 29; 5, 7, 13, 19, 31];\r\nassert(isequal(twin_primes(n),twins_corr))\r\n\r\n%%\r\nn = 1:10;\r\ntwins_corr = [3, 5, 11, 17, 29, 41, 59, 71, 101, 107; 5, 7, 13, 19, 31, 43, 61, 73, 103, 109];\r\nassert(isequal(twin_primes(n),twins_corr))\r\n\r\n%%\r\nn = 1:25;\r\ntwins_corr = [3, 5, 11, 17, 29, 41, 59, 71, 101, 107, 137, 149, 179, 191, 197, 227, 239, 269, 281, 311, 347, 419, 431, 461, 521; 5, 7, 13, 19, 31, 43, 61, 73, 103, 109, 139, 151, 181, 193, 199, 229, 241, 271, 283, 313, 349, 421, 433, 463, 523];\r\nassert(isequal(twin_primes(n),twins_corr))\r\n\r\n%%\r\nn = 1:51;\r\ntwins_corr = [3, 5, 11, 17, 29, 41, 59, 71, 101, 107, 137, 149, 179, 191, 197, 227, 239, 269, 281, 311, 347, 419, 431, 461, 521, 569, 599, 617, 641, 659, 809, 821, 827, 857, 881, 1019, 1031, 1049, 1061, 1091, 1151, 1229, 1277, 1289, 1301, 1319, 1427, 1451, 1481, 1487, 1607; 5, 7, 13, 19, 31, 43, 61, 73, 103, 109, 139, 151, 181, 193, 199, 229, 241, 271, 283, 313, 349, 421, 433, 463, 523, 571, 601, 619, 643, 661, 811, 823, 829, 859, 883, 1021, 1033, 1051, 1063, 1093, 1153, 1231, 1279, 1291, 1303, 1321, 1429, 1453, 1483, 1489, 1609];\r\nassert(isequal(twin_primes(n),twins_corr))\r\n\r\n%%\r\nn = 10:29;\r\ntwins_corr = [107, 137, 149, 179, 191, 197, 227, 239, 269, 281, 311, 347, 419, 431, 461, 521, 569, 599, 617, 641; 109, 139, 151, 181, 193, 199, 229, 241, 271, 283, 313, 349, 421, 433, 463, 523, 571, 601, 619, 643];\r\nassert(isequal(twin_primes(n),twins_corr))\r\n\r\n%%\r\nn = 2:8;\r\ntwins_corr = [5, 11, 17, 29, 41, 59, 71; 7, 13, 19, 31, 43, 61, 73];\r\nassert(isequal(twin_primes(n),twins_corr))\r\n\r\n%%\r\nn = 35:42;\r\ntwins_corr = [881, 1019, 1031, 1049, 1061, 1091, 1151, 1229; 883, 1021, 1033, 1051, 1063, 1093, 1153, 1231];\r\nassert(isequal(twin_primes(n),twins_corr))\r\n\r\n%%\r\nn = 34:47;\r\ntwins_corr = [857, 881, 1019, 1031, 1049, 1061, 1091, 1151, 1229, 1277, 1289, 1301, 1319, 1427; 859, 883, 1021, 1033, 1051, 1063, 1093, 1153, 1231, 1279, 1291, 1303, 1321, 1429];\r\nassert(isequal(twin_primes(n),twins_corr))\r\n\r\n%%\r\nn = 9:-1:4;\r\ntwins_corr = [101, 71, 59, 41, 29, 17; 103, 73, 61, 43, 31, 19];\r\nassert(isequal(twin_primes(n),twins_corr))\r\n","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":26769,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":98,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":30,"created_at":"2015-02-14T03:03:50.000Z","updated_at":"2026-03-16T14:18:09.000Z","published_at":"2015-02-14T03:03:50.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eTwin primes are pairs of primes that are immediately next to each other (difference of two). The lesser of twin primes are 3, 5, 11, 17, 29, ... (\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://oeis.org/A001359\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eref.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e ). The greater of twin primes are 5, 7, 13, 19, 31, ... (\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://oeis.org/A006512\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eref.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e ). Therefore, the first five twin primes are [3,5] [5,7] [11,13] [17,19] [29,31].\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor a given index range n, return the twin primes corresponding to that range as a two-row column array.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":44385,"title":"Extra safe primes","description":"Did you know that the number 5 is the first safe prime? A safe prime is a prime number that can be expressed as 2p+1, where p is also a prime.\r\n\r\nTo celebrate Cody's Five-Year Anniversary, write a function to determine if a positive integer n is a safe prime in which the prime p (such that n=2p+1) is *also a safe prime*.\r\n\r\n*Examples*\r\n\r\n  isextrasafe(5) = false % because 5=2*2+1 and 2 is not a safe prime\r\n  isextrasafe(23) = true % because 23=2*11+1 and 11 is also a safe prime (11=2*5+1)\r\n","description_html":"\u003cp\u003eDid you know that the number 5 is the first safe prime? A safe prime is a prime number that can be expressed as 2p+1, where p is also a prime.\u003c/p\u003e\u003cp\u003eTo celebrate Cody's Five-Year Anniversary, write a function to determine if a positive integer n is a safe prime in which the prime p (such that n=2p+1) is \u003cb\u003ealso a safe prime\u003c/b\u003e.\u003c/p\u003e\u003cp\u003e\u003cb\u003eExamples\u003c/b\u003e\u003c/p\u003e\u003cpre class=\"language-matlab\"\u003eisextrasafe(5) = false % because 5=2*2+1 and 2 is not a safe prime\r\nisextrasafe(23) = true % because 23=2*11+1 and 11 is also a safe prime (11=2*5+1)\r\n\u003c/pre\u003e","function_template":"function tf = isextrasafe(x)\r\n    tf = false;\r\nend","test_suite":"%%\r\nx = 0;\r\nassert(isequal(isextrasafe(x),false))\r\n\r\n%%\r\nx = 5;\r\nassert(isequal(isextrasafe(x),false))\r\n\r\n%%\r\nx = 7;\r\nassert(isequal(isextrasafe(x),false))\r\n\r\n%%\r\nx = 11;\r\nassert(isequal(isextrasafe(x),true))\r\n\r\n%%\r\nx = 15;\r\nassert(isequal(isextrasafe(x),false))\r\n\r\n%%\r\nx = 23;\r\nassert(isequal(isextrasafe(x),true))\r\n\r\n%%\r\nx = 71;\r\nassert(isequal(isextrasafe(x),false))\r\n\r\n%%\r\nx = 719;\r\nassert(isequal(isextrasafe(x),true))\r\n\r\n%%\r\nx = 2039;\r\nassert(isequal(isextrasafe(x),true))\r\n\r\n%%\r\nx = 2040;\r\nassert(isequal(isextrasafe(x),false))\r\n\r\n%%\r\nx = 5807;\r\nassert(isequal(isextrasafe(x),true))","published":true,"deleted":false,"likes_count":13,"comments_count":4,"created_by":4793,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":757,"test_suite_updated_at":"2017-10-19T17:09:19.000Z","rescore_all_solutions":true,"group_id":34,"created_at":"2017-10-13T20:02:13.000Z","updated_at":"2026-04-10T14:37:08.000Z","published_at":"2017-10-16T01:45:10.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eDid you know that the number 5 is the first safe prime? A safe prime is a prime number that can be expressed as 2p+1, where p is also a prime.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eTo celebrate Cody's Five-Year Anniversary, write a function to determine if a positive integer n is a safe prime in which the prime p (such that n=2p+1) is\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ealso a safe prime\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eExamples\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[isextrasafe(5) = false % because 5=2*2+1 and 2 is not a safe prime\\nisextrasafe(23) = true % because 23=2*11+1 and 11 is also a safe prime (11=2*5+1)]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":1204,"title":"Prime Time","description":"All you need to do here is submit your solution a prime number of seconds after the top of the hour.  Any hour at all...  Easy, right?","description_html":"\u003cp\u003eAll you need to do here is submit your solution a prime number of seconds after the top of the hour.  Any hour at all...  Easy, right?\u003c/p\u003e","function_template":"function y = prime_time(x)\r\n  y = x;\r\nend","test_suite":"%%\r\nfiletext = fileread('prime_time.m')\r\nassert(isempty(strfind(filetext, 'fopen')))\r\n%%\r\nx=123\r\ny = prime_time(x)\r\na=clock\r\nb=floor(polyval(a(5:6),60))\r\nif isprime(b)\r\ny_correct= y\r\nelse \r\ny_correct = NaN; \r\nend\r\nassert(isequal(y,y_correct),sprintf('%s%g%s','Time is ',datestr(now,13),', or ',b,' seconds after the hour.'))","published":true,"deleted":false,"likes_count":2,"comments_count":0,"created_by":1615,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":63,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2013-01-14T18:34:19.000Z","updated_at":"2025-11-22T17:34:33.000Z","published_at":"2013-01-14T18:34:19.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eAll you need to do here is submit your solution a prime number of seconds after the top of the hour. Any hour at all... Easy, right?\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":896,"title":"Sophie Germain prime","description":"In number theory, a prime number p is a *Sophie Germain prime* if 2p + 1 is also prime. For example, 23 is a Sophie Germain prime because it is a prime and 2 × 23 + 1 = 47, and 47 is also a prime number. These numbers are named after French mathematician Marie-Sophie Germain.\r\n\r\nSee \u003chttp://en.wikipedia.org/wiki/Sophie_Germain_prime Sophie Germain prime\u003e article on Wikipedia.\r\n\r\n\r\nIn this Problem , the input is a number and you must return true or false if this number is a Sophie Germain prime.","description_html":"\u003cp\u003eIn number theory, a prime number p is a \u003cb\u003eSophie Germain prime\u003c/b\u003e if 2p + 1 is also prime. For example, 23 is a Sophie Germain prime because it is a prime and 2 × 23 + 1 = 47, and 47 is also a prime number. These numbers are named after French mathematician Marie-Sophie Germain.\u003c/p\u003e\u003cp\u003eSee \u003ca href=\"http://en.wikipedia.org/wiki/Sophie_Germain_prime\"\u003eSophie Germain prime\u003c/a\u003e article on Wikipedia.\u003c/p\u003e\u003cp\u003eIn this Problem , the input is a number and you must return true or false if this number is a Sophie Germain prime.\u003c/p\u003e","function_template":"function tf = your_fcn_name(x)\r\n  tf = true;\r\nend","test_suite":"%%\r\np = 233;\r\ny_correct = true;\r\nassert(isequal(your_fcn_name(p),y_correct))\r\n\r\n%%\r\np = 23;\r\ny_correct14 = true;\r\nassert(isequal(your_fcn_name(p),y_correct14))\r\n\r\n%%\r\np = 22;\r\ny_correct14 = false;\r\nassert(isequal(your_fcn_name(p),y_correct14))\r\n\r\n%% \r\np = 1 % p must also be a prime number !!\r\ny_correct1t = false;\r\nassert(isequal(your_fcn_name(p),y_correct1t))\r\n\r\n%% \r\np = 14 % p must also be a prime number !!\r\ncorrect1t = false;\r\nassert(isequal(your_fcn_name(p),correct1t))\r\n\r\n%% \r\np = 29 \r\ncorrect1tp = true;\r\nassert(isequal(your_fcn_name(p),correct1tp))\r\n","published":true,"deleted":false,"likes_count":6,"comments_count":2,"created_by":639,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":1066,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":44,"created_at":"2012-08-10T13:04:11.000Z","updated_at":"2026-04-09T08:16:22.000Z","published_at":"2012-08-10T13:04:11.000Z","restored_at":"2018-10-10T14:57:27.000Z","restored_by":null,"spam":false,"simulink":false,"admin_reviewed":true,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIn number theory, a prime number p is a\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eSophie Germain prime\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e if 2p + 1 is also prime. For example, 23 is a Sophie Germain prime because it is a prime and 2 × 23 + 1 = 47, and 47 is also a prime number. These numbers are named after French mathematician Marie-Sophie Germain.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eSee\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://en.wikipedia.org/wiki/Sophie_Germain_prime\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eSophie Germain prime\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e article on Wikipedia.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIn this Problem , the input is a number and you must return true or false if this number is a Sophie Germain prime.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":2337,"title":"Sum of big primes without primes","description":"Inspired by Project Euler n°10 (I am quite obviously a fan).\r\nWith problem n°250 by Doug, you can find some global methods to compute the sum of all the primes below the input n.\r\nFor example, the sum of the primes below 10 is 2 + 3 + 5 + 7 = 17.\r\nBut how to proceed (in time) with big number and WITHOUT the primes function ?\r\nHINTS: sum(primes(n)) is possible here but why miss the wonderfull Sieve of Eratosthenes ?\r\nhttp://en.wikipedia.org/wiki/Sieve_of_Eratosthenes","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 171px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 85.5px; transform-origin: 407px 85.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 183px 8px; transform-origin: 183px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eInspired by Project Euler n°10 (I am quite obviously a fan).\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 376px 8px; transform-origin: 376px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWith problem n°250 by Doug, you can find some global methods to compute the sum of all the primes below the input n.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 208px 8px; transform-origin: 208px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eFor example, the sum of the primes below 10 is 2 + 3 + 5 + 7 = 17.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 255.5px 8px; transform-origin: 255.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eBut how to proceed (in time) with big number and WITHOUT the primes function ?\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 288.5px 8px; transform-origin: 288.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eHINTS: sum(primes(n)) is possible here but why miss the wonderfull Sieve of Eratosthenes ?\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"/#null\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003ehttp://en.wikipedia.org/wiki/Sieve_of_Eratosthenes\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = big_euler10(n)\r\n  y = x;\r\nend","test_suite":"%%\r\nfiletext = fileread('big_euler10.m');\r\nillegal = contains(filetext, 'assignin') || contains(filetext, 'regexp') || ...\r\n          contains(filetext, 'primes'); \r\nassert(~illegal)\r\n\r\n%%\r\nx = 1;\r\ny_correct = 0;\r\nassert(isequal(big_euler10(x),y_correct))\r\n%%\r\nx = 10;\r\ny_correct = 17;\r\nassert(isequal(big_euler10(x),y_correct))\r\n%%\r\nx = 100;\r\ny_correct = 1060;\r\nassert(isequal(big_euler10(x),y_correct))\r\n%%\r\nx = 1000;\r\ny_correct = 76127;\r\nassert(isequal(big_euler10(x),y_correct))\r\n%%\r\nx = 10000;\r\ny_correct = 5736396;\r\nassert(isequal(big_euler10(x),y_correct))\r\n%%\r\nx = 100000;\r\ny_correct = 454396537;\r\nassert(isequal(big_euler10(x),y_correct))\r\n%%\r\nx = 1000000;\r\ny_correct = 37550402023;\r\nassert(isequal(big_euler10(x),y_correct))\r\n%%\r\nx = 1000000-100;\r\ny_correct = 37542402433;\r\nassert(isequal(big_euler10(x),y_correct))\r\n%%\r\nx = 2000000-1000;\r\ny_correct = 142781862782;\r\nassert(isequal(big_euler10(x),y_correct))\r\n%% Solution of Project Euler 10 with n=2000000\r\n\r\n","published":true,"deleted":false,"likes_count":3,"comments_count":3,"created_by":5390,"edited_by":223089,"edited_at":"2023-06-05T10:25:19.000Z","deleted_by":null,"deleted_at":null,"solvers_count":239,"test_suite_updated_at":"2023-06-05T10:25:19.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2014-05-27T21:25:58.000Z","updated_at":"2026-03-29T22:02:38.000Z","published_at":"2014-05-27T21:51:18.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eInspired by Project Euler n°10 (I am quite obviously a fan).\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWith problem n°250 by Doug, you can find some global methods to compute the sum of all the primes below the input n.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor example, the sum of the primes below 10 is 2 + 3 + 5 + 7 = 17.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eBut how to proceed (in time) with big number and WITHOUT the primes function ?\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eHINTS: sum(primes(n)) is possible here but why miss the wonderfull Sieve of Eratosthenes ?\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"\\\"\u003e\u003cw:r\u003e\u003cw:t\u003ehttp://en.wikipedia.org/wiki/Sieve_of_Eratosthenes\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":44403,"title":"Goldbach's marginal conjecture - Write integer as sum of three primes","description":"Goldbach's strong conjecture states that every even integer greater than 2 can be expressed as the sum of two primes. For example: 4 = 2+2, 6 = 3+3, 8 = 3+5, 10 = 3+7 = 5+5, 12 = 5+7 etc.\r\n\r\nAs a corrollary, Goldbach's weak conjecture states that every odd integer greater than 7 can be expressed as the sum of three odd primes. For example: 9 = 3+3+3, 11 = 3+3+5, 13 = 3+3+7 = 3+5+5, 15 = 3+5+7 = 5+5+5 etc.\r\n\r\nA third conjecture was written by Goldbach in the margin of a letter, and (in its modern version) states that \r\n\r\n\" _Every integer greater than 5 can be expressed as the sum of three primes._ \"\r\n\r\nExamples:\r\n\r\n*  6 = 2 + 2 + 2\r\n*  7 = 2 + 2 + 3\r\n*  8 = 2 + 3 + 3 \r\n*  9 = 2 + 2 + 5 = 3 + 3 + 3 \r\n* 10 = 2 + 3 + 5\r\n* 11 = 2 + 2 + 7 = 3 + 3 + 5\r\n* 12 = 2 + 3 + 7 = 2 + 5 + 5\r\n* 13 = 3 + 3 + 7 = 3 + 5 + 5\r\n* 14 = 2 + 5 + 7\r\n* 15 = 2 + 2 + 11 = 3 + 5 + 7 = 5 + 5 + 5\r\n\r\nYour task is to write a function which takes a positive integer _n_ as input, and which returns a 1-by-3 vector _y_, which contains three numbers that are primes and whose sum equals _n_. If there exist multiple solutions for _y_, then any one of those solutions will suffice. However, _y_ must be in sorted order. You can assume that _n_ will be an integer greater than 5.\r\n\r\n","description_html":"\u003cp\u003eGoldbach's strong conjecture states that every even integer greater than 2 can be expressed as the sum of two primes. For example: 4 = 2+2, 6 = 3+3, 8 = 3+5, 10 = 3+7 = 5+5, 12 = 5+7 etc.\u003c/p\u003e\u003cp\u003eAs a corrollary, Goldbach's weak conjecture states that every odd integer greater than 7 can be expressed as the sum of three odd primes. For example: 9 = 3+3+3, 11 = 3+3+5, 13 = 3+3+7 = 3+5+5, 15 = 3+5+7 = 5+5+5 etc.\u003c/p\u003e\u003cp\u003eA third conjecture was written by Goldbach in the margin of a letter, and (in its modern version) states that\u003c/p\u003e\u003cp\u003e\" \u003ci\u003eEvery integer greater than 5 can be expressed as the sum of three primes.\u003c/i\u003e \"\u003c/p\u003e\u003cp\u003eExamples:\u003c/p\u003e\u003cul\u003e\u003cli\u003e6 = 2 + 2 + 2\u003c/li\u003e\u003cli\u003e7 = 2 + 2 + 3\u003c/li\u003e\u003cli\u003e8 = 2 + 3 + 3\u003c/li\u003e\u003cli\u003e9 = 2 + 2 + 5 = 3 + 3 + 3\u003c/li\u003e\u003cli\u003e10 = 2 + 3 + 5\u003c/li\u003e\u003cli\u003e11 = 2 + 2 + 7 = 3 + 3 + 5\u003c/li\u003e\u003cli\u003e12 = 2 + 3 + 7 = 2 + 5 + 5\u003c/li\u003e\u003cli\u003e13 = 3 + 3 + 7 = 3 + 5 + 5\u003c/li\u003e\u003cli\u003e14 = 2 + 5 + 7\u003c/li\u003e\u003cli\u003e15 = 2 + 2 + 11 = 3 + 5 + 7 = 5 + 5 + 5\u003c/li\u003e\u003c/ul\u003e\u003cp\u003eYour task is to write a function which takes a positive integer \u003ci\u003en\u003c/i\u003e as input, and which returns a 1-by-3 vector \u003ci\u003ey\u003c/i\u003e, which contains three numbers that are primes and whose sum equals \u003ci\u003en\u003c/i\u003e. If there exist multiple solutions for \u003ci\u003ey\u003c/i\u003e, then any one of those solutions will suffice. However, \u003ci\u003ey\u003c/i\u003e must be in sorted order. You can assume that \u003ci\u003en\u003c/i\u003e will be an integer greater than 5.\u003c/p\u003e","function_template":"function y = goldbach3(n)\r\n  y = [n,n,n];\r\nend","test_suite":"%%\r\nn = 6;\r\ny = goldbach3(n);\r\ny_correct = [2,2,2];\r\nassert(isequal(y,y_correct))\r\n\r\n%%\r\nn = 7;\r\ny = goldbach3(n);\r\ny_correct = [2,2,3];\r\nassert(isequal(y,y_correct))\r\n\r\n%%\r\nn = 8;\r\ny = goldbach3(n);\r\ny_correct = [2,3,3];\r\nassert(isequal(y,y_correct))\r\n\r\n%%\r\nn = 9;\r\ny = goldbach3(n);\r\ny_correct1 = [2,2,5];\r\ny_correct2 = [3,3,3];\r\nassert(isequal(y,y_correct1)|isequal(y,y_correct2))\r\n\r\n%%\r\nn = 10;\r\ny = goldbach3(n);\r\ny_correct = [2,3,5];\r\nassert(isequal(y,y_correct))\r\n\r\n%%\r\nn = 11;\r\ny = goldbach3(n);\r\ny_correct1 = [2,2,7];\r\ny_correct2 = [3,3,5];\r\nassert(isequal(y,y_correct1)|isequal(y,y_correct2))\r\n\r\n%%\r\nn = 12;\r\ny = goldbach3(n);\r\ny_correct1 = [2,3,7];\r\ny_correct2 = [2,5,5];\r\nassert(isequal(y,y_correct1)|isequal(y,y_correct2))\r\n\r\n%%\r\nn = 13;\r\ny = goldbach3(n);\r\ny_correct1 = [3,3,7];\r\ny_correct2 = [3,5,5];\r\nassert(isequal(y,y_correct1)|isequal(y,y_correct2))\r\n\r\n%%\r\nn = 14;\r\ny = goldbach3(n);\r\ny_correct = [2,5,7];\r\nassert(isequal(y,y_correct))\r\n\r\n%%\r\nn = 15;\r\ny = goldbach3(n);\r\ny_correct1 = [2,2,11];\r\ny_correct2 = [3,5,7];\r\ny_correct3 = [5,5,5];\r\nassert(isequal(y,y_correct1)|isequal(y,y_correct2)|isequal(y,y_correct3))\r\n\r\n%%\r\nn = 101;\r\ny = goldbach3(n);\r\nassert(isequal(y,sort(y)))\r\nassert(all(isprime(y)))\r\nassert(sum(y)==n)\r\n\r\n%%\r\nn = 102;\r\ny = goldbach3(n);\r\nassert(isequal(y,sort(y)))\r\nassert(all(isprime(y)))\r\nassert(sum(y)==n)\r\n\r\n%% \r\nfor n = 250:300\r\n    y = goldbach3(n);\r\n    assert(isequal(y,sort(y)));\r\n    assert(all(isprime(y)));\r\n    assert(sum(y)==n);\r\nend\r\n\r\n%%\r\nn = randi(2000)+5; % generate a random integer greater than 5 and smaller than 2006\r\ny = goldbach3(n);\r\nassert(isequal(y,sort(y)))\r\nassert(all(isprime(y)))\r\nassert(sum(y)==n)\r\n\r\n%% \r\nvalid = zeros(1,50);\r\nfor k = 1:50\r\n    n = randi(1000)+5; % generate a random integer greater than 5 and smaller than 1006\r\n    yk = goldbach3(n);\r\n    valid(k) = (isequal(yk,sort(yk)) \u0026 all(isprime(yk)) \u0026 sum(yk)==n);\r\nend\r\nassert(all(valid));\r\n","published":true,"deleted":false,"likes_count":3,"comments_count":2,"created_by":108199,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":75,"test_suite_updated_at":"2017-11-18T23:12:48.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2017-11-14T00:05:38.000Z","updated_at":"2026-03-16T15:38:02.000Z","published_at":"2017-11-14T01:21:48.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eGoldbach's strong conjecture states that every even integer greater than 2 can be expressed as the sum of two primes. For example: 4 = 2+2, 6 = 3+3, 8 = 3+5, 10 = 3+7 = 5+5, 12 = 5+7 etc.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eAs a corrollary, Goldbach's weak conjecture states that every odd integer greater than 7 can be expressed as the sum of three odd primes. For example: 9 = 3+3+3, 11 = 3+3+5, 13 = 3+3+7 = 3+5+5, 15 = 3+5+7 = 5+5+5 etc.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eA third conjecture was written by Goldbach in the margin of a letter, and (in its modern version) states that\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\"\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eEvery integer greater than 5 can be expressed as the sum of three primes.\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \\\"\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eExamples:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e6 = 2 + 2 + 2\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e7 = 2 + 2 + 3\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e8 = 2 + 3 + 3\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e9 = 2 + 2 + 5 = 3 + 3 + 3\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e10 = 2 + 3 + 5\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e11 = 2 + 2 + 7 = 3 + 3 + 5\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e12 = 2 + 3 + 7 = 2 + 5 + 5\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e13 = 3 + 3 + 7 = 3 + 5 + 5\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e14 = 2 + 5 + 7\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e15 = 2 + 2 + 11 = 3 + 5 + 7 = 5 + 5 + 5\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eYour task is to write a function which takes a positive integer\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e as input, and which returns a 1-by-3 vector\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ey\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, which contains three numbers that are primes and whose sum equals\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e. If there exist multiple solutions for\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ey\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, then any one of those solutions will suffice. However,\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ey\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e must be in sorted order. You can assume that\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e will be an integer greater than 5.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":2910,"title":"Mersenne Primes vs. All Primes","description":"A Mersenne prime (M) is a prime number of the form M = 2^p - 1, where p is another prime number. \u003chttps://www.mathworks.com/matlabcentral/cody/problems/525-mersenne-primes Problem 525\u003e asks the user to determine if a number is a Mersenne prime. In this problem, you are tasked with returning the number of primes numbers below the input number, n, that are Mersenne primes and the fraction of all primes below that input number that the Mersenne primes represent.\r\n\r\nFor example, for n = 100, there are 25 primes numbers: 2, 3, 5, 7, ..., 89, 97. As far as Mersenne primes go, there are only three that are less than 100: 2^2 - 1 = 3, 2^3 - 1 = 7, and 2^5 - 1 = 31. The corresponding fraction would be 3/25.","description_html":"\u003cp\u003eA Mersenne prime (M) is a prime number of the form M = 2^p - 1, where p is another prime number. \u003ca href = \"https://www.mathworks.com/matlabcentral/cody/problems/525-mersenne-primes\"\u003eProblem 525\u003c/a\u003e asks the user to determine if a number is a Mersenne prime. In this problem, you are tasked with returning the number of primes numbers below the input number, n, that are Mersenne primes and the fraction of all primes below that input number that the Mersenne primes represent.\u003c/p\u003e\u003cp\u003eFor example, for n = 100, there are 25 primes numbers: 2, 3, 5, 7, ..., 89, 97. As far as Mersenne primes go, there are only three that are less than 100: 2^2 - 1 = 3, 2^3 - 1 = 7, and 2^5 - 1 = 31. The corresponding fraction would be 3/25.\u003c/p\u003e","function_template":"function [y,f] = Mersenne_prime_comp(n)\r\n y = 1;\r\n f = 0;\r\nend","test_suite":"%%\r\nn = 1e2;\r\ny_correct = 3;\r\nf_correct = 3/25;\r\n[y,f] = Mersenne_prime_comp(n);\r\nassert(isequal(y,y_correct))\r\nassert(isequal(f,f_correct))\r\n\r\n%%\r\nn = 1e3;\r\ny_correct = 4;\r\nf_correct = 0.023809523809524;\r\n[y,f] = Mersenne_prime_comp(n);\r\nassert(isequal(y,y_correct))\r\nassert(abs(f-f_correct)\u003c(10*eps))\r\n\r\n%%\r\nn = 1e4;\r\ny_correct = 5;\r\nf_correct = 0.004068348250610;\r\n[y,f] = Mersenne_prime_comp(n);\r\nassert(isequal(y,y_correct))\r\nassert(abs(f-f_correct)\u003c(10*eps))\r\n\r\n%%\r\nn = 1e5;\r\ny_correct = 5;\r\nf_correct = 5.212677231025855e-04;\r\n[y,f] = Mersenne_prime_comp(n);\r\nassert(isequal(y,y_correct))\r\nassert(abs(f-f_correct)\u003c(10*eps))\r\n\r\n%%\r\nn = 1e6;\r\ny_correct = 7;\r\nf_correct = 8.917424647761727e-05;\r\n[y,f] = Mersenne_prime_comp(n);\r\nassert(isequal(y,y_correct))\r\nassert(abs(f-f_correct)\u003c(10*eps))\r\n","published":true,"deleted":false,"likes_count":8,"comments_count":2,"created_by":26769,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":846,"test_suite_updated_at":"2015-02-01T04:14:08.000Z","rescore_all_solutions":false,"group_id":29,"created_at":"2015-02-01T03:56:07.000Z","updated_at":"2026-04-01T10:02:01.000Z","published_at":"2015-02-01T04:14:08.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eA Mersenne prime (M) is a prime number of the form M = 2^p - 1, where p is another prime number.\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/525-mersenne-primes\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eProblem 525\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e asks the user to determine if a number is a Mersenne prime. In this problem, you are tasked with returning the number of primes numbers below the input number, n, that are Mersenne primes and the fraction of all primes below that input number that the Mersenne primes represent.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor example, for n = 100, there are 25 primes numbers: 2, 3, 5, 7, ..., 89, 97. As far as Mersenne primes go, there are only three that are less than 100: 2^2 - 1 = 3, 2^3 - 1 = 7, and 2^5 - 1 = 31. The corresponding fraction would be 3/25.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":2595,"title":"Polite numbers. Politeness.","description":"A polite number is an integer that sums of two or more consecutive positive integers.\r\nPoliteness of a positive integer is a number of nontrivial ways to write n as a sum of two or more consecutive positive integers.\r\n\r\nFor example _9 = 4+5 = 2+3+4_  and politeness of 9 is 2.\r\n\r\nGiven _N_ return politeness of _N_.\r\n\r\nSee also \u003chttp://www.mathworks.co.uk/matlabcentral/cody/problems/2593 2593\u003e","description_html":"\u003cp\u003eA polite number is an integer that sums of two or more consecutive positive integers.\r\nPoliteness of a positive integer is a number of nontrivial ways to write n as a sum of two or more consecutive positive integers.\u003c/p\u003e\u003cp\u003eFor example \u003ci\u003e9 = 4+5 = 2+3+4\u003c/i\u003e  and politeness of 9 is 2.\u003c/p\u003e\u003cp\u003eGiven \u003ci\u003eN\u003c/i\u003e return politeness of \u003ci\u003eN\u003c/i\u003e.\u003c/p\u003e\u003cp\u003eSee also \u003ca href = \"http://www.mathworks.co.uk/matlabcentral/cody/problems/2593\"\u003e2593\u003c/a\u003e\u003c/p\u003e","function_template":"function P = politeness(N)\r\n  P=N;\r\nend","test_suite":"%%\r\nx = 1;\r\ny_correct = 0;\r\nassert(isequal(politeness(x),y_correct))\r\n%%\r\nx = 2;\r\ny_correct = 0;\r\nassert(isequal(politeness(x),y_correct))\r\n%%\r\nx = 3;\r\ny_correct = 1;\r\nassert(isequal(politeness(x),y_correct))\r\n%%\r\nx = 7;\r\ny_correct = 1;\r\nassert(isequal(politeness(x),y_correct))\r\n%%\r\nx = 9;\r\ny_correct = 2;\r\nassert(isequal(politeness(x),y_correct))\r\n%%\r\nx = 15;\r\ny_correct = 3;\r\nassert(isequal(politeness(x),y_correct))\r\n%%\r\nx = 18;\r\ny_correct = 2;\r\nassert(isequal(politeness(x),y_correct))\r\n%%\r\nx = 21;\r\ny_correct = 3;\r\nassert(isequal(politeness(x),y_correct))\r\n%%\r\nx = 1024;\r\ny_correct = 0;\r\nassert(isequal(politeness(x),y_correct))\r\n%%\r\nx = 1025;\r\ny_correct = 5;\r\nassert(isequal(politeness(x),y_correct))\r\n%%\r\nx = 25215;\r\ny_correct = 11;\r\nassert(isequal(politeness(x),y_correct))\r\n%%\r\nx = 62;\r\ny_correct = 1;\r\nassert(isequal(politeness(x),y_correct))\r\n%%\r\nx = 63;\r\ny_correct = 5;\r\nassert(isequal(politeness(x),y_correct))\r\n%%\r\nx = 65;\r\ny_correct = 3;\r\nassert(isequal(politeness(x),y_correct))\r\n%%\r\n% anti-lookup \u0026 clue\r\nnums=primes(200);\r\npattern=[1 nums([false ~randi([0 25],1,45)])];\r\nx=prod(pattern)*2^randi([0 5]);\r\ny_correct=2^numel(pattern)/2-1;\r\nassert(isequal(politeness(x),y_correct))\r\n%%\r\nfor k=randi(2e4,1,20)\r\n  assert(isequal(politeness(k*(k-1))+1,(politeness(k)+1)*(politeness(k-1)+1)))\r\nend","published":true,"deleted":false,"likes_count":8,"comments_count":6,"created_by":14358,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":186,"test_suite_updated_at":"2014-09-17T15:38:21.000Z","rescore_all_solutions":false,"group_id":8,"created_at":"2014-09-16T22:47:12.000Z","updated_at":"2026-02-16T10:30:04.000Z","published_at":"2014-09-17T10:56:54.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eA polite number is an integer that sums of two or more consecutive positive integers. Politeness of a positive integer is a number of nontrivial ways to write n as a sum of two or more consecutive positive integers.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor example\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e9 = 4+5 = 2+3+4\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e and politeness of 9 is 2.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eGiven\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eN\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e return politeness of\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eN\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eSee also\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://www.mathworks.co.uk/matlabcentral/cody/problems/2593\\\"\u003e\u003cw:r\u003e\u003cw:t\u003e2593\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":1672,"title":"Leftovers?  Again?!","description":"I am thinking of a positive number X.  To determine what number I am thinking of, I will give you two 1xN vectors.  The first vector (V1) is several numbers, none of which will share a factor.  The second vector (V2) is the remainder of X when divided by each of the numbers in V1. Calculate what the lowest possible value of X can be given these criteria.  For example:\r\n\r\nV1 = [2 3] ; V2 = [1 2]\r\n\r\nwould give an X value of 5.  There are an infinite number of other values of X that would satisfy V1 and V2, but I want the lowest one.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: normal; text-decoration: none; white-space: normal; \"\u003e\u003cdiv style=\"display: block; min-width: 0px; padding-top: 0px; transform-origin: 332px 82.5px; vertical-align: baseline; perspective-origin: 332px 82.5px; \"\u003e\u003cdiv style=\"font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-bottom: 9px; margin-left: 4px; margin-right: 10px; margin-top: 2px; text-align: left; transform-origin: 309px 42px; white-space: pre-wrap; perspective-origin: 309px 42px; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"display: inline; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; transform-origin: 0px 0px; perspective-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eI am thinking of a positive number X. To determine what number I am thinking of, I will give you two 1xN vectors. The first vector (V1) is several numbers, none of which will share a factor. The second vector (V2) is the remainder of X when divided by each of the numbers in V1. Calculate what the lowest possible value of X can be given these criteria. For example:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-bottom: 9px; margin-left: 4px; margin-right: 10px; margin-top: 2px; text-align: left; transform-origin: 309px 10.5px; white-space: pre-wrap; perspective-origin: 309px 10.5px; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"display: inline; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; transform-origin: 0px 0px; perspective-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eV1 = [2 3] ; V2 = [1 2]\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-bottom: 9px; margin-left: 4px; margin-right: 10px; margin-top: 2px; text-align: left; transform-origin: 309px 21px; white-space: pre-wrap; perspective-origin: 309px 21px; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"display: inline; margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; transform-origin: 0px 0px; perspective-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003ewould give an X value of 5. There are an infinite number of other values of X that would satisfy V1 and V2, but I want the lowest one.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function X = leftovers(n,a)\r\n X = pi;\r\nend","test_suite":"%!cp leftovers.m safe\r\n%!rm *.*\r\n%!mv safe leftovers.m\r\n%!rm @*\r\n\r\n% Clean user's function from some known jailbreaking mechanisms\r\nfid = fopen('leftovers.m');\r\nst = regexprep(char(fread(fid)'), '!', 'error(''No shell commands!''); %');\r\nst = regexprep(st, 'feval', 'error(''No fancy functions!''); %');\r\nst = regexprep(st, 'str2func', 'error(''No fancy functions!''); %');\r\nst = regexprep(st, 'regex', 'error(''No fancy functions!''); %');\r\nst = regexprep(st, 'system', 'error(''No shell commands!''); %');\r\nst = regexprep(st, 'dos', 'error(''No shell commands!''); %');\r\nst = regexprep(st, 'unix', 'error(''No shell commands!''); %');\r\nst = regexprep(st, 'perl', 'error(''No external languages commands!''); %');\r\nst = regexprep(st, 'java', 'error(''No external languages commands!''); %');\r\nst = regexprep(st, 'assert', 'error(''No overwriting!''); %');\r\nfclose(fid)\r\n\r\nfid = fopen('leftovers.m' , 'w');\r\nfwrite(fid,st);\r\nfclose(fid)\r\n%%\r\nV1 = [2 3] ; V2 = [1 2];; y_correct = 5; assert(isequal(leftovers(V1,V2),y_correct))\r\n%%\r\nV1=[3 5 7] ; V2=[1 2 3]; y_correct = 52; assert(isequal(leftovers(V1,V2),y_correct))\r\n%%\r\nV1=[3 4 5] ; V2=[2 3 1]; y_correct = 11; assert(isequal(leftovers(V1,V2),y_correct))\r\n%%\r\nV1=[4 9 25] ; V2=[3 2 7]; y_correct = 407; assert(isequal(leftovers(V1,V2),y_correct))\r\n%%\r\nV1=[9 10 77] ; V2=[1 2 69]; y_correct = 6922; assert(isequal(leftovers(V1,V2),y_correct))\r\n%%\r\na=primes(30);\r\nb=ceil(8*rand()+2)\r\nV1=a(1:b);V2=1:b;\r\ny_correct=[23 53 1523 29243 299513 4383593 188677703 5765999453];\r\nassert(isequal(leftovers(V1,V2),y_correct(b-2)))\r\n%%\r\nV1=[leftovers([6 35],[3 9]) leftovers([3 5 7],[1 3 1])];\r\nassert(isequal(leftovers(V1,V1-8),379))\r\n%%\r\n% Discourage the for x=1:inf loops\r\nV1=[74 93 145 161 209 221]; V2=[66 85 137 153 201 213];\r\ny_correct=7420738134802;\r\nassert(isequal(leftovers(V1,V2),y_correct))\r\n%%\r\n% Discourage the for x=1:inf loops\r\nV1=[17 82 111 155 203 247 253] ; V2=[11 50 68 95 124 150 154];\r\ny_correct=59652745309190;\r\nassert(isequal(leftovers(V1,V2),y_correct))\r\n","published":true,"deleted":false,"likes_count":4,"comments_count":10,"created_by":1615,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":32,"test_suite_updated_at":"2020-09-29T13:24:49.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2013-06-21T18:45:36.000Z","updated_at":"2025-11-22T17:35:45.000Z","published_at":"2013-06-21T18:45:36.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eI am thinking of a positive number X. To determine what number I am thinking of, I will give you two 1xN vectors. The first vector (V1) is several numbers, none of which will share a factor. The second vector (V2) is the remainder of X when divided by each of the numbers in V1. Calculate what the lowest possible value of X can be given these criteria. For example:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eV1 = [2 3] ; V2 = [1 2]\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewould give an X value of 5. There are an infinite number of other values of X that would satisfy V1 and V2, but I want the lowest one.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":60,"title":"The Goldbach Conjecture","description":"The Goldbach conjecture asserts that every even integer greater than 2 can be expressed as the sum of two primes.\r\nGiven the even integer n, return primes p1 and p2 that satisfy the condition n = p1 + p2. Note that the primes are not always unique. The test is not sensitive to order or uniqueness. You just need to meet the appropriate conditions.\r\nExample:\r\n Input  n = 286\r\n Output (any of the following is acceptable) \r\n        [  3 283]\r\n        [283   3]\r\n        [  5 281]\r\n        [107 179]\r\n        [137 149]","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 255.033px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 127.517px; transform-origin: 407px 127.517px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 12.5px 8px; transform-origin: 12.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"/#null\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eGoldbach conjecture\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 287px 8px; transform-origin: 287px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e asserts that every even integer greater than 2 can be expressed as the sum of two primes.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 368.5px 8px; transform-origin: 368.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eGiven the even integer n, return primes p1 and p2 that satisfy the condition n = p1 + p2. Note that the primes are not always unique. The test is not sensitive to order or uniqueness. You just need to meet the appropriate conditions.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 28.5px 8px; transform-origin: 28.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eExample:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgb(247, 247, 247); block-size: 143.033px; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-end-end-radius: 4px; border-end-start-radius: 4px; border-start-end-radius: 4px; border-start-start-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; margin-block-end: 10px; margin-block-start: 10px; margin-bottom: 10px; margin-inline-end: 3px; margin-inline-start: 3px; margin-left: 3px; margin-right: 3px; margin-top: 10px; perspective-origin: 404px 71.5167px; transform-origin: 404px 71.5167px; margin-left: 3px; margin-top: 10px; margin-bottom: 10px; margin-right: 3px; \"\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 60px 8.5px; tab-size: 4; transform-origin: 60px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; perspective-origin: 32px 8.5px; transform-origin: 32px 8.5px; \"\u003e Input  \u003c/span\u003e\u003cspan style=\"border-block-end-color: rgb(170, 4, 249); border-block-start-color: rgb(170, 4, 249); border-bottom-color: rgb(170, 4, 249); border-inline-end-color: rgb(170, 4, 249); border-inline-start-color: rgb(170, 4, 249); border-left-color: rgb(170, 4, 249); border-right-color: rgb(170, 4, 249); border-top-color: rgb(170, 4, 249); caret-color: rgb(170, 4, 249); color: rgb(170, 4, 249); column-rule-color: rgb(170, 4, 249); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(170, 4, 249); perspective-origin: 28px 8.5px; text-decoration-color: rgb(170, 4, 249); text-emphasis-color: rgb(170, 4, 249); transform-origin: 28px 8.5px; \"\u003en = 286\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 180px 8.5px; tab-size: 4; transform-origin: 180px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e Output (any of the following is acceptable) \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 68px 8.5px; tab-size: 4; transform-origin: 68px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e        [  3 283]\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 68px 8.5px; tab-size: 4; transform-origin: 68px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e        [283   3]\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 68px 8.5px; tab-size: 4; transform-origin: 68px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e        [  5 281]\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 68px 8.5px; tab-size: 4; transform-origin: 68px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e        [107 179]\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 68px 8.5px; tab-size: 4; transform-origin: 68px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e        [137 149]\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function [p1,p2] = goldbach(n)\r\n  p1 = n;\r\n  p2 = n;\r\nend","test_suite":"%%\r\nfiletext = fileread('goldbach.m');\r\nillegal = contains(filetext, 'assignin') || contains(filetext, 'regexp'); \r\nassert(~illegal)\r\n\r\n%%\r\nnList = 28:6:76;\r\nfor i = 1:length(nList)\r\n  n = nList(i);\r\n  [p1,p2] = goldbach(n)\r\n  assert(isprime(p1) \u0026\u0026 isprime(p2) \u0026\u0026 (p1+p2==n));\r\nend\r\n\r\n%%\r\nnList = [18 20 22 100 102 114 1000 2000 36 3600];\r\nfor i = 1:length(nList)\r\n  n = nList(i);\r\n  [p1,p2] = goldbach(n)\r\n  assert(isprime(p1) \u0026\u0026 isprime(p2) \u0026\u0026 (p1+p2==n));\r\nend","published":true,"deleted":false,"likes_count":60,"comments_count":17,"created_by":1,"edited_by":223089,"edited_at":"2023-06-05T15:48:22.000Z","deleted_by":null,"deleted_at":null,"solvers_count":5973,"test_suite_updated_at":"2023-06-05T15:48:22.000Z","rescore_all_solutions":false,"group_id":2,"created_at":"2012-01-18T01:00:25.000Z","updated_at":"2026-04-09T08:17:56.000Z","published_at":"2012-01-18T01:00:25.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eGoldbach conjecture\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e asserts that every even integer greater than 2 can be expressed as the sum of two primes.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eGiven the even integer n, return primes p1 and p2 that satisfy the condition n = p1 + p2. Note that the primes are not always unique. The test is not sensitive to order or uniqueness. You just need to meet the appropriate conditions.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eExample:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[ Input  n = 286\\n Output (any of the following is acceptable) \\n        [  3 283]\\n        [283   3]\\n        [  5 281]\\n        [107 179]\\n        [137 149]]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":64,"title":"The Goldbach Conjecture, Part 2","description":"The \u003chttp://en.wikipedia.org/wiki/Goldbach's_conjecture Goldbach\nconjecture\u003e asserts that every even integer greater than 2 can be\nexpressed as the sum of two primes.\n \nGiven the even integer n, return c, the number of different ways two primes can be added to result in n. Only count a pair once; the order is unimportant.\n\nExample:\n\n Input  n = 10\n Output c is 2\n\nbecause of the prime pairs [3 7] and [5 5].\n\n Input  n = 50\n Output c is 4\n\nbecause of [3 47], [7 43], [13 37], and [19 31].\n","description_html":"\u003cp\u003eThe \u003ca href=\"http://en.wikipedia.org/wiki/Goldbach's_conjecture\"\u003eGoldbach\nconjecture\u003c/a\u003e asserts that every even integer greater than 2 can be\nexpressed as the sum of two primes.\u003c/p\u003e\u003cp\u003eGiven the even integer n, return c, the number of different ways two primes can be added to result in n. Only count a pair once; the order is unimportant.\u003c/p\u003e\u003cp\u003eExample:\u003c/p\u003e\u003cpre\u003e Input  n = 10\n Output c is 2\u003c/pre\u003e\u003cp\u003ebecause of the prime pairs [3 7] and [5 5].\u003c/p\u003e\u003cpre\u003e Input  n = 50\n Output c is 4\u003c/pre\u003e\u003cp\u003ebecause of [3 47], [7 43], [13 37], and [19 31].\u003c/p\u003e","function_template":"function c = goldbach2(n)\n  c = 1;\nend","test_suite":"%%\nn = 6;\nc_correct = 1;\nassert(isequal(goldbach2(n),c_correct))\n\n%%\nn = 10;\nc_correct = 2;\nassert(isequal(goldbach2(n),c_correct))\n\n%%\nn = 50;\nc_correct = 4;\nassert(isequal(goldbach2(n),c_correct))\n\n%%\nn = 480;\nc_correct = 29;\nassert(isequal(goldbach2(n),c_correct))","published":true,"deleted":false,"likes_count":18,"comments_count":4,"created_by":1,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":2413,"test_suite_updated_at":"2012-01-18T01:00:26.000Z","rescore_all_solutions":false,"group_id":2,"created_at":"2012-01-18T01:00:26.000Z","updated_at":"2026-02-04T16:46:15.000Z","published_at":"2012-01-18T01:00:26.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://en.wikipedia.org/wiki/Goldbach's_conjecture\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eGoldbach conjecture\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e asserts that every even integer greater than 2 can be expressed as the sum of two primes.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eGiven the even integer n, return c, the number of different ways two primes can be added to result in n. Only count a pair once; the order is unimportant.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eExample:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[ Input  n = 10\\n Output c is 2]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ebecause of the prime pairs [3 7] and [5 5].\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[ Input  n = 50\\n Output c is 4]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ebecause of [3 47], [7 43], [13 37], and [19 31].\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":83,"title":"Prime factor digits","description":"Consider the following number system. Calculate the prime factorization for each number n, then represent the prime factors in a vector like so:\r\n       13 11 7 5 3 2\r\n      ---------------\r\n    2:             1\r\n    3:           1 0\r\n    4:             2\r\n    5:         1 0 0\r\n    6:           1 1\r\n   12:           1 2\r\n   14:       1 0 0 1\r\n   18:           2 1\r\n   26:   1 0 0 0 0 1\r\n   60:         1 1 2\r\nEach \"place\" in the number system represents a prime number. Given n, return the vector p.\r\nAs shown above, if n = 26, then p = [1 0 0 0 0 1].\r\nThe input n is always an integer greater than 1. Suppress any leading zeros. The length of the vector is determined by the largest prime factor.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 409.2px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 204.6px; transform-origin: 407px 204.6px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 368.5px 8px; transform-origin: 368.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eConsider the following number system. Calculate the prime factorization for each number n, then represent the prime factors in a vector like so:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgb(247, 247, 247); block-size: 245.2px; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-end-end-radius: 4px; border-end-start-radius: 4px; border-start-end-radius: 4px; border-start-start-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; margin-block-end: 10px; margin-block-start: 10px; margin-bottom: 10px; margin-inline-end: 3px; margin-inline-start: 3px; margin-left: 3px; margin-right: 3px; margin-top: 10px; perspective-origin: 404px 122.6px; transform-origin: 404px 122.6px; margin-left: 3px; margin-top: 10px; margin-bottom: 10px; margin-right: 3px; \"\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 80px 8.5px; transform-origin: 80px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e       13 11 7 5 3 2\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 84px 8.5px; transform-origin: 84px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e      ---------------\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 80px 8.5px; transform-origin: 80px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e    2:             1\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 80px 8.5px; transform-origin: 80px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e    3:           1 0\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 80px 8.5px; transform-origin: 80px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e    4:             2\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 80px 8.5px; transform-origin: 80px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e    5:         1 0 0\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 80px 8.5px; transform-origin: 80px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e    6:           1 1\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 80px 8.5px; transform-origin: 80px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e   12:           1 2\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 80px 8.5px; transform-origin: 80px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e   14:       1 0 0 1\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 80px 8.5px; transform-origin: 80px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e   18:           2 1\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 80px 8.5px; transform-origin: 80px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e   26:   1 0 0 0 0 1\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 80px 8.5px; transform-origin: 80px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e   60:         1 1 2\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 10px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 10px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 10px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 289.5px 8px; transform-origin: 289.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eEach \"place\" in the number system represents a prime number. Given n, return the vector p.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 153px 8px; transform-origin: 153px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eAs shown above, if n = 26, then p = [1 0 0 0 0 1].\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 384px 8px; transform-origin: 384px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe input n is always an integer greater than 1. Suppress any leading zeros. The length of the vector is determined by the largest prime factor.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function p = factor_digits(n)\r\n  p = 0;\r\nend","test_suite":"%%\r\nn = 26;\r\np_correct = [1 0 0 0 0 1];\r\nassert(isequal(factor_digits(n),p_correct))\r\n\r\n%%\r\nn = 168;\r\np_correct = [1 0 1 3];\r\nassert(isequal(factor_digits(n),p_correct))\r\n\r\n%%\r\nn = 2;\r\np_correct = 1;\r\nassert(isequal(factor_digits(n),p_correct))\r\n\r\n%%\r\nn = 1444;\r\np_correct = 2*[1 0 0 0 0 0 0 1];\r\nassert(isequal(factor_digits(n),p_correct))\r\n\r\n%%\r\nn = 47;\r\np_correct = [1 0 0 0 0 0 0 0 0 0 0 0 0 0 0];\r\nassert(isequal(factor_digits(n),p_correct))\r\n\r\n%%\r\nn = 999;\r\np_correct = [1 0 0 0 0 0 0 0 0 0 3 0];\r\nassert(isequal(factor_digits(n),p_correct))\r\n\r\n%%\r\nn = 20;\r\np_correct = [1 0 2];\r\nassert(isequal(factor_digits(n),p_correct))","published":true,"deleted":false,"likes_count":28,"comments_count":6,"created_by":1,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":2540,"test_suite_updated_at":"2021-08-08T11:30:25.000Z","rescore_all_solutions":false,"group_id":2,"created_at":"2012-01-18T01:00:29.000Z","updated_at":"2026-04-10T14:34:09.000Z","published_at":"2012-01-18T01:00:29.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eConsider the following number system. Calculate the prime factorization for each number n, then represent the prime factors in a vector like so:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[       13 11 7 5 3 2\\n      ---------------\\n    2:             1\\n    3:           1 0\\n    4:             2\\n    5:         1 0 0\\n    6:           1 1\\n   12:           1 2\\n   14:       1 0 0 1\\n   18:           2 1\\n   26:   1 0 0 0 0 1\\n   60:         1 1 2]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eEach \\\"place\\\" in the number system represents a prime number. Given n, return the vector p.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eAs shown above, if n = 26, then p = [1 0 0 0 0 1].\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe input n is always an integer greater than 1. Suppress any leading zeros. The length of the vector is determined by the largest prime factor.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":1095,"title":"Circular Primes (based on Project Euler, problem 35)","description":"The number, 197, is called a circular prime because all rotations of the digits: 197, 971, and 719, are themselves prime.\r\n\r\nThere are thirteen such primes below 100: 2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 79, and 97.\r\n\r\nGiven a number x, write a MATLAB script that will tell you the number of circular primes less than or equal to x as well as a sorted list of what the circular prime numbers are.","description_html":"\u003cp\u003eThe number, 197, is called a circular prime because all rotations of the digits: 197, 971, and 719, are themselves prime.\u003c/p\u003e\u003cp\u003eThere are thirteen such primes below 100: 2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 79, and 97.\u003c/p\u003e\u003cp\u003eGiven a number x, write a MATLAB script that will tell you the number of circular primes less than or equal to x as well as a sorted list of what the circular prime numbers are.\u003c/p\u003e","function_template":"function [how_many what_numbers]=circular_prime(x)\r\n    how_many=3;\r\n    what_numbers=[2 3 5];\r\nend","test_suite":"%%\r\n[y numbers]=circular_prime(197)\r\nassert(isequal(y,16)\u0026\u0026isequal(numbers,[2 3 5 7 11 13 17 31 37 71 73 79 97 113 131 197]))\r\n%%\r\n[y numbers]=circular_prime(100)\r\nassert(isequal(y,13)\u0026\u0026isequal(numbers,[2 3 5 7 11 13 17 31 37 71 73 79 97]))\r\n%%\r\n[y numbers]=circular_prime(250)\r\nassert(isequal(y,17)\u0026\u0026isequal(numbers,[2 3 5 7 11 13 17 31 37 71 73 79 97 113 131 197 199]))\r\n%%\r\n[y numbers]=circular_prime(2000)\r\nassert(isequal(y,27)\u0026\u0026isequal(numbers,[2 3 5 7 11 13 17 31 37 71 73 79 97 113 131 197 199 311 337 373 719 733 919 971 991 1193 1931]))\r\n%%\r\n[y numbers]=circular_prime(10000)\r\nassert(isequal(y,33)\u0026\u0026isequal(numbers,[2 3 5 7 11 13 17 31 37 71 73 79 97 113 131 197 199 311 337 373 719 733 919 971 991 1193 1931 3119 3779 7793 7937 9311 9377]))\r\n%%\r\n[y numbers]=circular_prime(54321)\r\nassert(isequal(y,38)\u0026\u0026isequal(numbers,[2 3 5 7 11 13 17 31 37 71 73 79 97 113 131 197 199 311 337 373 719 733 919 971 991 1193 1931 3119 3779 7793 7937 9311 9377 11939 19391 19937 37199 39119]))\r\n","published":true,"deleted":false,"likes_count":10,"comments_count":6,"created_by":1615,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":652,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2012-12-05T18:02:09.000Z","updated_at":"2026-04-09T08:20:38.000Z","published_at":"2012-12-05T18:02:09.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe number, 197, is called a circular prime because all rotations of the digits: 197, 971, and 719, are themselves prime.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThere are thirteen such primes below 100: 2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 79, and 97.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eGiven a number x, write a MATLAB script that will tell you the number of circular primes less than or equal to x as well as a sorted list of what the circular prime numbers are.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":2342,"title":"Numbers spiral diagonals (Part 2)","description":"Inspired by Project Euler n°28 and 58.\r\nA n x n spiral matrix is obtained by starting with the number 1 and moving to the right in a clockwise direction.\r\nFor example with n=5, the spiral matrix is :\r\n                       21 22 23 24 25\r\n                       20  7  8  9 10\r\n                       19  6  1  2 11\r\n                       18  5  4  3 12\r\n                       17 16 15 14 13\r\nThe sum of the numbers on the diagonals is 101 (See problem 2340) and you have 5 primes (3, 5, 7, 13, 17) out of the 9 numbers lying along both diagonals. So the prime ratio is 5/9 ≈ 55%.\r\nWith a 7x7 spiral matrix, the ratio is 62% (8 primes out of the 13 diagonal numbers).\r\nWhat is the side length (always odd and greater than 1) of the square spiral for which the ratio of primes along both diagonals FIRST falls below p% ? (0\u003cp\u003c1)","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 326.167px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 163.083px; transform-origin: 407px 163.083px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 120px 8px; transform-origin: 120px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eInspired by Project Euler n°28 and 58.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 341px 8px; transform-origin: 341px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eA n x n spiral matrix is obtained by starting with the number 1 and moving to the right in a clockwise direction.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 131.5px 8px; transform-origin: 131.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eFor example with n=5, the spiral matrix is :\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgb(247, 247, 247); block-size: 102.167px; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-end-end-radius: 4px; border-end-start-radius: 4px; border-start-end-radius: 4px; border-start-start-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; margin-block-end: 10px; margin-block-start: 10px; margin-bottom: 10px; margin-inline-end: 3px; margin-inline-start: 3px; margin-left: 3px; margin-right: 3px; margin-top: 10px; perspective-origin: 404px 51.0833px; transform-origin: 404px 51.0833px; margin-left: 3px; margin-top: 10px; margin-bottom: 10px; margin-right: 3px; \"\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 148px 8.5px; tab-size: 4; transform-origin: 148px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e                       21 22 23 24 25\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 148px 8.5px; tab-size: 4; transform-origin: 148px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e                       20  7  8  9 10\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 148px 8.5px; tab-size: 4; transform-origin: 148px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e                       19  6  1  2 11\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 148px 8.5px; tab-size: 4; transform-origin: 148px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e                       18  5  4  3 12\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 148px 8.5px; tab-size: 4; transform-origin: 148px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e                       17 16 15 14 13\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 10px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 10px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 10px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 382.5px 8px; transform-origin: 382.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe sum of the numbers on the diagonals is 101 (See problem 2340) and you have 5 primes (3, 5, 7, 13, 17) out of the 9 numbers lying along both diagonals. So the prime ratio is 5/9 ≈ 55%.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 262px 8px; transform-origin: 262px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWith a 7x7 spiral matrix, the ratio is 62% (8 primes out of the 13 diagonal numbers).\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 364.5px 8px; transform-origin: 364.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"perspective-origin: 74px 8px; transform-origin: 74px 8px; \"\u003eWhat is the side length \u003c/span\u003e\u003cspan style=\"border-block-end-style: solid; border-block-end-width: 1px; border-bottom-style: solid; border-bottom-width: 1px; perspective-origin: 2.5px 8.5px; transform-origin: 2.5px 8.5px; \"\u003e(\u003c/span\u003e\u003cspan style=\"perspective-origin: 96px 8px; transform-origin: 96px 8px; \"\u003ealways odd and greater than 1\u003c/span\u003e\u003cspan style=\"border-block-end-style: solid; border-block-end-width: 1px; border-bottom-style: solid; border-bottom-width: 1px; perspective-origin: 2.5px 8.5px; transform-origin: 2.5px 8.5px; \"\u003e)\u003c/span\u003e\u003cspan style=\"perspective-origin: 189.5px 8px; transform-origin: 189.5px 8px; \"\u003e of the square spiral for which the ratio of primes along both diagonals FIRST falls below p% ? (0\u0026lt;p\u0026lt;1)\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function res=spiral_ratio(pourcentage)\r\nres=pourcentage*2;\r\nend","test_suite":"%%\r\nx = 0.8;\r\ny_correct = 3;\r\nassert(isequal(spiral_ratio(x),y_correct))\r\n%%\r\nx = 0.5;\r\ny_correct = 11;\r\nassert(isequal(spiral_ratio(x),y_correct))\r\n%%\r\nx = 0.4;\r\ny_correct = 31;\r\nassert(isequal(spiral_ratio(x),y_correct))\r\n%%\r\nx = 0.3;\r\ny_correct = 49;\r\nassert(isequal(spiral_ratio(x),y_correct))\r\n%%\r\nx = 0.25;\r\ny_correct = 99;\r\nassert(isequal(spiral_ratio(x),y_correct))\r\n%%\r\nx = 0.2;\r\ny_correct = 309;\r\nassert(isequal(spiral_ratio(x),y_correct))\r\n%%\r\nx = 0.15;\r\ny_correct = 981;\r\nassert(isequal(spiral_ratio(x),y_correct))\r\n%%\r\nx = 0.14;\r\ny_correct = 1883;\r\nassert(isequal(spiral_ratio(x),y_correct))\r\n%%\r\nx = 0.13;\r\ny_correct = 3593;\r\nassert(isequal(spiral_ratio(x),y_correct))\r\n%%\r\nx = 0.12;\r\ny_correct = 6523;\r\nassert(isequal(spiral_ratio(x),y_correct))\r\n%%\r\nx = 0.11;\r\ny_correct = 12201;\r\nassert(isequal(spiral_ratio(x),y_correct))","published":true,"deleted":false,"likes_count":7,"comments_count":5,"created_by":5390,"edited_by":223089,"edited_at":"2022-09-26T17:42:20.000Z","deleted_by":null,"deleted_at":null,"solvers_count":196,"test_suite_updated_at":"2022-07-09T19:28:50.000Z","rescore_all_solutions":false,"group_id":31,"created_at":"2014-05-31T18:36:25.000Z","updated_at":"2026-03-04T11:06:42.000Z","published_at":"2014-05-31T18:53:35.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eInspired by Project Euler n°28 and 58.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eA n x n spiral matrix is obtained by starting with the number 1 and moving to the right in a clockwise direction.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor example with n=5, the spiral matrix is :\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[                       21 22 23 24 25\\n                       20  7  8  9 10\\n                       19  6  1  2 11\\n                       18  5  4  3 12\\n                       17 16 15 14 13]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe sum of the numbers on the diagonals is 101 (See problem 2340) and you have 5 primes (3, 5, 7, 13, 17) out of the 9 numbers lying along both diagonals. So the prime ratio is 5/9 ≈ 55%.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWith a 7x7 spiral matrix, the ratio is 62% (8 primes out of the 13 diagonal numbers).\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWhat is the side length (always odd and greater than 1) of the square spiral for which the ratio of primes along both diagonals FIRST falls below p% ? (0\u0026lt;p\u0026lt;1)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":1394,"title":"Prime Ladders","description":"A \u003chttp://en.wikipedia.org/wiki/Word_ladder word ladder\u003e transforms one word to another by means of single-letter mutations. So COLD can become WARM like so (there are often multiple solutions):\r\n\r\n COLD\r\n CORD\r\n CARD\r\n WARD\r\n WARM\r\n\r\nA number ladder does much the same thing, changing one digit at a time. A *prime ladder* is a number ladder with the additional constraint that each element is a prime number. Here is a prime ladder that connects 757 and 139\r\n\r\n 757 \r\n 157\r\n 137\r\n 139\r\n\r\nGiven two numbers p1 and p2, construct a prime ladder column vector in which p1 is the first element, p2 is the last element, and each successive row differs by exactly one digit from the preceding element. \r\n\r\nTo restate the above example, consider\r\n\r\n p1 = 757\r\n p2 = 139\r\n\r\nfor which an acceptable answer is\r\n\r\n ladder = [757; 157; 137; 139]\r\n\r\nYou can assume that p1 and p2 contain the same number of digits. I am not looking for a unique answer. I will only check that the conditions of a prime ladder are met.\r\n\r\n","description_html":"\u003cp\u003eA \u003ca href = \"http://en.wikipedia.org/wiki/Word_ladder\"\u003eword ladder\u003c/a\u003e transforms one word to another by means of single-letter mutations. So COLD can become WARM like so (there are often multiple solutions):\u003c/p\u003e\u003cpre\u003e COLD\r\n CORD\r\n CARD\r\n WARD\r\n WARM\u003c/pre\u003e\u003cp\u003eA number ladder does much the same thing, changing one digit at a time. A \u003cb\u003eprime ladder\u003c/b\u003e is a number ladder with the additional constraint that each element is a prime number. Here is a prime ladder that connects 757 and 139\u003c/p\u003e\u003cpre\u003e 757 \r\n 157\r\n 137\r\n 139\u003c/pre\u003e\u003cp\u003eGiven two numbers p1 and p2, construct a prime ladder column vector in which p1 is the first element, p2 is the last element, and each successive row differs by exactly one digit from the preceding element.\u003c/p\u003e\u003cp\u003eTo restate the above example, consider\u003c/p\u003e\u003cpre\u003e p1 = 757\r\n p2 = 139\u003c/pre\u003e\u003cp\u003efor which an acceptable answer is\u003c/p\u003e\u003cpre\u003e ladder = [757; 157; 137; 139]\u003c/pre\u003e\u003cp\u003eYou can assume that p1 and p2 contain the same number of digits. I am not looking for a unique answer. I will only check that the conditions of a prime ladder are met.\u003c/p\u003e","function_template":"function ladder = prime_ladder(p1,p2)\r\n  ladder = 0;\r\nend","test_suite":"%%\r\n\r\np1 = 13;\r\np2 = 29;\r\nladder = prime_ladder(p1,p2);\r\n\r\nassert(all(isprime(ladder)))\r\nassert(iscolumn(ladder))\r\nassert(ladder(1)==p1)\r\nassert(ladder(end)==p2)\r\nassert(all(sum(diff(num2str(ladder))~=0,2)==1))\r\n\r\n%%\r\n\r\np1 = 389;\r\np2 = 269;\r\nladder = prime_ladder(p1,p2);\r\n\r\nassert(all(isprime(ladder)))\r\nassert(iscolumn(ladder))\r\nassert(ladder(1)==p1)\r\nassert(ladder(end)==p2)\r\nassert(all(sum(diff(num2str(ladder))~=0,2)==1))\r\n\r\n%%\r\n\r\np1 = 761;\r\np2 = 397;\r\nladder = prime_ladder(p1,p2);\r\n\r\nassert(all(isprime(ladder)))\r\nassert(iscolumn(ladder))\r\nassert(ladder(1)==p1)\r\nassert(ladder(end)==p2)\r\nassert(all(sum(diff(num2str(ladder))~=0,2)==1))\r\n\r\n%%\r\n\r\np1 = 983;\r\np2 = 239;\r\nladder = prime_ladder(p1,p2);\r\n\r\nassert(all(isprime(ladder)))\r\nassert(iscolumn(ladder))\r\nassert(ladder(1)==p1)\r\nassert(ladder(end)==p2)\r\nassert(all(sum(diff(num2str(ladder))~=0,2)==1))\r\n\r\n%%\r\n\r\np1 = 271;\r\np2 = 439;\r\nladder = prime_ladder(p1,p2);\r\n\r\nassert(all(isprime(ladder)))\r\nassert(iscolumn(ladder))\r\nassert(ladder(1)==p1)\r\nassert(ladder(end)==p2)\r\nassert(all(sum(diff(num2str(ladder))~=0,2)==1))\r\n\r\n%%\r\n\r\np1 = 877;\r\np2 = 733;\r\nladder = prime_ladder(p1,p2);\r\n\r\nassert(all(isprime(ladder)))\r\nassert(iscolumn(ladder))\r\nassert(ladder(1)==p1)\r\nassert(ladder(end)==p2)\r\nassert(all(sum(diff(num2str(ladder))~=0,2)==1))\r\n\r\n%%\r\n\r\np1 = 2267;\r\np2 = 1153;\r\nladder = prime_ladder(p1,p2);\r\n\r\nassert(all(isprime(ladder)))\r\nassert(iscolumn(ladder))\r\nassert(ladder(1)==p1)\r\nassert(ladder(end)==p2)\r\nassert(all(sum(diff(num2str(ladder))~=0,2)==1))\r\n","published":true,"deleted":false,"likes_count":10,"comments_count":3,"created_by":7,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":55,"test_suite_updated_at":"2013-03-27T21:24:26.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2013-03-26T22:51:16.000Z","updated_at":"2026-01-03T14:28:57.000Z","published_at":"2013-03-27T15:28:59.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eA\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://en.wikipedia.org/wiki/Word_ladder\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eword ladder\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e transforms one word to another by means of single-letter mutations. So COLD can become WARM like so (there are often multiple solutions):\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[ COLD\\n CORD\\n CARD\\n WARD\\n WARM]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eA number ladder does much the same thing, changing one digit at a time. A\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eprime ladder\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e is a number ladder with the additional constraint that each element is a prime number. Here is a prime ladder that connects 757 and 139\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[ 757 \\n 157\\n 137\\n 139]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eGiven two numbers p1 and p2, construct a prime ladder column vector in which p1 is the first element, p2 is the last element, and each successive row differs by exactly one digit from the preceding element.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eTo restate the above example, consider\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[ p1 = 757\\n p2 = 139]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003efor which an acceptable answer is\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[ ladder = [757; 157; 137; 139]]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eYou can assume that p1 and p2 contain the same number of digits. I am not looking for a unique answer. I will only check that the conditions of a prime ladder are met.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":44377,"title":"Five steps to enlightenment","description":"This problem asks you to identify valid variations of the famous \u003chttps://en.wikipedia.org/wiki/Sum_and_Product_Puzzle sum and product puzzle\u003e.\r\n\r\n\u003c\u003chttp://www.alfnie.com/software/printnumbers.jpg\u003e\u003e\r\n\r\nThe original _sum and product_ puzzle goes somewhat like this:\r\n\r\nScott and Priscilla are asked to guess two numbers X and Y, *ranging between 2 and 100* and with X\u003cY. Scott is told their sum S (=X+Y) and Priscilla is told their product P (=X*Y). After this, they have the following conversation:\r\n\r\n  Scott:      I don't know X and Y                                                (Step 1)\r\n  Priscilla:  Neither do I                                                        (Step 2)\r\n  Scott:      Before our conversation I already knew that you didn't know         (Step 3)\r\n  Priscilla:  But now I do know X and Y                                           (Step 4)\r\n  Scott:      And so do I!                                                        (Step 5)\r\n\r\nThe original puzzle asks you to deduce the value of X and Y from the above information\r\n\r\nTo solve this puzzle you may assume that: a) Scott and Priscilla are perfect mathematicians/logicians and always speak the truth; b) they are aware of each other circumstances (Scott knows that Priscilla has been told the product of X and Y, and Priscilla knows that Scott has been told the sum of X and Y before the beginning of their conversation); and c) the third sentence refers to Scott and Priscilla state before the beginning of the conversation (i.e. Scott already knew -at the outset of this conversation- that Priscilla did not know the solution -at the time-)\r\n\r\nBriefly, the deduction for this seemingly impossible puzzle starts with all possible pairs of values of X and Y, and uses  the information in the sentences above to sequentially reduce the range of possible (X,Y) pairs to those that would be consistent with each sentence (see figure below; blue dots represent (X,Y) pairs still possible after each of the sentences/steps above; plot uses matrix convention with origin at the upper-left corner, X in vertical axis, and Y in horizontal axis; upper triangular segment represents all possible X\u003cY pairs before the start of S\u0026P conversation). \r\n\r\n\u003c\u003chttp://www.alfnie.com/software/printnumbers_b.jpg\u003e\u003e\r\n\r\nFor example, in *Step 1* after S says _\"I don't know X and Y\"_ we learn that the pair (X,Y) = (2,3) (and a few others) are no longer possible, since otherwise S would have known the solution as soon as he was told their sum (X+Y=5), which takes a unique value among all possible (X,Y) pairs. Similarly, in *Step 2*, after P says _\"Neither do I\"_, the pair (X,Y) = (5,7) (and many others) are no longer possible, since otherwise P would have known the solution as soon as she was told their product X*Y=35, which turns out to be unique among all remaining X,Y pairs. In *Step 3*, S asserting _\"I knew that you didn't know\"_ tells us that, knowing only X+Y, he was able to determine beforehand that it would be _impossible_ for P to know both X and Y from their product X*Y alone, which, again, rules out a considerable number of (X,Y) pairs (e.g. all solutions with X+Y=12 can be ruled out, because if S was told that X+Y=12 he could not have possibly dismissed beforehand the possibility that perhaps X=5 and Y=7 which would have allowed P to know both X and Y as soon as she was told their product X*Y=35). In *Step 4*, P suddenly become aware of the solution after S revelation informs us that, unlike in Step 2, she is now (after step-3 crop in possible X,Y pairs) able to uniquely determine the solution (X,Y) from their product (X*Y). This allows us to rule out many (X,Y) pairs among the remaining possible values, such as (X,Y)=(2,15) or (5,6), because P would not have been able to uniquely identify the solution at this point if she was told that the product X*Y was 30. Last, in *Step 5*, S suddenly becoming aware of the solution after P revelation, again allows us to rule out any remaining (X,Y) pair where knowing the sum S would still not suffice to uniquely identify the solution.\r\n\r\nThis puzzle is very neat because, somewhat surprisingly, after sequentially reducing the range of possible (X,Y) pairs from the five sentences/steps above, only one possible pair remains. The solution X=4 and Y=13, which Priscilla learns after the third sentence, Scott learns after the fourth sentence, and you, the reader, learn after the fifth sentence. \r\n\r\nKey to the existence of a unique solution to this puzzle is the initial range of possible (X,Y) pairs that we are told to consider. If, for example, instead of considering all X,Y values between 2 and 100, we were told to consider all X,Y values between 1 and 100, the puzzle would not be solvable, as in this case there will be multiple possible solutions that would all be consistent with the five sentences above (see figure below; at Step 5 there still exist 6 different possible solutions to this puzzle). \r\n\r\n\u003c\u003chttp://www.alfnie.com/software/printnumbers_c.jpg\u003e\u003e\r\n\r\nSimilarly, if we were told to consider all X,Y values between 2 and 50, the puzzle would again not be solvable, as in this case there would be _no solution_ consistent with all five sentences (perhaps surprisingly, since the X=4 Y=13 solution above is in fact within the stated range). \r\n\r\n\u003c\u003chttp://www.alfnie.com/software/printnumbers_d.jpg\u003e\u003e\r\n\r\nOn the other hand, if we were told to consider X and Y values ranging between 1 and 24, for example, the puzzle would again become solvable, now with a new unique solution X=1 and Y=6. \r\n\r\n\u003c\u003chttp://www.alfnie.com/software/printnumbers_e.jpg\u003e\u003e\r\n\r\nIn this problem you are tasked to create a function that would determine whether a particular variation of this puzzle would work or not. Specifically, given an initial set of possible (X,Y) pairs (entered as a Nx2 matrix and representing the full set of possible X,Y pairs that Scott and Priscilla are told to consider), you should determine whether it is possible to solve this puzzle (i.e. whether one, and only one, (X,Y) pair is consistent with the five sequential sentences above). Your function should simply return 1 (or true) if the puzzle is solvable, and 0 (or false) otherwise. \r\n\r\nGood luck!","description_html":"\u003cp\u003eThis problem asks you to identify valid variations of the famous \u003ca href = \"https://en.wikipedia.org/wiki/Sum_and_Product_Puzzle\"\u003esum and product puzzle\u003c/a\u003e.\u003c/p\u003e\u003cimg src = \"http://www.alfnie.com/software/printnumbers.jpg\"\u003e\u003cp\u003eThe original \u003ci\u003esum and product\u003c/i\u003e puzzle goes somewhat like this:\u003c/p\u003e\u003cp\u003eScott and Priscilla are asked to guess two numbers X and Y, \u003cb\u003eranging between 2 and 100\u003c/b\u003e and with X\u0026lt;Y. Scott is told their sum S (=X+Y) and Priscilla is told their product P (=X*Y). After this, they have the following conversation:\u003c/p\u003e\u003cpre class=\"language-matlab\"\u003eScott:      I don't know X and Y                                                (Step 1)\r\nPriscilla:  Neither do I                                                        (Step 2)\r\nScott:      Before our conversation I already knew that you didn't know         (Step 3)\r\nPriscilla:  But now I do know X and Y                                           (Step 4)\r\nScott:      And so do I!                                                        (Step 5)\r\n\u003c/pre\u003e\u003cp\u003eThe original puzzle asks you to deduce the value of X and Y from the above information\u003c/p\u003e\u003cp\u003eTo solve this puzzle you may assume that: a) Scott and Priscilla are perfect mathematicians/logicians and always speak the truth; b) they are aware of each other circumstances (Scott knows that Priscilla has been told the product of X and Y, and Priscilla knows that Scott has been told the sum of X and Y before the beginning of their conversation); and c) the third sentence refers to Scott and Priscilla state before the beginning of the conversation (i.e. Scott already knew -at the outset of this conversation- that Priscilla did not know the solution -at the time-)\u003c/p\u003e\u003cp\u003eBriefly, the deduction for this seemingly impossible puzzle starts with all possible pairs of values of X and Y, and uses  the information in the sentences above to sequentially reduce the range of possible (X,Y) pairs to those that would be consistent with each sentence (see figure below; blue dots represent (X,Y) pairs still possible after each of the sentences/steps above; plot uses matrix convention with origin at the upper-left corner, X in vertical axis, and Y in horizontal axis; upper triangular segment represents all possible X\u0026lt;Y pairs before the start of S\u0026P conversation).\u003c/p\u003e\u003cimg src = \"http://www.alfnie.com/software/printnumbers_b.jpg\"\u003e\u003cp\u003eFor example, in \u003cb\u003eStep 1\u003c/b\u003e after S says \u003ci\u003e\"I don't know X and Y\"\u003c/i\u003e we learn that the pair (X,Y) = (2,3) (and a few others) are no longer possible, since otherwise S would have known the solution as soon as he was told their sum (X+Y=5), which takes a unique value among all possible (X,Y) pairs. Similarly, in \u003cb\u003eStep 2\u003c/b\u003e, after P says \u003ci\u003e\"Neither do I\"\u003c/i\u003e, the pair (X,Y) = (5,7) (and many others) are no longer possible, since otherwise P would have known the solution as soon as she was told their product X*Y=35, which turns out to be unique among all remaining X,Y pairs. In \u003cb\u003eStep 3\u003c/b\u003e, S asserting \u003ci\u003e\"I knew that you didn't know\"\u003c/i\u003e tells us that, knowing only X+Y, he was able to determine beforehand that it would be \u003ci\u003eimpossible\u003c/i\u003e for P to know both X and Y from their product X*Y alone, which, again, rules out a considerable number of (X,Y) pairs (e.g. all solutions with X+Y=12 can be ruled out, because if S was told that X+Y=12 he could not have possibly dismissed beforehand the possibility that perhaps X=5 and Y=7 which would have allowed P to know both X and Y as soon as she was told their product X*Y=35). In \u003cb\u003eStep 4\u003c/b\u003e, P suddenly become aware of the solution after S revelation informs us that, unlike in Step 2, she is now (after step-3 crop in possible X,Y pairs) able to uniquely determine the solution (X,Y) from their product (X*Y). This allows us to rule out many (X,Y) pairs among the remaining possible values, such as (X,Y)=(2,15) or (5,6), because P would not have been able to uniquely identify the solution at this point if she was told that the product X*Y was 30. Last, in \u003cb\u003eStep 5\u003c/b\u003e, S suddenly becoming aware of the solution after P revelation, again allows us to rule out any remaining (X,Y) pair where knowing the sum S would still not suffice to uniquely identify the solution.\u003c/p\u003e\u003cp\u003eThis puzzle is very neat because, somewhat surprisingly, after sequentially reducing the range of possible (X,Y) pairs from the five sentences/steps above, only one possible pair remains. The solution X=4 and Y=13, which Priscilla learns after the third sentence, Scott learns after the fourth sentence, and you, the reader, learn after the fifth sentence.\u003c/p\u003e\u003cp\u003eKey to the existence of a unique solution to this puzzle is the initial range of possible (X,Y) pairs that we are told to consider. If, for example, instead of considering all X,Y values between 2 and 100, we were told to consider all X,Y values between 1 and 100, the puzzle would not be solvable, as in this case there will be multiple possible solutions that would all be consistent with the five sentences above (see figure below; at Step 5 there still exist 6 different possible solutions to this puzzle).\u003c/p\u003e\u003cimg src = \"http://www.alfnie.com/software/printnumbers_c.jpg\"\u003e\u003cp\u003eSimilarly, if we were told to consider all X,Y values between 2 and 50, the puzzle would again not be solvable, as in this case there would be \u003ci\u003eno solution\u003c/i\u003e consistent with all five sentences (perhaps surprisingly, since the X=4 Y=13 solution above is in fact within the stated range).\u003c/p\u003e\u003cimg src = \"http://www.alfnie.com/software/printnumbers_d.jpg\"\u003e\u003cp\u003eOn the other hand, if we were told to consider X and Y values ranging between 1 and 24, for example, the puzzle would again become solvable, now with a new unique solution X=1 and Y=6.\u003c/p\u003e\u003cimg src = \"http://www.alfnie.com/software/printnumbers_e.jpg\"\u003e\u003cp\u003eIn this problem you are tasked to create a function that would determine whether a particular variation of this puzzle would work or not. Specifically, given an initial set of possible (X,Y) pairs (entered as a Nx2 matrix and representing the full set of possible X,Y pairs that Scott and Priscilla are told to consider), you should determine whether it is possible to solve this puzzle (i.e. whether one, and only one, (X,Y) pair is consistent with the five sequential sentences above). Your function should simply return 1 (or true) if the puzzle is solvable, and 0 (or false) otherwise.\u003c/p\u003e\u003cp\u003eGood luck!\u003c/p\u003e","function_template":"function valid = fivesteps(XY)\r\n  % XY is a Nx2 matrix, where each row represents a possible X,Y combination\r\n  % valid is 1 if the puzzle is solvable or 0 otherwise\r\n  valid = true;\r\nend","test_suite":"%%\r\nassessFunctionAbsence({'regexp','regexpi','regexprep','str2num'},'FileName','fivesteps.m')\r\nassert(isempty(regexp(fileread('fivesteps.m'),'assert')));\r\n[~,~]=system('rm freepass*');\r\n%%\r\n%lines=textread('fivesteps.m','%s'); \r\n%id=str2num(regexp(lines{end},'\\d+','match','once'));\r\n%assert(~ismember(id,[3430216]),'Please submit a valid non-cheating solution and ask the problem author to manually evaluate it'); % [3931805,3397427,3430216]\r\n%%\r\n% X,Y X\u003cY between 2 and 100\r\n[x,y]=find(triu(ones(100),1));\r\nz=[x y];\r\nvalid=all(z\u003e1,2);\r\nassert(fivesteps(z(valid,:))==true);\r\n%%\r\n% X,Y X\u003cY between 2 and 60\r\n[x,y]=find(triu(ones(60),1));\r\nz=[x y];\r\nvalid=all(z\u003e1,2);\r\nassert(fivesteps(z(valid,:))==false);\r\n%%\r\n% X,Y X\u003cY between 2 and 200\r\n[x,y]=find(triu(ones(200),1));\r\nz=[x y];\r\nvalid=all(z\u003e1,2);\r\nassert(fivesteps(z(valid,:))==true);\r\n%%\r\n% X,Y X\u003cY between 2 and 1680\r\n[x,y]=find(triu(ones(1680),1));\r\nz=[x y];\r\nvalid=all(z\u003e1,2);\r\nassert(fivesteps(z(valid,:))==true);\r\n%%\r\n% X,Y X\u003cY between 2 and 1700\r\n[x,y]=find(triu(ones(1700),1));\r\nz=[x y];\r\nvalid=all(z\u003e1,2);\r\nassert(fivesteps(z(valid,:))==false);\r\n%%\r\n% X,Y X\u003cY between 1 and 20\r\n[x,y]=find(triu(ones(20),1));\r\nz=[x y];\r\nvalid=all(z\u003e=1,2);\r\nassert(fivesteps(z(valid,:))==false);\r\n%%\r\n% X,Y X\u003cY between 1 and 30\r\n[x,y]=find(triu(ones(30),1));\r\nz=[x y];\r\nvalid=all(z\u003e=1,2);\r\nassert(fivesteps(z(valid,:))==true);\r\n%%\r\n% X,Y X\u003cY between 1 and 40\r\n[x,y]=find(triu(ones(40),1));\r\nz=[x y];\r\nvalid=all(z\u003e=1,2);\r\nassert(fivesteps(z(valid,:))==false);\r\n%%\r\n% X,Y X\u003cY between 3 and 5000\r\n[x,y]=find(triu(ones(3000),1));\r\nz=[x y];\r\nvalid=all(z\u003e=3,2);\r\nassert(fivesteps(z(valid,:))==true);\r\n%%\r\n% X,Y X\u003cY between 3 and 100\r\n[x,y]=find(triu(ones(100),1));\r\nz=[x y];\r\nvalid=all(z\u003e=3,2);\r\nassert(fivesteps(z(valid,:))==false);\r\n%%\r\n% X,Y X\u003cY even between 2 and 40\r\n[x,y]=meshgrid(2:2:40);\r\nz=[x(:) y(:)];\r\nvalid=y\u003ex;\r\nassert(fivesteps(z(valid,:))==false);\r\n%%\r\n% X,Y X\u003cY odd between 1 and 1000\r\n[x,y]=meshgrid(1:2:1000);\r\nz=[x(:) y(:)];\r\nvalid=y\u003ex;\r\nassert(fivesteps(z(valid,:))==true);\r\n%%\r\n% X,Y X\u003cY non-primes between 1 and 50\r\n[x,y]=meshgrid(setdiff(1:50,primes(50)));\r\nz=[x(:) y(:)];\r\nvalid=y\u003ex;\r\nassert(fivesteps(z(valid,:))==true);\r\n%%\r\n% X,Y X\u003cY primes between 1 and 50\r\n[x,y]=meshgrid(primes(50));\r\nz=[x(:) y(:)];\r\nvalid=y\u003ex;\r\nassert(fivesteps(z(valid,:))==false);\r\n%%\r\n% X,Y X\u003cY between 3 and 100\r\n[x,y]=find(triu(ones(randi([10,100])),1));\r\nz=[x y];\r\nvalid=all(z\u003e2,2);\r\nassert(fivesteps(z(valid,:))==false);\r\n%%\r\n% X,Y X\u003cY between 1 and 100\r\n[x,y]=find(triu(ones(randi([40,100])),1));\r\nz=[x y];\r\nvalid=all(z\u003e=1,2);\r\nassert(fivesteps(z(valid,:))==false);\r\n%%\r\n% X,Y X\u003cY between 2 and 100\r\n[x,y]=find(triu(ones(randi([62,100])),1));\r\nz=[x y];\r\nvalid=all(z\u003e1,2);\r\nassert(fivesteps(z(valid,:))==true);\r\n%%\r\n% X,Y X\u003cY between 2 and 100\r\n[x,y]=find(triu(ones(randi([10,61])),1));\r\nz=[x y];\r\nvalid=all(z\u003e1,2);\r\nassert(fivesteps(z(valid,:))==false); \r\n%%\r\n% X,Y X\u003cY in 3 5 12 18 20 28 30\r\n% Possible solutions after step 1: [3,20] [3,30] [5,18] [5,28] [18,30] [20,28] \r\n% Possible solutions after step 2: [3,30] [5,18] \r\n% Possible solutions after step 3: [5,18] \r\n% Possible solutions after step 4: [5,18] \r\n% Possible solutions after step 5: [5,18] \r\n[x,y]=meshgrid([3 5 12 18 20 28 30]);\r\nz=[x(:) y(:)];\r\nvalid=y\u003ex;\r\nassert(fivesteps(z(valid,:))==true);\r\n%%\r\n% X,Y X\u003cY in 3 5 12 15 28 30\r\n% Possible solutions after step 1: [3,30] [5,28] \r\n% Possible solutions after step 2: \r\n[x,y]=meshgrid([3 5 12 15 28 30]);\r\nz=[x(:) y(:)];\r\nvalid=y\u003ex;\r\nassert(fivesteps(z(valid,:))==false);\r\n%%\r\n% X,Y X\u003cY in 2 4 6 8 12 16 18\r\n% Possible solutions after step 1: [2,8] [2,12] [2,16] [2,18] [4,6] [4,16] [4,18] [6,8] [6,12] [6,16] [6,18] [8,12] [8,16] \r\n% Possible solutions after step 2: [2,12] [4,6] [4,18] [6,12] [6,16] [8,12] \r\n% Possible solutions after step 3: [2,12] [4,18] [6,12] [6,16] \r\n% Possible solutions after step 4: [2,12] [6,16] \r\n% Possible solutions after step 5: [2,12] [6,16] \r\n[x,y]=meshgrid([2 4 6 8 12 16 18]);\r\nz=[x(:) y(:)];\r\nvalid=y\u003ex;\r\nassert(fivesteps(z(valid,:))==false);\r\n%%\r\n% X,Y X\u003cY in 4 6 8 12 16 18\r\n% Possible solutions after step 1: [4,16] [4,18] [6,16] [6,18] [8,12] [8,16] \r\n% Possible solutions after step 2: [6,16] [8,12] \r\n% Possible solutions after step 3: [6,16] \r\n% Possible solutions after step 4: [6,16] \r\n% Possible solutions after step 5: [6,16] \r\n[x,y]=meshgrid([4 6 8 12 16 18]);\r\nz=[x(:) y(:)];\r\nvalid=y\u003ex;\r\nassert(fivesteps(z(valid,:))==true);\r\n%%\r\n% X,Y X\u003cY in 2 3 10 14 15 20 21\r\n% Possible solutions after step 1: [2,15] [2,21] [3,14] [3,20] [3,21] [10,14] [14,21] [15,20] \r\n% Possible solutions after step 2: [2,21] [3,14] \r\n% Possible solutions after step 3: [3,14] \r\n% Possible solutions after step 4: [3,14] \r\n% Possible solutions after step 5: [3,14] \r\n[x,y]=meshgrid([2 3 10 14 15 20 21]);\r\nz=[x(:) y(:)];\r\nvalid=y\u003ex;\r\nassert(fivesteps(z(valid,:))==true);\r\n%%\r\n% X,Y X\u003cY in 2 3 5 8 10 14 15 20 21\r\n% Possible solutions after step 1: [2,15] [2,20] [2,21] [3,10] [3,14] [3,15] [3,20] [3,21] [5,8] [5,20] [8,10] [8,14] [8,15] [8,21] [10,14] [10,15] [14,15] [14,21] [15,20] \r\n% Possible solutions after step 2: [2,15] [2,20] [2,21] [3,10] [3,14] [5,8] \r\n% Possible solutions after step 3: [2,15] [3,10] [3,14] [5,8] \r\n% Possible solutions after step 4: [3,14] [5,8] \r\n% Possible solutions after step 5: [3,14] [5,8] \r\n[x,y]=meshgrid([2 3 5 8 10 14 15 20 21]);\r\nz=[x(:) y(:)];\r\nvalid=y\u003ex;\r\nassert(fivesteps(z(valid,:))==false);\r\n%%\r\n% a few random cases to discourage look-up table solutions\r\n    data={[2 5 6 7 9 10 12 16 23 25 26 27 28 29 31],[49 94 97 109 112 113 154 157 158 161 184 187 188 191 193 199 202 203 206 208 209 214 217 218 221 223 224 225];...\r\n        [2 4 6 9 10 11 12 13 14 20 22 23 25 26 28],[49 64 79 81 109 111 113 124 125 126 128 129 154 156 158 159 161 169 171 173 174 176 177 184 186 188 189 191 192 193 199 201 203 204 206 207 208 209 214 216 218 219 221 222 223 224 225];...\r\n        [2 3 4 6 7 9 10 11 12 14 19 20 21 24 25 28 31],[19 35 87 89 108 109 121 124 126 127 142 144 145 155 156 171 174 175 177 180 206 215 223 230 240 241 245 247 253 257 262 263 264 270 271 274 278 279 280 281 285 288 289];...\r\n        [2 4 5 6 8 9 10 11 15 16 18 19 20 21 23 24 25 26 27 28 29],[45 66 67 87 109 111 150 152 153 171 174 177 214 219 221 234 235 237 240 242 243 256 258 261 263 264 265 276 277 279 282 284 285 286 287 297 298 300 303 305 306 307 308 309 319 321 324 326 327 328 329 330 331 339 340 342 345 347 348 349 350 351 352 353 360 361 363 366 368 369 370 371 372 373 374 375 381 382 384 387 389 390 391 392 393 394 395 396 397 402 403 405 408 410 411 412 413 414 415 416 417 418 419 423 424 426 429 431 432 433 434 435 436 437 438 439 440 441];...\r\n        [2 4 5 6 8 10 11 12 14 16 17 25 27 28 29 30 31],[1 54 89 109 125 137 140 144 145 154 156 171 177 179 181 188 194 196 198 199 205 208 211 213 216 217 222 225 228 229 230 232 233 234 235 239 245 247 248 250 251 252 253 256 262 264 266 267 268 269 270 271 273 279 281 283 284 285 286 287 288 289];...\r\n        [2 5 6 7 10 14 15 16 17 20 21 23 24 25 29 30],[120 152 154 168 170 171 184 185 186 187 188 200 202 203 204 205 216 218 219 220 221 222 232 234 235 236 237 238 239 248 250 251 252 253 254 255 256];...\r\n        [4 5 7 8 9 11 12 15 16 17 20 21 24 27 28 30],[18 68 82 85 86 98 100 102 103 116 118 119 120 129 130 136 146 150 151 152 153 162 166 168 169 171 194 196 198 199 202 203 205 212 226 230 231 232 233 235 237 239 244 249 250];...\r\n        [2 3 4 6 8 9 11 12 13 15 17 19 20 21 22 23 25 28 29 31],[22 43 102 122 123 127 147 148 162 163 168 169 182 187 188 190 202 203 207 208 209 210 211 222 223 227 228 229 230 231 232 242 243 247 248 249 250 251 252 253 262 263 267 268 269 270 271 272 273 274 282 283 287 288 289 290 291 292 293 294 295 302 303 307 308 309 310 311 312 313 314 315 316 322 323 327 328 329 330 331 332 333 334 335 336 337 342 343 347 348 349 350 351 352 353 354 355 356 357 358 362 363 367 368 369 370 371 372 373 374 375 376 377 378 379 382 383 387 388 389 390 391 392 393 394 395 396 397 398 399 400];...\r\n        [3 4 5 6 8 9 10 11 13 14 18 19 21 22 23 25 26 28 29 30 31],[111 150 153 154 155 174 176 177 195 197 198 199 216 218 219 220 221 237 239 240 242 243 258 260 261 262 263 264 265 279 281 282 283 284 285 286 287 300 302 303 304 305 306 307 308 309 321 323 324 325 326 327 328 329 330 331 342 344 345 346 347 348 349 350 351 352 353 363 365 366 367 368 369 370 371 372 373 374 375 384 386 387 388 389 390 391 392 393 394 395 396 397 405 407 408 409 410 411 412 413 414 415 416 417 418 419 426 428 429 430 431 432 433 434 435 436 437 438 439 440 441];...\r\n        [2 4 5 6 8 10 11 13 14 15 16 17 20 22 28 31],[51 52 83 100 103 132 135 136 137 148 151 153 154 168 183 185 187 188 196 199 200 201 202 204 205 212 215 217 218 220 221 222 228 231 232 233 234 236 237 238 239 244 247 249 250 252 253 254 255 256];...\r\n        [2 4 5 8 10 11 12 15 20 21 22 26 28 29 30 31],[1 65 82 103 119 120 129 145 151 152 154 177 183 184 186 188 193 199 200 204 205 209 215 216 218 220 221 222 225 231 232 234 236 237 238 239 241 247 248 250 252 253 254 255 256];...\r\n        [2 4 5 7 8 12 13 17 18 19 22 26 28 29 30],[49 64 65 109 110 113 124 125 128 129 139 140 143 144 145 154 155 158 159 160 161 184 185 188 189 191 193 199 200 203 204 205 206 208 209];...\r\n        [2 3 6 8 9 12 15 17 18 19 21 22 23 24 25 27 28 29 30],[81 96 119 121 137 138 140 141 157 159 160 161 176 180 195 197 198 199 201 214 216 217 218 220 221 233 235 237 239 240 241 254 255 256 258 259 260 261 271 273 274 275 277 278 279 280 281 290 292 293 294 296 297 298 299 300 301 309 311 312 313 315 316 317 318 319 320 321 328 330 331 332 334 335 336 337 338 339 340 341 347 349 350 351 352 353 354 355 356 357 358 359 360 361];...\r\n        [6 7 9 11 17 18 21 22 23 27 28 29],[53 101 105 125 129 131 137 141 143 144];...\r\n        [2 3 4 6 8 9 11 12 15 16 18 19 20 22 24 25 27 28 29 30],[22 81 83 102 103 121 122 127 147 148 161 164 166 167 168 188 189 190 202 203 208 210 211 222 231 232 242 247 248 250 251 252 253 262 263 267 268 269 270 271 272 273 274 285 287 288 290 291 292 293 294 295 310 314 315 316 322 323 327 328 330 331 332 333 334 335 336 337 347 348 350 351 352 353 354 355 356 357 358 362 367 368 370 371 372 373 374 375 376 377 378 379 382 387 388 390 391 392 393 394 395 396 397 398 399 400];...\r\n        [3 4 5 6 10 11 13 17 20 21 25 26 30],[57 70 71 96 97 99 161 162 164 169];...\r\n        [2 3 4 10 12 15 16 18 19 20 21 22 25 26 28 30 31],[1 52 55 120 123 127 140 144 145 157 161 162 163 171 174 178 179 180 181 188 191 195 196 197 198 199 205 208 212 213 214 215 216 217 222 225 229 230 231 232 233 234 235 239 242 246 247 248 249 250 251 252 253 256 259 263 264 266 267 268 269 270 271 273 276 280 281 282 283 284 285 286 287 288 289];...\r\n        [2 4 5 7 10 15 16 18 20 24 25 27 28 29 30 31],[17 18 82 86 98 102 103 114 118 119 120 146 150 151 152 154 178 182 183 186 187 188 209 210 214 215 216 218 220 222 226 230 231 232 234 236 238 239 242 246 247 248 250 252 254 255 256];...\r\n        [4 6 7 10 11 12 18 20 21 22 24 28],[26 27 40 51 52 88 97 99 101 103 105 112 113 117 123 124 131 133 134 136 139 141 143];...\r\n        [2 3 4 6 8 10 12 14 15 17 18 19 20 21 22 24 26 28 30],[20 41 79 80 117 118 119 121 137 138 141 156 157 160 174 178 181 193 197 198 200 201 212 213 214 220 221 229 231 232 233 235 238 239 240 241 251 252 255 269 272 274 276 277 279 281 288 292 293 295 296 297 298 300 301 307 311 312 314 315 316 317 319 320 321 326 330 331 333 334 335 336 338 339 340 341 345 349 350 352 353 354 355 357 358 359 360 361];...\r\n        [2 3 4 6 8 10 12 13 15 16 18 19 20 22 25 26 29],[18 35 37 70 72 88 91 105 106 107 121 124 139 140 141 144 145 160 161 163 173 176 179 180 181 190 195 197 207 208 209 211 214 215 216 217 224 227 230 231 232 234 235 241 247 248 249 251 252 253 258 259 260 275 278 281 282 283 285 286 287 289];...\r\n        [3 4 5 8 9 12 14 21 23 24 28 30],[1 49 53 97 101 105 121 125 129 131];...\r\n        [2 3 6 7 8 9 12 13 15 16 18 19 22 23 24 26 28],[52 55 89 103 120 124 140 145 157 159 174 179 181 191 196 198 199 208 213 215 216 217 230 232 235 242 245 247 249 250 251 252 253 259 261 276 278];...\r\n        [2 4 6 8 9 10 11 12 21 22 24 25 26 27 28 29 31],[1 19 37 55 73 86 87 89 90 104 121 123 124 125 127 137 139 140 141 144 145 155 159 172 174 178 179 181 188 189 190 192 194 195 198 199 205 206 207 208 212 215 216 217 222 224 226 227 239 241 242 243 246 247 249 250 251 253 256 257 258 259 260 263 264 266 267 268 270 271 273 274 275 276 277 280 281 283 284 285 287 288 289]};\r\n    for ndata=randi(size(data,1),1,20)\r\n        a=data{ndata,1};\r\n        N=numel(a);\r\n        A=zeros(N);\r\n        A(data{ndata,2})=1;A=A|A';\r\n        if rand\u003c.5, idx=find(A); else idx=find(~A); end\r\n        [b,c]=ind2sub(size(A),idx(randi(numel(idx))));\r\n        d=a;d([b c])=[];\r\n        [x,y]=meshgrid(d);\r\n        z=[x(:) y(:)];\r\n        valid=y\u003ex;\r\n        assert(fivesteps(z(valid,:))==A(b,c),'failed on d = %s (correct output = %s)',mat2str(d),mat2str(A(b,c)));\r\n    end\r\n","published":true,"deleted":false,"likes_count":6,"comments_count":36,"created_by":43,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":42,"test_suite_updated_at":"2018-01-02T21:35:48.000Z","rescore_all_solutions":true,"group_id":35,"created_at":"2017-10-11T19:04:33.000Z","updated_at":"2026-01-17T20:25:15.000Z","published_at":"2017-10-16T01:51:01.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/image\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/media/image1.JPEG\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/image\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/media/image2.JPEG\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/image\",\"targetMode\":\"\",\"relationshipId\":\"rId3\",\"target\":\"/media/image3.JPEG\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/image\",\"targetMode\":\"\",\"relationshipId\":\"rId4\",\"target\":\"/media/image4.JPEG\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/image\",\"targetMode\":\"\",\"relationshipId\":\"rId5\",\"target\":\"/media/image5.JPEG\"}],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThis problem asks you to identify valid variations of the famous\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://en.wikipedia.org/wiki/Sum_and_Product_Puzzle\\\"\u003e\u003cw:r\u003e\u003cw:t\u003esum and product puzzle\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe original\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003esum and product\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e puzzle goes somewhat like this:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eScott and Priscilla are asked to guess two numbers X and Y,\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eranging between 2 and 100\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e and with X\u0026lt;Y. Scott is told their sum S (=X+Y) and Priscilla is told their product P (=X*Y). After this, they have the following conversation:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[Scott:      I don't know X and Y                                                (Step 1)\\nPriscilla:  Neither do I                                                        (Step 2)\\nScott:      Before our conversation I already knew that you didn't know         (Step 3)\\nPriscilla:  But now I do know X and Y                                           (Step 4)\\nScott:      And so do I!                                                        (Step 5)]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe original puzzle asks you to deduce the value of X and Y from the above information\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eTo solve this puzzle you may assume that: a) Scott and Priscilla are perfect mathematicians/logicians and always speak the truth; b) they are aware of each other circumstances (Scott knows that Priscilla has been told the product of X and Y, and Priscilla knows that Scott has been told the sum of X and Y before the beginning of their conversation); and c) the third sentence refers to Scott and Priscilla state before the beginning of the conversation (i.e. Scott already knew -at the outset of this conversation- that Priscilla did not know the solution -at the time-)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eBriefly, the deduction for this seemingly impossible puzzle starts with all possible pairs of values of X and Y, and uses the information in the sentences above to sequentially reduce the range of possible (X,Y) pairs to those that would be consistent with each sentence (see figure below; blue dots represent (X,Y) pairs still possible after each of the sentences/steps above; plot uses matrix convention with origin at the upper-left corner, X in vertical axis, and Y in horizontal axis; upper triangular segment represents all possible X\u0026lt;Y pairs before the start of S\u0026amp;P conversation).\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId2\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor example, in\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eStep 1\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e after S says\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\\\"I don't know X and Y\\\"\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e we learn that the pair (X,Y) = (2,3) (and a few others) are no longer possible, since otherwise S would have known the solution as soon as he was told their sum (X+Y=5), which takes a unique value among all possible (X,Y) pairs. Similarly, in\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eStep 2\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, after P says\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\\\"Neither do I\\\"\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, the pair (X,Y) = (5,7) (and many others) are no longer possible, since otherwise P would have known the solution as soon as she was told their product X*Y=35, which turns out to be unique among all remaining X,Y pairs. In\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eStep 3\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, S asserting\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\\\"I knew that you didn't know\\\"\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e tells us that, knowing only X+Y, he was able to determine beforehand that it would be\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eimpossible\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e for P to know both X and Y from their product X*Y alone, which, again, rules out a considerable number of (X,Y) pairs (e.g. all solutions with X+Y=12 can be ruled out, because if S was told that X+Y=12 he could not have possibly dismissed beforehand the possibility that perhaps X=5 and Y=7 which would have allowed P to know both X and Y as soon as she was told their product X*Y=35). In\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eStep 4\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, P suddenly become aware of the solution after S revelation informs us that, unlike in Step 2, she is now (after step-3 crop in possible X,Y pairs) able to uniquely determine the solution (X,Y) from their product (X*Y). This allows us to rule out many (X,Y) pairs among the remaining possible values, such as (X,Y)=(2,15) or (5,6), because P would not have been able to uniquely identify the solution at this point if she was told that the product X*Y was 30. Last, in\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eStep 5\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, S suddenly becoming aware of the solution after P revelation, again allows us to rule out any remaining (X,Y) pair where knowing the sum S would still not suffice to uniquely identify the solution.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThis puzzle is very neat because, somewhat surprisingly, after sequentially reducing the range of possible (X,Y) pairs from the five sentences/steps above, only one possible pair remains. The solution X=4 and Y=13, which Priscilla learns after the third sentence, Scott learns after the fourth sentence, and you, the reader, learn after the fifth sentence.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eKey to the existence of a unique solution to this puzzle is the initial range of possible (X,Y) pairs that we are told to consider. If, for example, instead of considering all X,Y values between 2 and 100, we were told to consider all X,Y values between 1 and 100, the puzzle would not be solvable, as in this case there will be multiple possible solutions that would all be consistent with the five sentences above (see figure below; at Step 5 there still exist 6 different possible solutions to this puzzle).\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId3\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eSimilarly, if we were told to consider all X,Y values between 2 and 50, the puzzle would again not be solvable, as in this case there would be\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eno solution\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e consistent with all five sentences (perhaps surprisingly, since the X=4 Y=13 solution above is in fact within the stated range).\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId4\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eOn the other hand, if we were told to consider X and Y values ranging between 1 and 24, for example, the puzzle would again become solvable, now with a new unique solution X=1 and Y=6.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId5\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIn this problem you are tasked to create a function that would determine whether a particular variation of this puzzle would work or not. Specifically, given an initial set of possible (X,Y) pairs (entered as a Nx2 matrix and representing the full set of possible X,Y pairs that Scott and Priscilla are told to consider), you should determine whether it is possible to solve this puzzle (i.e. whether one, and only one, (X,Y) pair is consistent with the five sequential sentences above). Your function should simply return 1 (or true) if the puzzle is solvable, and 0 (or false) otherwise.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eGood luck!\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray 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