24 Game Solver

Back in college, I was taught how to play this little math game by a Chinese friend of mine. She told me that they used to play this game quite regularly as kids. As a group of four, we would each plunk down a randomly generated card to have four numbers that may be manipulated in any way using +, *, -, or ÷. For example let's say the cards show 1, 5, 7 and12. One possible solution is 1*(7-5)*12. Recently, I thought it would be fun to write a Matlab program that would find all possible solutions to a group of four numbers. My solution uses a series of nested loops to test for different ordering of the vector in question, different operation combinations, and operation groupings using parentheses. I tried to make the code more efficient by trimming repeated vectors. For example, if the test vector is [10 10 4 4], there are only six unique permutations, instead of the expected 24 (4!). I would love to see if anyone out there can make this code more efficient.

I wanted to test my code to see if it found all possible solutions. For a deck of 13 cards, it turns out that there are nchoosek(16,4) four card combinations, or 1820. From the Wikipedia page, I saw that Cheng Chang published all distinct solutions to the puzzle at http://www.ocf.berkeley.edu/~chechang/24/puzzle_sol.html. Initial versions of the program did not find all solutions. For example it missed the obvious [1 1 2 7], whose solution could simply be (1+2)*(1+7). I added different parentheses combinations hoping to catch more of the solutions. Finally, the algorithm was able to catch all the solutions except for one: [3 3 8 8]. The solution turned out to be tricky: 8/(3-8/3)=24. After eliminating a rounding error, the algorithm now finds all the solutions published on Cheng's page.