# 通过整数规划求解数独谜题：基于求解器

### 初始谜题

```B = [1,2,2; 1,5,3; 1,8,4; 2,1,6; 2,9,3; 3,3,4; 3,7,5; 4,4,8; 4,6,6; 5,1,8; 5,5,1; 5,9,6; 6,4,7; 6,6,5; 7,3,7; 7,7,6; 8,1,4; 8,9,8; 9,2,3; 9,5,4; 9,8,2]; drawSudoku(B) % For the listing of this program, see the end of this example.```

2009 年发布的 Cleve's Corner 中介绍了此谜题以及一种备选的 MATLAB® 求解方法。

### 将数独规则表示为约束

`$\sum _{k=1}^{9}x\left(i,j,k\right)=1.$`

`$\sum _{j=1}^{9}x\left(i,j,k\right)=1.$`

`$\sum _{i=1}^{9}x\left(i,j,k\right)=1.$`

3×3 粗线网格具有类似的约束。对于 $1\le i\le 3$$1\le j\le 3$ 的网格元素，对于每层 $1\le k\le 9$ 都满足：

`$\sum _{i=1}^{3}\sum _{j=1}^{3}x\left(i,j,k\right)=1.$`

$\sum _{i=1}^{3}\sum _{j=1}^{3}x\left(i+U,j+V,k\right)=1,$ 其中 $U,V\phantom{\rule{0.2em}{0ex}}ϵ\phantom{\rule{0.2em}{0ex}}\left\{0,3,6\right\}.$

### 编写数独的规则

`type sudokuEngine`
```function [S,eflag] = sudokuEngine(B) % This function sets up the rules for Sudoku. It reads in the puzzle % expressed in matrix B, calls intlinprog to solve the puzzle, and returns % the solution in matrix S. % % The matrix B should have 3 columns and at least 17 rows (because a Sudoku % puzzle needs at least 17 entries to be uniquely solvable). The first two % elements in each row are the i,j coordinates of a clue, and the third % element is the value of the clue, an integer from 1 to 9. If B is a % 9-by-9 matrix, the function first converts it to 3-column form. % Copyright 2014 The MathWorks, Inc. if isequal(size(B),[9,9]) % 9-by-9 clues % Convert to 81-by-3 [SM,SN] = meshgrid(1:9); % make i,j entries B = [SN(:),SM(:),B(:)]; % i,j,k rows % Now delete zero rows [rrem,~] = find(B(:,3) == 0); B(rrem,:) = []; end if size(B,2) ~= 3 || length(size(B)) > 2 error('The input matrix must be N-by-3 or 9-by-9') end if sum([any(B ~= round(B)),any(B < 1),any(B > 9)]) % enforces entries 1-9 error('Entries must be integers from 1 to 9') end %% The rules of Sudoku: N = 9^3; % number of independent variables in x, a 9-by-9-by-9 array M = 4*9^2; % number of constraints, see the construction of Aeq Aeq = zeros(M,N); % allocate equality constraint matrix Aeq*x = beq beq = ones(M,1); % allocate constant vector beq f = (1:N)'; % the objective can be anything, but having nonconstant f can speed the solver lb = zeros(9,9,9); % an initial zero array ub = lb+1; % upper bound array to give binary variables counter = 1; for j = 1:9 % one in each row for k = 1:9 Astuff = lb; % clear Astuff Astuff(1:end,j,k) = 1; % one row in Aeq*x = beq Aeq(counter,:) = Astuff(:)'; % put Astuff in a row of Aeq counter = counter + 1; end end for i = 1:9 % one in each column for k = 1:9 Astuff = lb; Astuff(i,1:end,k) = 1; Aeq(counter,:) = Astuff(:)'; counter = counter + 1; end end for U = 0:3:6 % one in each square for V = 0:3:6 for k = 1:9 Astuff = lb; Astuff(U+(1:3),V+(1:3),k) = 1; Aeq(counter,:) = Astuff(:)'; counter = counter + 1; end end end for i = 1:9 % one in each depth for j = 1:9 Astuff = lb; Astuff(i,j,1:end) = 1; Aeq(counter,:) = Astuff(:)'; counter = counter + 1; end end %% Put the particular puzzle in the constraints % Include the initial clues in the |lb| array by setting corresponding % entries to 1. This forces the solution to have |x(i,j,k) = 1|. for i = 1:size(B,1) lb(B(i,1),B(i,2),B(i,3)) = 1; end %% Solve the Puzzle % The Sudoku problem is complete: the rules are represented in the |Aeq| % and |beq| matrices, and the clues are ones in the |lb| array. Solve the % problem by calling |intlinprog|. Ensure that the integer program has all % binary variables by setting the intcon argument to |1:N|, with lower and % upper bounds of 0 and 1. intcon = 1:N; [x,~,eflag] = intlinprog(f,intcon,[],[],Aeq,beq,lb,ub); %% Convert the Solution to a Usable Form % To go from the solution x to a Sudoku grid, simply add up the numbers at % each \$(i,j)\$ entry, multiplied by the depth at which the numbers appear: if eflag > 0 % good solution x = reshape(x,9,9,9); % change back to a 9-by-9-by-9 array x = round(x); % clean up non-integer solutions y = ones(size(x)); for k = 2:9 y(:,:,k) = k; % multiplier for each depth k end S = x.*y; % multiply each entry by its depth S = sum(S,3); % S is 9-by-9 and holds the solved puzzle else S = []; end ```

### 调用数独求解器

`S = sudokuEngine(B); % Solves the puzzle pictured at the start`
```LP: Optimal objective value is 29565.000000. Optimal solution found. Intlinprog stopped at the root node because the objective value is within a gap tolerance of the optimal value, options.AbsoluteGapTolerance = 0. The intcon variables are integer within tolerance, options.IntegerTolerance = 1e-05. ```
`drawSudoku(S)`

### 用于绘制数独谜题的函数

`type drawSudoku`
```function drawSudoku(B) % Function for drawing the Sudoku board % Copyright 2014 The MathWorks, Inc. figure;hold on;axis off;axis equal % prepare to draw rectangle('Position',[0 0 9 9],'LineWidth',3,'Clipping','off') % outside border rectangle('Position',[3,0,3,9],'LineWidth',2) % heavy vertical lines rectangle('Position',[0,3,9,3],'LineWidth',2) % heavy horizontal lines rectangle('Position',[0,1,9,1],'LineWidth',1) % minor horizontal lines rectangle('Position',[0,4,9,1],'LineWidth',1) rectangle('Position',[0,7,9,1],'LineWidth',1) rectangle('Position',[1,0,1,9],'LineWidth',1) % minor vertical lines rectangle('Position',[4,0,1,9],'LineWidth',1) rectangle('Position',[7,0,1,9],'LineWidth',1) % Fill in the clues % % The rows of B are of the form (i,j,k) where i is the row counting from % the top, j is the column, and k is the clue. To place the entries in the % boxes, j is the horizontal distance, 10-i is the vertical distance, and % we subtract 0.5 to center the clue in the box. % % If B is a 9-by-9 matrix, convert it to 3 columns first if size(B,2) == 9 % 9 columns [SM,SN] = meshgrid(1:9); % make i,j entries B = [SN(:),SM(:),B(:)]; % i,j,k rows end for ii = 1:size(B,1) text(B(ii,2)-0.5,9.5-B(ii,1),num2str(B(ii,3))) end hold off end ```