Question

Hi there I need help writing an algorithm to solve Sudoku puzzles through integer linear programming using MATLAB.

If you could please provide the code and an explanation of how it works I'd be really grateful.

Any assistance would be much appreciated, many thanks in advance!

Answer 1

Solving Sudoku Using Recursive Backtracking

function X = sudoku(X) 
% SUDOKU  Solve Sudoku using recursive backtracking. 
%   sudoku(X), expects a 9-by-9 array X. 
 % Fill in all “singletons”. 
 % C is a cell array of candidate vectors for each cell. 
 % s is the first cell, if any, with one candidate. 
 % e is the first cell, if any, with no candidates. 
 [C,s,e] = candidates(X); 
 while ~isempty(s) && isempty(e) 
    X(s) = C{s}; 
    [C,s,e] = candidates(X); 
 end 
 % Return for impossible puzzles. 
 if ~isempty(e) 
    return 
 end 
 % Recursive backtracking. 
 if any(X(:) == 0) 
    Y = X; 
    z = find(X(:) == 0,1);    % The first unfilled cell. 
    for r = [C{z}]            % Iterate over candidates. 
       X = Y; 
       X(z) = r;              % Insert a tentative value. 
       X = sudoku(X);         % Recursive call. 
       if all(X(:) > 0)       % Found a solution. 
          return 
       end 
     end 
   end 
% ------------------------------ 
   function [C,s,e] = candidates(X) 
      C = cell(9,9); 
      tri = @(k) 3*ceil(k/3-1) + (1:3); 
      for j = 1:9 
         for i = 1:9 
            if X(i,j)==0 
               z = 1:9; 
               z(nonzeros(X(i,:))) = 0; 
               z(nonzeros(X(:,j))) = 0; 
               z(nonzeros(X(tri(i),tri(j)))) = 0; 
               C{i,j} = nonzeros(z)’; 
            end 
         end 
      end 
 L = cellfun(@length,C);   % Number of candidates. 
 s = find(X==0 & L==1,1); 
 e = find(X==0 & L==0,1); 
 end % candidates 
end % sudoku