Optimization Toolbox 

This example shows how to solve a Sudoku puzzle using binary integer programming.
You probably have seen Sudoku puzzles. A puzzle is to fill a 9by9 grid with integers from 1 through 9 so that each integer appears only once in each row, column, and major 3by3 square. The grid is partially populated with clues, and your task is to fill in the rest of the grid.
On this page… 

Binary Integer Programming Approach 
Here is a data matrix B of clues. The first row, B(1,2,2), means row 1, column 2 has a clue 2. The second row, B(1,5,3), means row 1, column 5 has a clue 3. Here is the entire matrix B.
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.
This puzzle, and an alternative MATLAB® solution technique, was featured in Cleve's Corner in 2009.
There are many approaches to solving Sudoku puzzles manually, as well as many programmatic approaches. This example shows a straightforward approach using binary integer programming.
This approach is particularly simple because you do not give a solution algorithm. Just express the rules of Sudoku, express the clues as constraints on the solution, and then intlinprog produces the solution.
Binary Integer Programming Approach
The key idea is to transform a puzzle from a square 9by9 grid to a cubic 9by9by9 array of binary values (0 or 1). Think of the cubic array as being 9 square grids stacked on top of each other. The top grid, a square layer of the array, has a 1 wherever the solution or clue has a 1. The second layer has a 1 wherever the solution or clue has a 2. The ninth layer has a 1 wherever the solution or clue has a 9.
This formulation is precisely suited for binary integer programming.
The objective function is not needed here, and might as well be 0. The problem is really just to find a feasible solution, meaning one that satisfies all the constraints. However, for tiebreaking in the internals of the integer programming solver, giving increased solution speed, use a nonconstant objective function.
Express the Rules for Sudoku as Constraints
Suppose a solution is represented in a 9by9by9 binary array. What properties does have? First, each square in the 2D grid (i,j) has exactly one value, so there is exactly one nonzero element among the 3D array entries . In other words, for every and ,
Similarly, in each row of the 2D grid, there is exactly one value out of each of the digits from 1 to 9. In other words, for each and ,
And each column in the 2D grid has the same property: for each and ,
The major 3by3 grids have a similar constraint. For the grid elements and , and for each ,
To represent all nine major grids, just add 3 or 6 to each and index:
where
Each initial value (clue) can be expressed as a constraint. Suppose that the clue is for some . Then . The constraint ensures that all other for .
Although the Sudoku rules are conveniently expressed in terms of a 9by9by9 solution array x, linear constraints are given in terms of a vector solution matrix x(:). Therefore, when you write a Sudoku program, you have to use constraint matrices derived from 9by9by9 initial arrays.
Here is one approach to set up Sudoku rules, and also include the clues as constraints. The sudokuEngine file comes with your software.
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 % 9by9 matrix, the function first converts it to 3column form. % Copyright 2014 The MathWorks, Inc. if isequal(size(B),[9,9]) % 9by9 clues % Convert to 81by3 [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 Nby3 or 9by9') end if sum([any(B ~= round(B)),any(B < 1),any(B > 9)]) % enforces entries 19 error('Entries must be integers from 1 to 9') end %% The rules of Sudoku: N = 9^3; % number of independent variables in x, a 9by9by9 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 9by9by9 array x = round(x); % clean up noninteger 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 9by9 and holds the solved puzzle else S = []; end
S = sudokuEngine(B); % Solves the puzzle pictured at the start
drawSudoku(S)
LP: Optimal objective value is 29565.000000. Cut Generation: Applied 2 strong CG cuts, and 1 zerohalf cut. Lower bound is 29565.000000. Relative gap is 0.00%. Optimal solution found. Intlinprog stopped at the root node because the objective value is within a gap tolerance of the optimal value; options.TolGapAbs = 0 (the default value). The intcon variables are integer within tolerance, options.TolInteger = 1e05 (the default value).
You can easily check that the solution is correct.
Function to Draw the Sudoku Puzzle
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, 10i is the vertical distance, and % we subtract 0.5 to center the clue in the box. % % If B is a 9by9 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)0.5,9.5B(ii,1),num2str(B(ii,3))) end hold off end