# Solve Sudoku Puzzles Via Integer Programming: Problem-Based

This example shows how to solve a Sudoku puzzle using binary integer programming. For the solver-based approach, see Solve Sudoku Puzzles Via Integer Programming: Solver-Based.

You probably have seen Sudoku puzzles. A puzzle is to fill a 9-by-9 grid with integers from 1 through 9 so that each integer appears only once in each row, column, and major 3-by-3 square. The grid is partially populated with clues, and your task is to fill in the rest of the grid.

### Initial Puzzle

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 MATLAB produces the solution.

### Binary Integer Programming Approach

The key idea is to transform a puzzle from a square 9-by-9 grid to a cubic 9-by-9-by-9 array of binary values (0 or 1). Think of the cubic array as being 9 square grids stacked on top of each other, where each layer corresponds to an integer from 1 through 9. 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 a constant term 0. The problem is really just to find a feasible solution, meaning one that satisfies all the constraints. However, for tie breaking 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 $x$ is represented in a 9-by-9-by-9 binary array. What properties does $x$ have? First, each square in the 2-D grid (i,j) has exactly one value, so there is exactly one nonzero element among the 3-D array entries $x\left(i,j,1\right),...,x\left(i,j,9\right)$. In other words, for every $i$ and $j$,

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

Similarly, in each row $i$ of the 2-D grid, there is exactly one value out of each of the digits from 1 to 9. In other words, for each $i$ and $k$,

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

And each column $j$ in the 2-D grid has the same property: for each $j$ and $k$,

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

The major 3-by-3 grids have a similar constraint. For the grid elements $1\le i\le 3$ and $1\le j\le 3$, and for each $1\le k\le 9$,

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

To represent all nine major grids, just add 3 or 6 to each $i$ and $j$ index:

$\sum _{i=1}^{3}\sum _{j=1}^{3}x\left(i+U,j+V,k\right)=1,$ where $U,V\phantom{\rule{0.5em}{0ex}}ϵ\phantom{\rule{0.5em}{0ex}}\left\{0,3,6\right\}.$

### Express Clues

Each initial value (clue) can be expressed as a constraint. Suppose that the $\left(i,j\right)$ clue is $m$ for some $1\le m\le 9$. Then $x\left(i,j,m\right)=1$. The constraint $\sum _{k=1}^{9}x\left(i,j,k\right)=1$ ensures that all other $x\left(i,j,k\right)=0$ for $k\ne m$.

### Sudoku in Optimization Problem Form

Create an optimization variable x that is binary and of size 9-by-9-by-9.

x = optimvar('x',9,9,9,'Type','integer','LowerBound',0,'UpperBound',1);

Create an optimization problem with a rather arbitrary objective function. The objective function can help the solver by destroying the inherent symmetry of the problem.

sudpuzzle = optimproblem;
mul = ones(1,1,9);
mul = cumsum(mul,3);
sudpuzzle.Objective = sum(sum(sum(x,1),2).*mul);

Represent the constraints that the sums of x in each coordinate direction are one.

sudpuzzle.Constraints.consx = sum(x,1) == 1;
sudpuzzle.Constraints.consy = sum(x,2) == 1;
sudpuzzle.Constraints.consz = sum(x,3) == 1;

Create the constraints that the sums of the major grids sum to one as well.

majorg = optimconstr(3,3,9);

for u = 1:3
for v = 1:3
arr = x(3*(u-1)+1:3*(u-1)+3,3*(v-1)+1:3*(v-1)+3,:);
majorg(u,v,:) = sum(sum(arr,1),2) == ones(1,1,9);
end
end
sudpuzzle.Constraints.majorg = majorg;

Include the initial clues by setting lower bounds of 1 at the clue entries. This setting fixes the value of the corresponding entry to be 1, and so sets the solution at each clued value to be the clue entry.

for u = 1:size(B,1)
x.LowerBound(B(u,1),B(u,2),B(u,3)) = 1;
end

Solve the Sudoku puzzle.

sudsoln = solve(sudpuzzle)
Solving problem using intlinprog.
LP:                Optimal objective value is 405.000000.

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 default
value). The intcon variables are integer within tolerance,
options.IntegerTolerance = 1e-05 (the default value).
sudsoln = struct with fields:
x: [9x9x9 double]

Round the solution to ensure that all entries are integers, and display the solution.

sudsoln.x = round(sudsoln.x);

y = ones(size(sudsoln.x));
for k = 2:9
y(:,:,k) = k; % multiplier for each depth k
end
S = sudsoln.x.*y; % multiply each entry by its depth
S = sum(S,3); % S is 9-by-9 and holds the solved puzzle
drawSudoku(S)

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, 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