# 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 $$x(i,j,1),...,x(i,j,9)$$. In other words, for every $$i$$ and $$j$$,

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

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 $$k$$,

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

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

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

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 $$1\le k\le 9$$,

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

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

$$\sum _{i=1}^{3}\sum _{j=1}^{3}x(i+U,j+V,k)=1,$$ where $$U,V\phantom{\rule{0.2em}{0ex}}\u03f5\phantom{\rule{0.2em}{0ex}}\{0,3,6\}.$$

### Express Clues

Each initial value (clue) can be expressed as a constraint. Suppose that the $$(i,j)$$ clue is $$m$$ for some $$1\le m\le 9$$. Then $$x(i,j,m)=1$$. The constraint $$\sum _{k=1}^{9}x(i,j,k)=1$$ ensures that all other $$x(i,j,k)=0$$ 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. Running HiGHS 1.7.0: Copyright (c) 2024 HiGHS under MIT licence terms Coefficient ranges: Matrix [1e+00, 1e+00] Cost [1e+00, 9e+00] Bound [1e+00, 1e+00] RHS [1e+00, 1e+00] Presolving model 221 rows, 246 cols, 990 nonzeros 0s 42 rows, 28 cols, 192 nonzeros 0s 22 rows, 18 cols, 86 nonzeros 0s Objective function is integral with scale 1 Solving MIP model with: 22 rows 18 cols (18 binary, 0 integer, 0 implied int., 0 continuous) 86 nonzeros Nodes | B&B Tree | Objective Bounds | Dynamic Constraints | Work Proc. InQueue | Leaves Expl. | BestBound BestSol Gap | Cuts InLp Confl. | LpIters Time 0 0 0 0.00% 372 inf inf 0 0 0 0 0.0s 0 0 0 0.00% 405 inf inf 0 0 2 7 0.0s Solving report Status Optimal Primal bound 405 Dual bound 405 Gap 0% (tolerance: 0.01%) Solution status feasible 405 (objective) 0 (bound viol.) 0 (int. viol.) 0 (row viol.) Timing 0.01 (total) 0.01 (presolve) 0.00 (postsolve) Nodes 1 LP iterations 7 (total) 0 (strong br.) 0 (separation) 0 (heuristics) Optimal solution found. Intlinprog stopped at the root node because the objective value is within a gap tolerance of the optimal value, options.AbsoluteGapTolerance = 1e-06. The intcon variables are integer within tolerance, options.ConstraintTolerance = 1e-06.

`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