Try our new documentation site.


sudoku.m


function sudoku(filename)

%  Copyright 2019, Gurobi Optimization, LLC */
%
% Sudoku example.
%
% The Sudoku board is a 9x9 grid, which is further divided into a 3x3 grid
% of 3x3 grids.  Each cell in the grid must take a value from 0 to 9.
% No two grid cells in the same row, column, or 3x3 subgrid may take the
% same value.
%
% In the MIP formulation, binary variables x[i,j,v] indicate whether
% cell <i,j> takes value 'v'.  The constraints are as follows:
%   1. Each cell must take exactly one value (sum_v x[i,j,v] = 1)
%   2. Each value is used exactly once per row (sum_i x[i,j,v] = 1)
%   3. Each value is used exactly once per column (sum_j x[i,j,v] = 1)
%   4. Each value is used exactly once per 3x3 subgrid (sum_grid x[i,j,v] = 1)
%
% Input datasets for this example can be found in examples/data/sudoku*.
%

SUBDIM = 3;
DIM    = SUBDIM*SUBDIM;

fileID = fopen(filename);
if fileID == -1
    fprintf('Could not read file %s, quit\n', filename);
    return;
end

board = repmat(-1, DIM, DIM);
for i = 1:DIM
    s = fgets(fileID, 100);
    if length(s) <= DIM
        fprintf('Error: not enough board positions specified, quit\n');
        return;
    end
    for j = 1:DIM
        if s(j) ~= '.'
            board(i, j) = str2double(s(j));
            if board(i,j) < 1 || board(i,j) > DIM
                fprintf('Error: Unexpected character in Input line %d, quit\n', i);
                return;
            end
        end
    end
end

% Map X(i,j,k) into an index variable in the model
nVars = DIM * DIM * DIM;


% Build model
model.vtype    = repmat('B', nVars, 1);
model.lb       = zeros(nVars, 1);
model.ub       = ones(nVars, 1);

for i = 1:DIM
    for j = 1:DIM
        for v = 1:DIM
            var = (i-1)*DIM*DIM + (j-1)*DIM + v;
            model.varnames{var} = sprintf('x[%d,%d,%d]', i, j, v);
        end
    end
end

% Create constraints:
nRows = 4 * DIM * DIM;
model.A = sparse(nRows, nVars);
model.rhs = ones(nRows, 1);
model.sense = repmat('=', nRows, 1);

Row = 1;

% Each cell gets a value */
for i = 1:DIM
    for j = 1:DIM
        for v = 1:DIM
            if board(i,j) == v
                model.lb((i-1)*DIM*DIM + (j-1)*DIM + v) = 1;
            end
            model.A(Row, (i-1)*DIM*DIM + (j-1)*DIM + v) = 1;
        end
        Row = Row + 1;
    end
end

% Each value must appear once in each row
for v = 1:DIM
    for j = 1:DIM
        for i = 1:DIM
            model.A(Row, (i-1)*DIM*DIM + (j-1)*DIM + v) = 1;
        end
        Row = Row + 1;
    end
end

% Each value must appear once in each column
for v = 1:DIM
    for i = 1:DIM
        for j = 1:DIM
            model.A(Row, (i-1)*DIM*DIM + (j-1)*DIM + v) = 1;
        end
        Row = Row + 1;
    end
end

% Each value must appear once in each subgrid
for v = 1:DIM
    for ig = 0: SUBDIM-1
        for jg = 0: SUBDIM-1
            for i = ig*SUBDIM+1:(ig+1)*SUBDIM
                for j = jg*SUBDIM+1:(jg+1)*SUBDIM
                    model.A(Row, (i-1)*DIM*DIM + (j-1)*DIM + v) = 1;
                end
            end
            Row = Row + 1;
        end
    end
end

% Save model
gurobi_write(model, 'sudoku_m.lp');

% Optimize model
params.logfile = 'sudoku_m.log';
result = gurobi(model, params);

if strcmp(result.status, 'OPTIMAL')
    fprintf('Solution:\n');
    for i = 1:DIM
        for j = 1:DIM
            for v = 1:DIM
                var = (i-1)*DIM*DIM + (j-1)*DIM + v;
                if result.x(var) > 0.99
                    fprintf('%d', v);
                end
            end
        end
        fprintf('\n');
    end
else
    fprintf('Problem was infeasible\n')
end

Try Gurobi for Free

Choose the evaluation license that fits you best, and start working with our Expert Team for technical guidance and support.

Evaluation License
Get a free, full-featured license of the Gurobi Optimizer to experience the performance, support, benchmarking and tuning services we provide as part of our product offering.
Academic License
Gurobi supports the teaching and use of optimization within academic institutions. We offer free, full-featured copies of Gurobi for use in class, and for research.
Cloud Trial

Request free trial hours, so you can see how quickly and easily a model can be solved on the cloud.

Search