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## Copyright (C) 2009 Jaroslav Hajek <highegg@gmail.com>
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## This program is free software; you can redistribute it and/or modify it under
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## the terms of the GNU General Public License as published by the Free Software
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## Foundation; either version 3 of the License, or (at your option) any later
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## This program is distributed in the hope that it will be useful, but WITHOUT
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## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
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## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more
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## You should have received a copy of the GNU General Public License along with
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## this program; if not, see <http://www.gnu.org/licenses/>.
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## @deftypefn{Function File} [@var{x}, @var{ntrial}] = solvesudoku (@var{s})
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## Solves a classical 9x9 sudoku. @var{s} should be a 9x9 array with
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## numbers from 0:9. 0 indicates empty field.
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## Returns the filled table or empty matrix if no solution exists.
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## If requested, @var{ntrial} returns the number of trial-and-error steps needed.
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## This uses a recursive backtracking technique combined with revealing new singleton
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## fields by logic. The beauty of it is that it is completely vectorized.
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function [x, ntrial] = solvesudoku (s)
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if (! (ismatrix (s) && ndims (s) == 2 && all (size (s) == [9, 9])))
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error ("needs a 9x9 matrix");
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if (! ismember (unique (s(:)), 0:9))
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error ("matrix must contain values from 0:9");
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if (! verifysudoku (s))
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error ("matrix is not a valid sudoku grid");
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[x, ntrial] = solvesudoku_rec (s);
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function ok = verifysudoku (s)
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b(sub2ind ([9, 9, 9], i, j, k)) = true;
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okc = sum (b, 1) <= 1;
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okr = sum (b, 2) <= 1;
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b = reshape (b, [3, 3, 3, 3, 9]);
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ok3 = sum (sum (b, 1), 3) <= 1;
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ok = all (okc(:) & okr(:) & ok3(:));
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function [x, ntrial] = solvesudoku_rec (s)
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## Run until the logic is exhausted.
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x = getsingletons (b, x);
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finished = isempty (x) || all (x(:));
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until (finished || all ((x == s)(:)));
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## Find the field with minimum possibilities.
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[msb, i] = min (sb(:));
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[i, j] = ind2sub ([9, 9], i);
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for k = find (b(i,j,:))'
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[x, ntrial1] = solvesudoku_rec (s);
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ntrial += 1 + ntrial1;
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## Given a 9x9x9 logical array of allowed values, get the logical singletons.
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function s = getsingletons (b, s)
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## Check for fields with only one option.
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if (any (sb(:) == 0))
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## We want to return as soon as some new singletons are found.
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[s(s1), xx] = find (reshape (b, [], 9)(s1, :).');
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if (sum (s(:) != 0) > n0)
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## Check for columns where a number has only one field left.
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sb = squeeze (sum (b, 1));
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if (any (sb(:) == 0))
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[i, xx] = find (b(:, s1));
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s(sub2ind ([9, 9], i, j)) = k;
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if (sum (s(:) != 0) > n0)
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sb = squeeze (sum (b, 2));
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if (any (sb(:) == 0))
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[j, xx] = find (permute (b, [2, 1, 3])(:, s1));
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s(sub2ind ([9, 9], i, j)) = k;
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if (sum (s(:) != 0) > n0)
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bb = reshape (b, [3, 3, 3, 3, 9]);
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sb = squeeze (sum (sum (bb, 1), 3));
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if (any (sb(:) == 0))
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s1 = reshape (sb == 1, 9, 9);
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[i, xx] = find (reshape (permute (bb, [1, 3, 2, 4, 5]), 9, 9*9)(:, s1));
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[i1, i2] = ind2sub ([3, 3], i);
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[j1, j2] = ind2sub ([3, 3], j);
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s(sub2ind ([3, 3, 3, 3], i1, j1, i2, j2)) = k;
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if (sum (s(:) != 0) > n0)
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## Given known values (singletons), calculate options.
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function b = getoptions (s)
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[i, j, s] = find (s);
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bc(:, sub2ind ([9, 9], j, s)) = false;
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br(:, sub2ind ([9, 9], i, s)) = false;
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b3 = true (3, 3, 3, 3, 9);
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b3(:, :, sub2ind ([3, 3, 9], ceil (i/3), ceil (j/3), s)) = false;
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## Permute elements to correct order.
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br = permute (br, [2, 1, 3]);
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b3 = reshape (permute (b3, [1, 3, 2, 4, 5]), [9, 9, 9]);
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## The singleton fields themselves.
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bb(sub2ind ([9, 9], i, j), :) = false;
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bb = reshape (bb, [9, 9, 9]);
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b = bc & br & b3 & bb;
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## Correct singleton fields.
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b = reshape (b, 9, 9, 9);
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b(sub2ind ([9, 9, 9], i, j, s)) = true;
added
removed
inst/solvesudoku.m
