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alpar@9
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1 /* QUEENS, a classic combinatorial optimization problem */
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2
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alpar@9
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3 /* Written in GNU MathProg by Andrew Makhorin <mao@gnu.org> */
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4
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alpar@9
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5 /* The Queens Problem is to place as many queens as possible on the 8x8
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6 (or more generally, nxn) chess board in a way that they do not fight
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7 each other. This problem is probably as old as the chess game itself,
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8 and thus its origin is not known, but it is known that Gauss studied
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9 this problem. */
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10
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11 param n, integer, > 0, default 8;
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12 /* size of the chess board */
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13
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14 var x{1..n, 1..n}, binary;
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15 /* x[i,j] = 1 means that a queen is placed in square [i,j] */
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16
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17 s.t. a{i in 1..n}: sum{j in 1..n} x[i,j] <= 1;
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18 /* at most one queen can be placed in each row */
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19
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20 s.t. b{j in 1..n}: sum{i in 1..n} x[i,j] <= 1;
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21 /* at most one queen can be placed in each column */
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22
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23 s.t. c{k in 2-n..n-2}: sum{i in 1..n, j in 1..n: i-j == k} x[i,j] <= 1;
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24 /* at most one queen can be placed in each "\"-diagonal */
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25
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26 s.t. d{k in 3..n+n-1}: sum{i in 1..n, j in 1..n: i+j == k} x[i,j] <= 1;
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27 /* at most one queen can be placed in each "/"-diagonal */
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28
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29 maximize obj: sum{i in 1..n, j in 1..n} x[i,j];
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30 /* objective is to place as many queens as possible */
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31
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32 /* solve the problem */
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33 solve;
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34
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35 /* and print its optimal solution */
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36 for {i in 1..n}
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37 { for {j in 1..n} printf " %s", if x[i,j] then "Q" else ".";
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38 printf("\n");
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39 }
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40
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41 end;
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