alpar@1: /* QUEENS, a classic combinatorial optimization problem */
alpar@1: 
alpar@1: /* Written in GNU MathProg by Andrew Makhorin <mao@gnu.org> */
alpar@1: 
alpar@1: /* The Queens Problem is to place as many queens as possible on the 8x8
alpar@1:    (or more generally, nxn) chess board in a way that they do not fight
alpar@1:    each other. This problem is probably as old as the chess game itself,
alpar@1:    and thus its origin is not known, but it is known that Gauss studied
alpar@1:    this problem. */
alpar@1: 
alpar@1: param n, integer, > 0, default 8;
alpar@1: /* size of the chess board */
alpar@1: 
alpar@1: var x{1..n, 1..n}, binary;
alpar@1: /* x[i,j] = 1 means that a queen is placed in square [i,j] */
alpar@1: 
alpar@1: s.t. a{i in 1..n}: sum{j in 1..n} x[i,j] <= 1;
alpar@1: /* at most one queen can be placed in each row */
alpar@1: 
alpar@1: s.t. b{j in 1..n}: sum{i in 1..n} x[i,j] <= 1;
alpar@1: /* at most one queen can be placed in each column */
alpar@1: 
alpar@1: 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;
alpar@1: /* at most one queen can be placed in each "\"-diagonal */
alpar@1: 
alpar@1: 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;
alpar@1: /* at most one queen can be placed in each "/"-diagonal */
alpar@1: 
alpar@1: maximize obj: sum{i in 1..n, j in 1..n} x[i,j];
alpar@1: /* objective is to place as many queens as possible */
alpar@1: 
alpar@1: /* solve the problem */
alpar@1: solve;
alpar@1: 
alpar@1: /* and print its optimal solution */
alpar@1: for {i in 1..n}
alpar@1: {  for {j in 1..n} printf " %s", if x[i,j] then "Q" else ".";
alpar@1:    printf("\n");
alpar@1: }
alpar@1: 
alpar@1: end;