alpar@1: /* Numbrix, Number Placement Puzzle */
alpar@1: 
alpar@1: /* Written in GNU MathProg by Robert Wood <rwood@targus.com>  */
alpar@1: 
alpar@1: /* Numbrix is a logic-based number-placement puzzle.[1]
alpar@1:  * The objective is to fill the grid so that each cell contains
alpar@1:  * digits in sequential order taking a horizontal or vertical
alpar@1:  * path; diagonal paths are not allowed. The puzzle setter
alpar@1:  * provides a grid often with the outer most cells completed.
alpar@1:  *
alpar@1:  * Completed Numbrix puzzles are usually a square of numbers
alpar@1:  * in order from 1 to 64 (8x8 grid) or from 1 to 81 (9x9 grid),
alpar@1:  * following a continuous path in sequence.
alpar@1:  *
alpar@1:  * The modern puzzle was invented by Marilyn vos Savant in 2008
alpar@1:  * and published by Parade Magazine under the name "Numbrix",
alpar@1:  * near her weekly Ask Marilyn article.
alpar@1:  *
alpar@1:  *    http://en.wikipedia.org/wiki/Numbrix  */
alpar@1: 
alpar@1: set I := {1..9};
alpar@1: set J := {1..9};
alpar@1: set VALS := {1..81};
alpar@1: 
alpar@1: param givens{I, J}, integer, >= 0, <= 81, default 0;
alpar@1: /* the "givens" */
alpar@1: 
alpar@1: param neighbors{i in I,j in J, i2 in I, j2 in J} , binary :=
alpar@1: (if abs(i - i2) + abs(j -j2) == 1 then
alpar@1:      1
alpar@1:  else
alpar@1:      0
alpar@1: );
alpar@1: /*  defines which spots are the boards are neighbors */
alpar@1: 
alpar@1: var x{i in I, j in J, k in VALS}, binary;
alpar@1: /* x[i,j,k] = 1 means cell [i,j] is assigned number k */
alpar@1: 
alpar@1: s.t. fa{i in I, j in J, k in VALS: givens[i,j] != 0}:
alpar@1:      x[i,j,k] = (if givens[i,j] = k then 1 else 0);
alpar@1: /* assign pre-defined numbers using the "givens" */
alpar@1: 
alpar@1: s.t. fb{i in I, j in J}: sum{k in VALS} x[i,j,k] = 1;
alpar@1: /* each cell must be assigned exactly one number */
alpar@1: 
alpar@1: s.t. singleNum {k in VALS}:  sum{i in I, j in J} x[i,j,k] = 1;
alpar@1: /*  a value can only occur once */
alpar@1: 
alpar@1: s.t. neighborContraint {i in I, j in J, k in 1..80}:
alpar@1:         x[i,j,k] <= sum{i2 in I, j2 in J} x[i2,j2,k+1] * neighbors[i,j,i2,j2];
alpar@1: /* each cell must have a neighbor with the next higher value */
alpar@1: 
alpar@1: 
alpar@1: /* there is no need for an objective function here */
alpar@1: 
alpar@1: 
alpar@1: solve;
alpar@1: 
alpar@1: for {i in I}
alpar@1: {  for {0..0: i = 1 or i = 4 or i = 7}
alpar@1:       printf " +----------+----------+----------+\n";
alpar@1:    for {j in J}
alpar@1:    {  for {0..0: j = 1 or j = 4 or j = 7} printf(" |");
alpar@1:       printf " %2d", sum{k in VALS} x[i,j,k] * k;
alpar@1:       for {0..0: j = 9} printf(" |\n");
alpar@1:    }
alpar@1:    for {0..0: i = 9}
alpar@1:       printf " +----------+----------+----------+\n";
alpar@1: }
alpar@1: 
alpar@1: data;
alpar@1: 
alpar@1: param givens : 1 2 3 4 5 6 7 8 9 :=
alpar@1:            1   . . . . . . . . .
alpar@1:            2   . 11 12 15 18 21 62 61 .
alpar@1:            3   .  6 . . . . . 60 .
alpar@1:            4   . 33 . . . . . 57 .
alpar@1:            5   . 32 . . . . . 56 .
alpar@1:            6   . 37 . . . . . 73 .
alpar@1:            7   . 38 . . . . . 72 .
alpar@1:            8   . 43 44 47 48 51 76 77 .
alpar@1:            9   . . . . . . . . . ;
alpar@1: 
alpar@1: end;