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