/* PENTOMINO, a geometric placement puzzle */
/* Written in GNU MathProg by Andrew Makhorin <mao@gnu.org> */
/* A pentomino is a plane geometric figure by joining five equal
squares edge to edge. It is a polyomino with five cells. Pentominoes
were defined by Prof. Solomon W. Golomb in his book "Polyominoes:
Puzzles, Patterns, Problems, and Packings."
There are twelve pentominoes, not counting rotations and reflections
as distinct:
+---+
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+---+ +---+ +---+
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+---+---+ +---+ +---+ +---+
| | | | | | | | |
+---+---+---+ +---+ +---+ +---+---+
| | | | | | | | | |
+---+---+ +---+ +---+---+ +---+---+
| | | | | | | | |
+---+ +---+ +---+---+ +---+
F I L N
+---+---+ +---+---+---+ +---+
| | | | | | | | |
+---+---+ +---+---+---+ +---+ +---+ +---+
| | | | | | | | | | |
+---+---+ +---+ +---+---+---+ +---+---+---+
| | | | | | | | | | | |
+---+ +---+ +---+---+---+ +---+---+---+
P T U V
+---+
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+---+ +---+ +---+---+ +---+---+
| | | | | | | | | |
+---+---+ +---+---+---+ +---+---+ +---+---+
| | | | | | | | | | |
+---+---+---+ +---+---+---+ +---+ +---+---+
| | | | | | | | | |
+---+---+ +---+ +---+ +---+---+
W X Y Z
A classic pentomino puzzle is to tile a given outline, i.e. cover
it without overlap and without gaps. Each of 12 pentominoes has an
area of 5 unit squares, so the outline must have area of 60 units.
Note that it is allowed to rotate and reflect the pentominoes.
(From Wikipedia, the free encyclopedia.) */
set A;
check card(A) = 12;
/* basic set of pentominoes */
set B{a in A};
/* B[a] is a set of distinct versions of pentomino a obtained by its
rotations and reflections */
set C := setof{a in A, b in B[a]} b;
check card(C) = 63;
/* set of distinct versions of all pentominoes */
set D{c in C}, within {0..4} cross {0..4};
/* D[c] is a set of squares (i,j), relative to (0,0), that constitute
a distinct version of pentomino c */
param m, default 6;
/* number of rows in the outline */
param n, default 10;
/* number of columns in the outline */
set R, default {1..m} cross {1..n};
/* set of squares (i,j), relative to (1,1), of the outline to be tiled
with the pentominoes */
check card(R) = 60;
/* the outline must have exactly 60 squares */
set S := setof{c in C, i in 1..m, j in 1..n:
forall{(ii,jj) in D[c]} ((i+ii,j+jj) in R)} (c,i,j);
/* set of all possible placements, where triplet (c,i,j) means that
the base square (0,0) of a distinct version of pentomino c is placed
at square (i,j) of the outline */
var x{(c,i,j) in S}, binary;
/* x[c,i,j] = 1 means that placement (c,i,j) is used in the tiling */
s.t. use{a in A}: sum{(c,i,j) in S: substr(c,1,1) = a} x[c,i,j] = 1;
/* every pentomino must be used exactly once */
s.t. cov{(i,j) in R}:
sum{(c,ii,jj) in S: (i-ii, j-jj) in D[c]} x[c,ii,jj] = 1;
/* every square of the outline must be covered exactly once */
/* this is a feasibility problem, so no objective is needed */
solve;
for {i in 1..m}
{ for {j in 1..n}
{ for {0..0: (i,j) in R}
{ for {(c,ii,jj) in S: (i-ii,j-jj) in D[c] and x[c,ii,jj]}
printf " %s", substr(c,1,1);
}
for {0..0: (i,j) not in R}
printf " .";
}
printf "\n";
}
data;
/* These data correspond to a puzzle from the book "Pentominoes" by
Jon Millington */
param m := 8;
param n := 15;
set R : 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 :=
1 - - - - - - - + - - - - - - -
2 - - - - - - + + + - - - - - -
3 - - - - - + + + + + - - - - -
4 - - - - + + + - + + + - - - -
5 - - - + + + + - + + + + - - -
6 - - + + + + + - + + + + + - -
7 - + + + + + + - + + + + + + -
8 + + + + + + + + + + + + + + + ;
/* DO NOT CHANGE ANY DATA BELOW! */
set A := F, I, L, N, P, T, U, V, W, X, Y, Z;
set B[F] := F1, F2, F3, F4, F5, F6, F7, F8;
set B[I] := I1, I2;
set B[L] := L1, L2, L3, L4, L5, L6, L7, L8;
set B[N] := N1, N2, N3, N4, N5, N6, N7, N8;
set B[P] := P1, P2, P3, P4, P5, P6, P7, P8;
set B[T] := T1, T2, T3, T4;
set B[U] := U1, U2, U3, U4;
set B[V] := V1, V2, V3, V4;
set B[W] := W1, W2, W3, W4;
set B[X] := X;
set B[Y] := Y1, Y2, Y3, Y4, Y5, Y6, Y7, Y8;
set B[Z] := Z1, Z2, Z3, Z4;
set D[F1] : 0 1 2 :=
0 - + +
1 + + -
2 - + - ;
set D[F2] : 0 1 2 :=
0 - + -
1 + + +
2 - - + ;
set D[F3] : 0 1 2 :=
0 - + -
1 - + +
2 + + - ;
set D[F4] : 0 1 2 :=
0 + - -
1 + + +
2 - + - ;
set D[F5] : 0 1 2 :=
0 + + -
1 - + +
2 - + - ;
set D[F6] : 0 1 2 :=
0 - - +
1 + + +
2 - + - ;
set D[F7] : 0 1 2 :=
0 - + -
1 + + -
2 - + + ;
set D[F8] : 0 1 2 :=
0 - + -
1 + + +
2 + - - ;
set D[I1] : 0 :=
0 +
1 +
2 +
3 +
4 + ;
set D[I2] : 0 1 2 3 4 :=
0 + + + + + ;
set D[L1] : 0 1 :=
0 + -
1 + -
2 + -
3 + + ;
set D[L2] : 0 1 2 3 :=
0 + + + +
1 + - - - ;
set D[L3] : 0 1 :=
0 + +
1 - +
2 - +
3 - + ;
set D[L4] : 0 1 2 3 :=
0 - - - +
1 + + + + ;
set D[L5] : 0 1 :=
0 - +
1 - +
2 - +
3 + + ;
set D[L6] : 0 1 2 3 :=
0 + - - -
1 + + + + ;
set D[L7] : 0 1 :=
0 + +
1 + -
2 + -
3 + - ;
set D[L8] : 0 1 2 3 :=
0 + + + +
1 - - - + ;
set D[N1] : 0 1 :=
0 + -
1 + -
2 + +
3 - + ;
set D[N2] : 0 1 2 3 :=
0 - + + +
1 + + - - ;
set D[N3] : 0 1 :=
0 + -
1 + +
2 - +
3 - + ;
set D[N4] : 0 1 2 3 :=
0 - - + +
1 + + + - ;
set D[N5] : 0 1 :=
0 - +
1 - +
2 + +
3 + - ;
set D[N6] : 0 1 2 3 :=
0 + + - -
1 - + + + ;
set D[N7] : 0 1 :=
0 - +
1 + +
2 + -
3 + - ;
set D[N8] : 0 1 2 3 :=
0 + + + -
1 - - + + ;
set D[P1] : 0 1 :=
0 + +
1 + +
2 + - ;
set D[P2] : 0 1 2 :=
0 + + +
1 - + + ;
set D[P3] : 0 1 :=
0 - +
1 + +
2 + + ;
set D[P4] : 0 1 2 :=
0 + + -
1 + + + ;
set D[P5] : 0 1 :=
0 + +
1 + +
2 - + ;
set D[P6] : 0 1 2 :=
0 - + +
1 + + + ;
set D[P7] : 0 1 :=
0 + -
1 + +
2 + + ;
set D[P8] : 0 1 2 :=
0 + + +
1 + + - ;
set D[T1] : 0 1 2 :=
0 + + +
1 - + -
2 - + - ;
set D[T2] : 0 1 2 :=
0 - - +
1 + + +
2 - - + ;
set D[T3] : 0 1 2 :=
0 - + -
1 - + -
2 + + + ;
set D[T4] : 0 1 2 :=
0 + - -
1 + + +
2 + - - ;
set D[U1] : 0 1 2 :=
0 + - +
1 + + + ;
set D[U2] : 0 1 :=
0 + +
1 + -
2 + + ;
set D[U3] : 0 1 2 :=
0 + + +
1 + - + ;
set D[U4] : 0 1 :=
0 + +
1 - +
2 + + ;
set D[V1] : 0 1 2 :=
0 - - +
1 - - +
2 + + + ;
set D[V2] : 0 1 2 :=
0 + - -
1 + - -
2 + + + ;
set D[V3] : 0 1 2 :=
0 + + +
1 + - -
2 + - - ;
set D[V4] : 0 1 2 :=
0 + + +
1 - - +
2 - - + ;
set D[W1] : 0 1 2 :=
0 - - +
1 - + +
2 + + - ;
set D[W2] : 0 1 2 :=
0 + - -
1 + + -
2 - + + ;
set D[W3] : 0 1 2 :=
0 - + +
1 + + -
2 + - - ;
set D[W4] : 0 1 2 :=
0 + + -
1 - + +
2 - - + ;
set D[X] : 0 1 2 :=
0 - + -
1 + + +
2 - + - ;
set D[Y1] : 0 1 :=
0 + -
1 + -
2 + +
3 + - ;
set D[Y2] : 0 1 2 3 :=
0 + + + +
1 - + - - ;
set D[Y3] : 0 1 :=
0 - +
1 + +
2 - +
3 - + ;
set D[Y4] : 0 1 2 3 :=
0 - - + -
1 + + + + ;
set D[Y5] : 0 1 :=
0 - +
1 - +
2 + +
3 - + ;
set D[Y6] : 0 1 2 3 :=
0 - + - -
1 + + + + ;
set D[Y7] : 0 1 :=
0 + -
1 + +
2 + -
3 + - ;
set D[Y8] : 0 1 2 3 :=
0 + + + +
1 - - + - ;
set D[Z1] : 0 1 2 :=
0 - + +
1 - + -
2 + + - ;
set D[Z2] : 0 1 2 :=
0 + - -
1 + + +
2 - - + ;
set D[Z3] : 0 1 2 :=
0 + + -
1 - + -
2 - + + ;
set D[Z4] : 0 1 2 :=
0 - - +
1 + + +
2 + - - ;
end;