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+/* 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:
+
+ +---+
+ | |
+ +---+ +---+ +---+
+ | | | | | |
+ +---+---+ +---+ +---+ +---+
+ | | | | | | | | |
+ +---+---+---+ +---+ +---+ +---+---+
+ | | | | | | | | | |
+ +---+---+ +---+ +---+---+ +---+---+
+ | | | | | | | | |
+ +---+ +---+ +---+---+ +---+
+ F I L N
+
+ +---+---+ +---+---+---+ +---+
+ | | | | | | | | |
+ +---+---+ +---+---+---+ +---+ +---+ +---+
+ | | | | | | | | | | |
+ +---+---+ +---+ +---+---+---+ +---+---+---+
+ | | | | | | | | | | | |
+ +---+ +---+ +---+---+---+ +---+---+---+
+ P T U V
+
+ +---+
+ | |
+ +---+ +---+ +---+---+ +---+---+
+ | | | | | | | | | |
+ +---+---+ +---+---+---+ +---+---+ +---+---+
+ | | | | | | | | | | |
+ +---+---+---+ +---+---+---+ +---+ +---+---+
+ | | | | | | | | | |
+ +---+---+ +---+ +---+ +---+---+
+ 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;
generated by cgit v1.2.1 (git 2.18.0) at 2025-05-04 05:20:39 +0000