Jip J. Dekker
f2a1c4e389
git-subtree-dir: software/mza git-subtree-split: f970a59b177c13ca3dd8aaef8cc6681d83b7e813
67 lines
2.0 KiB
MiniZinc
67 lines
2.0 KiB
MiniZinc
% RUNS ON mzn20_fd
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% RUNS ON mzn-fzn_fd
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% RUNS ON mzn20_mip
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%% Bennett quasigroup existence problem (QG5).
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%%
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%% A quasigroup is just a groupoid whose "multiplication table" is a
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%% Latin square: each row and each column is a permutation of the
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%% elements. Bennett's equation (not invented by Frank Bennett but
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%% studied by him) is (yx.y)y = x which forces the quasigroup to have
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%% interesting properties such as being orthogonal to certain of its
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%% conjugates. Idempotent ones are more important than the others.
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%%
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%% Bennett quasigroups exist for all positive N except for 2, 6, 10
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%% and with the possible exceptions of 14, 18, 26, 30, 38, 42 and 158.
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%%
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%% Idempotent ones exist for all positive N not in {2, 3, 4, 6, 9, 10,
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%% 12, 13, 14, 15, 16} except possibly for N in {18, 20, 22, 24, 26,
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%% 28, 30, 34, 38, 39, 42, 44, 46, 51, 52, 58, 60, 62, 66, 68, 70, 72,
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%% 74, 75, 76, 86, 87, 90, 94, 96, 98, 99, 100, 102, 106, 108, 110,
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%% 114, 116, 118, 122, 132, 142, 146, 154, 158, 164, 170, 174}
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%%
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%% The existence of smallish open problems makes this algebraic
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%% question attractive from the viewpoint of automated reasoning and
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%% constraint satisfaction.
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%%
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%% Reference:
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%% F. E. Bennett
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%% Quasigroups
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%% C. J. Colbourn (ed)
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%% The CRC Handbook of Combinatorial Designs
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%% Chapter 36, pp .424-429.
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include "globals.mzn";
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int: N;
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array[1..N,1..N] of var 1..N: q;
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constraint
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forall( x in 1..N )
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( q[x, x] = x );
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constraint
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forall( x in 1..N )
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( alldifferent([q[x,y] | y in 1..N]) );
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constraint
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forall( y in 1..N )
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( alldifferent([q[x,y] | x in 1..N]) );
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constraint
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forall( x in 1..N )
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( q[x, 1] < x + 2 );
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constraint
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forall( x in 1..N, y in 1..N )
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( q[q[q[y, x], y], y] = x );
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solve :: int_search(array1d(1..N*N, q), input_order, indomain, complete)
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satisfy;
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output ["Bennett quasigroup of size " ++ show(N) ++ ":\n"] ++
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[ if y=1 then "\n " else " " endif ++ show(q[x,y]) | x,y in 1..N ] ++
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[ "\n" ];
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N = 5; % Solutions also exist for N=7, N=8 and N=11.
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