%
% Quasigroup problem in MiniZinc.
%
% This model is a translation of the EssencePrime model quasiGroup7.eprime
% from the Minion Translator examples.
% """
% The quasiGroup existence problem (CSP lib problem 3)
%
% An m order quasigroup  is an mxm multiplication table of integers 1..m,
% where each element occurrs exactly once in each row and column and certain
% multiplication axioms hold (in this case, we want axiom 7 to hold).
% """

% See
% http://www.dcs.st-and.ac.uk/~ianm/CSPLib/prob/prob003/index.html
% http://www.dcs.st-and.ac.uk/~ianm/CSPLib/prob/prob003/spec.html
% Axiom 7:
% """
% QG7.m problems are order m quasigroups for which (b*a)*b = a*(b*a).
% """
%
% Model created by Hakan Kjellerstrand, hakank@gmail.com

% Licenced under CC-BY-4.0 : http://creativecommons.org/licenses/by/4.0/

include "globals.mzn";

int: n;
set of int: nDomain = 0..n-1;

array[nDomain, nDomain] of var nDomain: quasiGroup;

solve :: int_search([quasiGroup[row, col] | row, col in nDomain],
first_fail, indomain_min, complete) satisfy;
% solve satisfy;

constraint

% assign the "reflected" quasigroup to qGColumns to access its columns
%  forall row,col : nDomain .
%    quasiGroupColumns[col,row] = quasiGroup[row,col],

% All rows have to be different
forall(row in nDomain) (
all_different([quasiGroup[row,col] | col in nDomain])
)
/\
% all values in the diagonals
forall( i in nDomain ) (
quasiGroup[i,i] = i
)
/\

% All columns have to be different
forall(col in nDomain) (
all_different([quasiGroup[row, col] | row in nDomain])
)
/\

% this strange constraint
% corresponds to:
% quasiGroup[i, quasiGroup[j,i]] = quasiGroup[quasiGroup[j,i], j]
forall(i,j in nDomain) (
quasiGroup[i, quasiGroup[j,i]] = quasiGroup[quasiGroup[j,i],j]
)
/\
% some implied? constraint
forall(i in nDomain) (
quasiGroup[i,n-1] + 2 >= i
)
;

output [
if col = 0 then "\n" else " " endif ++
show(quasiGroup[row, col])
| row, col in nDomain
] ++ ["\n"];

%
% data
%
n = 5;