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/* MAGIC, Magic Square */
/* Written in GNU MathProg by Andrew Makhorin <mao@gnu.org> */
/* In recreational mathematics, a magic square of order n is an arrangement of n^2 numbers, usually distinct integers, in a square, such that n numbers in all rows, all columns, and both diagonals sum to the same constant. A normal magic square contains the integers from 1 to n^2.
(From Wikipedia, the free encyclopedia.) */
param n, integer, > 0, default 4;/* square order */
set N := 1..n^2;/* integers to be placed */
var x{i in 1..n, j in 1..n, k in N}, binary;/* x[i,j,k] = 1 means that cell (i,j) contains integer k */
s.t. a{i in 1..n, j in 1..n}: sum{k in N} x[i,j,k] = 1;/* each cell must be assigned exactly one integer */
s.t. b{k in N}: sum{i in 1..n, j in 1..n} x[i,j,k] = 1;/* each integer must be assigned exactly to one cell */
var s;/* the magic sum */
s.t. r{i in 1..n}: sum{j in 1..n, k in N} k * x[i,j,k] = s;/* the sum in each row must be the magic sum */
s.t. c{j in 1..n}: sum{i in 1..n, k in N} k * x[i,j,k] = s;/* the sum in each column must be the magic sum */
s.t. d: sum{i in 1..n, k in N} k * x[i,i,k] = s;/* the sum in the diagonal must be the magic sum */
s.t. e: sum{i in 1..n, k in N} k * x[i,n-i+1,k] = s;/* the sum in the co-diagonal must be the magic sum */
solve;
printf "\n";printf "Magic sum is %d\n", s;printf "\n";for{i in 1..n}{ printf{j in 1..n} "%3d", sum{k in N} k * x[i,j,k]; printf "\n";}printf "\n";
end;
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