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########################################################################
##
## Copyright (C) 1999-2024 The Octave Project Developers
##
## See the file COPYRIGHT.md in the top-level directory of this
## distribution or <https://octave.org/copyright/>.
##
## This file is part of Octave.
##
## Octave is free software: you can redistribute it and/or modify it
## under the terms of the GNU General Public License as published by
## the Free Software Foundation, either version 3 of the License, or
## (at your option) any later version.
##
## Octave is distributed in the hope that it will be useful, but
## WITHOUT ANY WARRANTY; without even the implied warranty of
## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
## GNU General Public License for more details.
##
## You should have received a copy of the GNU General Public License
## along with Octave; see the file COPYING. If not, see
## <https://www.gnu.org/licenses/>.
##
########################################################################
## -*- texinfo -*-
## @deftypefn {} {@var{M} =} magic (@var{n})
##
## Create an @var{n}-by-@var{n} magic square.
##
## A magic square is an arrangement of the integers @code{1:n^2} such that the
## row sums, column sums, and diagonal sums are all equal to the same value.
##
## Note: @var{n} must be a scalar greater than or equal to 3. If you supply
## @var{n} less than 3, magic returns either a nonmagic square, or else the
## degenerate magic squares 1 and [].
## @end deftypefn
function M = magic (n)
if (nargin < 1)
print_usage ();
endif
n = fix (n);
if (n < 0)
error ("magic: N must be non-negative");
elseif (n < 1)
M = [];
elseif (mod (n, 2) == 1)
shift = floor ((0:n*n-1)/n);
c = mod ([1:n*n] - shift + (n-3)/2, n);
r = mod ([n*n:-1:1] + 2*shift, n);
M(c*n+r+1) = 1:n*n;
M = reshape (M, n, n);
elseif (mod (n, 4) == 0)
M = reshape (1:n*n, n, n)';
I = [1:4:n, 4:4:n];
J = fliplr (I);
M(I,I) = M(J,J);
I = [2:4:n, 3:4:n];
J = fliplr (I);
M(I,I) = M(J,J);
elseif (mod (n, 4) == 2)
m = n/2;
M = magic (m);
M = [M, M+2*m*m; M+3*m*m, M+m*m];
k = (m-1)/2;
if (k > 1)
I = 1:m;
J = [2:k, n-k+2:n];
M([I,I+m],J) = M([I+m,I],J);
endif
I = [1:k, k+2:m];
M([I,I+m],1) = M([I+m,I],1);
I = k + 1;
M([I,I+m],I) = M([I+m,I],I);
endif
endfunction
%!test
%! for i = 3:30
%! A = magic (i);
%! assert (norm(diff([sum(diag(A)),sum(diag(flipud(A))),sum(A),sum(A')])),0);
%! endfor
## Not a magic square but we must return something (bug #46672).
## While one day we may change the actual return of magic (2),
## this properties still must be true.
%!test <*46672>
%! m = magic (2);
%! assert (size (m), [2 2]);
%! assert (m, [4 3; 1 2]);
%!assert (isempty (magic (0)))
%!assert (magic (1), 1)
%!assert (magic (1.5), 1)
## Test input validation
%!error <Invalid call> magic ()
%!error <N must be non-negative> magic (-5)
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