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## -*- 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)