Panmagic Square
If all the diagonals (including those obtained by ``wrapping around'' the edges) of a Magic Square, as well as theusual rows, columns, and main diagonals sum to the Magic Constant, the square is said to be a PanmagicSquare (also called Diabolical Square, Nasik Square, or Pandiagonal Square). No panmagic squaresexist of order 3 or any order 
for 
an Integer. The Siamese method for generatingMagic Squares produces panmagic squares for orders 
with ordinary vector (2, 1) and breakvector (1, 
).
The Lo Shu is not panmagic, but it is an Associative Magic Square. Order four squares can be panmagic orAssociative, but not both. Order five squares are the smallest which can be bothAssociative and panmagic, and 16 distinct Associativepanmagic squares exist, one of which is illustrated above (Gardner 1988).
The number of distinct panmagic squares of order 1, 2, ... are 1, 0, 0, 384, 3600, 0, ... (Sloane's A027567, Hunter andMadachy 1975). Panmagic squares are related to Hypercubes.
See also Associative Magic Square, Hypercube, Franklin Magic Square, Magic Square
References
Gardner, M. The Second Scientific American Book of Mathematical Puzzles & Diversions: A New Selection. New York: Simon and Schuster, pp. 135-137, 1961. Gardner, M. ``Magic Squares and Cubes.'' Ch. 17 in Time Travel and Other Mathematical Bewilderments. New York: W. H. Freeman, pp. 213-225, 1988. Hunter, J. A. H. and Madachy, J. S. ``Mystic Arrays.'' Ch. 3 in Mathematical Diversions. New York: Dover, pp. 24-25, 1975. Kraitchik, M. ``Panmagic Squares.'' §7.9 in Mathematical Recreations. New York: W. W. Norton, pp. 174-176, 1942. Madachy, J. S. Madachy's Mathematical Recreations. New York: Dover, p. 87, 1979. Rosser, J. B. and Walker, R. J. ``The Algebraic Theory of Diabolical Squares.'' Duke Math. J. 5, 705-728, 1939. Sloane, N. J. A. Sequence A027567in ``The On-Line Version of the Encyclopedia of Integer Sequences.''http://www.research.att.com/~njas/sequences/eisonline.html.
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