Mathpages: Solving Magic Squares
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Description: An attempt to solve for all possible magic squares, a mathematical treatment.
What's the most efficient way of generating all possible NxN magic squares? Obviously trying all (N^2)! possible arrangements and checking for "magicness" would be prohibitive. Presumably this was not the method used by Bernard Frenicle de Bessy back in 1693 to determine that there are exactly 880 distinct 4x4 magic squares, not counting rotations and reflections. (Does anyone know what method he actually used?) There are several more modern treatments of the subject, such as in the book "Winning Ways" by Conway, but they mainly seem to approach the question in terms of equivalence classes and transformations of various kinds. This approach is certainly illuminating, but it's also interesting to try just "solving" the problem algebraically. In general, when dealing with NxN squares, where N is odd, it's helpful to subtract (N^2 + 1)/2 from each of the numbers 1 through N^2, since this makes all the common sums zero. For example, with a typical 3x3 square we subtract 5 from each number to give a b c 1 -4 3 d e f = 2 0 -2 g h i -3 4 -1 which makes it immediately obvious that there is only one possible magic square of order 3, up to rotations and reflections. This can also be seen by combining the algebraic conditions on the variables to eliminate all but two of the variables and arrive at the conic 2a^2 + 2ab + b^2 = 10 (1) (See
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| Page title: | Solving Magic Squares |
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| Date | Creation Date: 08-aug-1999 Expiration Date: 07-oct-2021 |