A normal magic square of order n contains the integers 1, 2, ..., n² arranged such that the sum of the numbers in any row, column, or main diagonal is the same:
n(n²+1)/2
Subtracting the average of the numbers in the square, (n²+1)/2, from each number, yields a skeleton square of normalized numbers, "the bones", consisting of positive and negative numbers, and, for odd order squares, 0. This is a convenient way of seeing symmetry in the square. It may also facilitate square construction, transforms, and analysis.
There is 1 distinct, that is, not including rotations and reflections, magic square of order 3. It was known to the ancient Chinese, who called it the Lo Shu.
Notice that the complementary pairs of the square, i.e., pairs of numbers whose sum is n² + 1, appear as ± pairs in the bones.
There are 880 distinct magic squares of order 4. These were enumerated by Bernard Frénicle de Bessy in 1693. The square shown here is called Dürer's magic square.
There are 275,305,224 distinct magic squares of order 5. The number was first computed by Richard Schroeppel in 1973.
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