Associative magic square
An associative magic square is a magic square for which each pair of numbers symmetrically opposite to the center sum up to the same value. For an square, filled with the numbers from to , this common sum must equal . These squares are also called associated magic squares, regular magic squares, regmagic squares, or symmetric magic squares.
For instance, the Lo Shu Square, the unique magic square, is associative, because each pair of opposite points form a line of the square together with the center point, so the sum of the two opposite points equals the sum of a line minus the value of the center point regardless of which two opposite points are chosen. The magic square from Albrecht Dürer's 1514 engraving Melencolia I, also found in a 1765 letter of Benjamin Franklin, is also associative, with each pair of opposite numbers summing to 17.
Existence and enumeration
The numbers of possible associative magic squares for , counting two squares as the same whenever they differ only by a rotation or reflection, are:
The number zero in the position for associative magic squares is an example of a more general phenomenon: these squares do not exist for values of that are singly even (that is, equal to 2 modulo 4). Every associative magic square of even order forms a singular matrix, but associative magic squares of odd order can be singular or nonsingular.
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