An associative magic square is a

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for which each pair of numbers symmetrically opposite to the center sum up to the same value. For an ''n'' × ''n'' square, filled with the numbers from 1 to ''n''², this common sum must equal ''n''² + 1. These squares are also called associated magic squares, regular magic squares, regmagic squares, or symmetric magic squares.

Examples

For instance, the

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– the unique 3 × 3 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 4 × 4 magic square from

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1514 engraving – also found in a 1765 letter of

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– is also associative, with each pair of opposite numbers summing to 17.

Existence and enumeration

The numbers of possible associative ''n'' × ''n'' magic squares for ''n'' = 3,4,5,..., counting two squares as the same whenever they differ only by a rotation or reflection, are: :1, 48, 48544, 0, 1125154039419854784, ... The number zero for ''n'' = 6 is an example of a more general phenomenon: associative magic squares do not exist for values of ''n'' that are singly even (equal to 2

modulo In computing and mathematics, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another, the latter being called the '' modulus'' of the operation. Given two positive numbers and , mo ...

4). Every associative magic square of even order forms a singular matrix, but associative magic squares of odd order can be singular or nonsingular.

References

External links

* {{Magic polygons Magic squares