# Associative magic square Detail from Melencolia I showing a $4\times 4$ associative 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 $n\times n$ 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 Lo Shu Square, the unique $3\times 3$ 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 $4\times 4$ 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 $n\times n$ magic squares for $n=3,4,5,\dots$ , counting two squares as the same whenever they differ only by a rotation or reflection, are:

1, 48, 48544, 0, 1125154039419854784, ... (sequence A081262 in the OEIS)

The number zero in the position for $6\times 6$ associative magic squares is an example of a more general phenomenon: these squares do not exist for values of $n$ 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.