Enumeration of associative magic squares of order 7

Abstract

An associative magic square is a magic square such that the sum of any 2 cells at symmetric positions with respect to the center is constant. The total number of associative magic squares of order 7 is enormous, and thus, it is not realistic to obtain the number by simple backtracking. As a recent result, Artem Ripatti reported the number of semi-magic squares of order 6 (the magic squares of 6x6 without diagonal sum conditions) in 2018. In this research, with reference to Ripatti's method of enumerating semi-magic squares, we have calculated the total number of associative magic squares of order 7. There are exactly 1,125,154,039,419,854,784 associative magic squares of order 7, excluding symmetric patterns.