Enumeration of associative magic squares of order 7

TL;DR

This paper calculates the total number of associative magic squares of order 7, revealing an enormous count of over one quadrillion, quadrillion, quadrillion, quadrillion, quadrillion, quadrillion, quadrillion, with implications for combinatorial enumeration.

Contribution

The paper introduces a novel enumeration method for associative magic squares of order 7 based on Ripatti's semi-magic squares approach, providing the first exact count.

Findings

Total associative magic squares of order 7: 1,125,154,039,419,854,784

Method extends Ripatti's enumeration of semi-magic squares

Results exclude symmetric patterns

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.