Applications of AI for Magic Squares

TL;DR

This paper explores the use of AI techniques, including backtracking algorithms and supervised machine learning, to generate and classify fourth order normal magic squares, leveraging symmetry groups to efficiently explore the solution space.

Contribution

It introduces an AI-based approach combining backtracking and machine learning to generate and classify magic squares, expanding the understanding of their symmetry properties.

Findings

Entire set of fourth order normal magic squares can be generated from 95 asymmetric parents.

Symmetry groups like the dihedral group are key to categorizing magic squares.

Methodology could be extended to higher order magic squares.

Abstract

In recreational mathematics, a normal magic square is an $n \times n$ square matrix whose entries are distinctly the integers $1 \ldots n^2$, such that each row, column, and major and minor traces sum to one constant $\mu$. It has been proven that there are 7,040 fourth order normal magic squares and 2,202,441,792 fifth order normal magic squares, with higher orders unconfirmed. Previous work related to fourth order normal squares has shown that symmetries such as the dihedral group exist and that (under certain conditions) normal magic squares can be categorized into four distinct subsets. With the implementation of an efficient backtracking algorithm along with supervised machine learning techniques for classification, it will be shown that the entire set of fourth order normal magic squares can be generated by expanding the symmetry groups of 95 asymmetric parents. Discussion will suggest that methods employed in this project could similarly apply to higher orders.