Enumeration of $4 \times 4$ Magic Squares

TL;DR

This paper counts the number of 4x4 magic squares using advanced mathematical techniques, extending the enumeration methods previously applied only to 3x3 squares.

Contribution

It introduces a novel approach combining inside-out polytopes and Ehrhart theory to enumerate 4x4 magic squares based on sum and entry bounds.

Findings

Exact count of 4x4 magic squares as a function of sum

Enumeration method applicable to larger n

Extension of previous 3x3 enumeration results

Abstract

A \emph{magic square} is an $n \times n$ array of distinct positive integers whose sum along any row, column, or main diagonal is the same number. We compute the number of such squares for $n=4$, as a function of either the magic sum or an upper bound on the entries. The previous record for both functions was the $n=3$ case. Our methods are based on inside-out polytopes, i.e., the combination of hyperplane arrangements and Ehrhart's theory of lattice-point enumeration.