Squaring the magic squares of order 4

TL;DR

This paper determines the exact count of normal multiplicative magic squares of order 4, introduces a new representation method, and highlights the role of symmetric group actions in their structure.

Contribution

It provides the first complete proof for the number of such magic squares and a novel construction approach using only five specific matrices.

Findings

Exact number of normal multiplicative magic squares of order 4 established

New representation simplifies construction of these magic squares

Role of symmetric group action clarified in their structure

Abstract

In this paper, we present the problem of counting magic squares and we focus on the case of multiplicative magic squares of order 4. We give the exact number of normal multiplicative magic squares of order 4 with an original and complete proof, pointing out the role of the action of the symmetric group. Moreover, we provide a new representation for magic squares of order 4. Such representation allows the construction of magic squares in a very simple way, using essentially only five particular 4X4 matrices.