Pseudo Magic Squares

TL;DR

This paper introduces pseudo magic squares, a generalization of traditional magic squares over the integers without the distinctness constraint, and explores their structure using group and ring theory to find new non-isomorphic examples.

Contribution

It proposes the concept of pseudo magic squares and extends it with algebraic structures, offering new methods to classify and generate non-isomorphic solutions.

Findings

Defined pseudo magic squares over integers

Introduced group and ring structures to analyze them

Provided tools to find new non-isomorphic pseudo magic squares

Abstract

A magic square of order n is an nxn square (matrix) whose entries are distinct nonnegative integers such that the sum of the numbers of any row and column is the same number, the magic constant. In this paper we introduce the concept of pseudo magic squares, i.e., magic squares defined over the ring of integers, without the restriction of distinct numbers. Additionally, we generalize this new concept by introducing a group (ring) structure over it. This new approach can provide useful tools in order to find new non-isomorphic pseudo magic squares.