A Travelling Salesman Paths within nxn (n = 3, 4, 5) Magic Squares

TL;DR

This paper analyzes the symmetries and path length distributions of all magic squares of orders 3, 4, and 5, drawing parallels with the travelling salesman problem and revealing symmetry-based characteristics.

Contribution

It uncovers symmetry-related properties of total path distances in magic squares of orders 3 to 5, providing new insights into their structural characteristics.

Findings

Symmetries characterize total path distance distributions.

Distinct configurations for each order: 1,880, 275,305,224.

Open questions on higher-order magic squares and path length extremities.

Abstract

Intriguing symmetries are uncovered regarding all magic squares of orders 3, 4, and 5, with 1, 880, and 275,305,224 distinct configurations, respectively. In analogy with the travelling salesman problem, the distributions of the total topological distances of the paths travelled by passing through all the vertices (matrix elements) only once and spanning all elements of the matrix are analyzed. Symmetries are found to characterise the distributions of the total topological distances in these instances. These results raise open questions about the symmetries found in higher-order magic squares and the formulation of their minimum and maximum total path lengths.