Classifying Magic Squares and Their Associated Symmetries Using a Chord Diagram Approach

Overview

Dr. Vassilev University of New Mexico Magic squares are defined as matrices whose rows, columns, and main diagonals all add to the same number. They have existed and have been a topic of great interest since as early as 190 BCE, appearing in several different cultures. Some of the greatest minds in mathematics dedicated time to their exploration and construction including Euler, Ramanujan, and Fermat. Even with their prevalence throughout the mathematical world, few general theorems have been proven about magic squares and their properties. Through the course of this project, a novel approach to studying the squares was developed that represents the matrix elements as connected points on a circle with n2 equally spaced points. Furthermore, this visual representation can be used to classify the squares by the symmetries they produce. Unlike previously used visual aids, this method of classification extends to all orders of magic squares. Using this method of circular graphing, generalized theorems about the reflection and rotational symmetries of magic squares were developed as well. It was also proven using these tools that the maximum number of possible axes of symmetry of a given n×n magic square is precisely n/2. Inconsistencies were found in prior work by other authors on classification of 4 × 4 magic squares using permutation matrices were discovered, and further relations to this theory were developed. Finally, techniques for magic square multiplication using Kronecker Tensor Products were applied to enumeration of magic squares - one of the largest existing problems related to magic squares.

Awards (1)

• 3rd Place Mathematics & Computer Science

Competition history

• JSHS 2020 Category not listed

Resources

• National JSHS abstract book ↗

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