Within this paper, we will briefly review the history of a collection of number puzzles which take the shape of squares, polygons, and polyhedra in both modular and nonmodular arithmetic. Among other results, we develop construction techniques for solutions of both Modulo and regular Magic Squares. For other polygons in nonmodular arithmetic, specifically of order 3, we present a proof of why there are only four Magic Triangles using linear algebra, disprove the existence of the Magic Tetrahedron in two ways, and utilizing the infamous 3-SUM combinatorics problem we disprove the existence of the Magic Octahedron.

Author Bio

Nicasio Velez graduated from Maryville College in 2018 with a Bachelor's in Mathematics. The experience of completing this research inspired him to pursue a Doctorate in Mathematics to allow him to pursue both his passions for teaching and research. Nicasio is currently a Graduate Teaching Assistant at the University of Tennessee, Knoxville.