Magic Triangles

TL;DR

This paper introduces the concept of magic triangles, explores their properties for small sizes, and proposes a simulated annealing method to find larger magic triangles, filling a gap in the study of magic shapes.

Contribution

It defines magic triangles, analyzes small cases, and develops a simulated annealing approach for constructing larger magic triangles, which was previously overlooked.

Findings

Counted magic triangles for small sizes

Analyzed integer distributions within triangles

Proposed a simulated annealing method for larger sizes

Abstract

Magic squares are well-known arrangements of integers with common row, column, and diagonal sums. Various other magic shapes have been proposed, but triangles have been somewhat overlooked. We introduce certain triangular arrangements of integers with common sums in three directions, which we call magic triangles. For small sizes of these triangles, we count the number of unique magic triangles and examine distributions of integers at different positions within them. While we cannot enumerate the number of magic triangles at larger sizes, we offer a simulated annealing method for finding magic triangles.