Magic Polygons and Combinatorial Algorithms: A Detailed Mathematical Approach

Abstract

In this chapter, we analyze the Magic Polygons of order 3 (P(n, 2)) and present certain properties that were helpful in the implementation of an algorithm to determine the number of magic polygons for regular polygons up to 24 sides. First, is made an equivalence between Magic Polygons and elements of the Symmetric Group, such equivalence is clear once the Magic Polygons is a finite arrangement of numbers. Made such equivalence, it's trivial that in order to find all Magic Polygons for a regular polygon of n sides, it's enough to generate all permutations of the set {1,2. . . 2n+1} and verify which ones satisfies the definition. But this is not the best way, because the same permutation would be returned many times, due the action of the Dihedral Group in the regular polygon. Therefore, a mathematical approach is needed in order to simplify the computational process. This way, we reach the concept of Equivalents Magic Polygons, and based in some properties here approached, we avoid some of them. Yet, is introduced the concept of Derivatives Magic Polygons because a Magic Polygon can be built from any Arithmetic Progression, and is not restricted to the natural sequence.