Magic Polygons and Combinatorial Algorithms: A Detailed Mathematical Approach

In this branch, we analyze the Magic Polygons of order 3 (P(n, 2)) and present sure properties that were advantageous in the implementation of an treasure to determine the number of magic polygons for orderly polygons up to 24 parts. First, is made an sameness between Magic Polygons and items of the Symmetric Group, such sameness is clear once the Magic Polygons is a fixed arrangement of numbers. Made specific equivalence, it’s inconsequential that in order to find all Magic Polygons for a regular shape of n sides, it’s enough to create all permutations of the set {1,2. . . 2n+1} and verify that ones answer the definition. But this is not highest in rank way, cause the same change would be restored many times, due the action of the Dihedral Group in the normal polygon. Therefore, a analytical approach is needed in consideration of simplify the computational process. This habit, we reach the concept of Equivalents Magic Polygons, and located in some features here began, we avoid few of them. Yet, is introduced the idea of Derivatives Magic Polygons because a Magic Polygon maybe built from some Arithmetic Progression, and is not restricted to the instinctive sequence.

Author(s) Details:

Danniel Dias Augusto,
Departamento de Matematica – Universidade de Brasilia (UnB), Distrito Federal- 70297-400, Brasil.

Please see the link here: https://stm.bookpi.org/RHMCS-V4/article/view/9159

Keywords: Magic polygons, combinatorial algorithms, symmetric group, dihedral group