The Self-Similarity of the Josephus Problem and its Variants

Masakazu Naito, Sohtaro Doro, Daisuke Minematsu, Ryohei Miyadera

In this article the authors are going to study the self-similarity of the graphs that are made from the Josephus Problem and its variants. Here the authors present the graphs of the traditional Josephus Problem, the Josephus Problem in both directions, the linear Josephus Problem and the linear Josephus Problem in both directions, and the authors calculated the ratio of self-similarity. The authors are going to present recursive relations for the linear Josephus problem in both directions, and by these recursive relations the authors can prove the self-similarity of the graph. Although this article is quite long, only a small part of it is used for the mathematical theory. The authors present the mathematical theory of the Josephus Problem only in Section 5. Section 1 is a brief introduction, and in Section 2, Section 3 and Section 4 the authors present many interesting graphs that are created by the variants of the Josephus Problem. Therefore those who are interested in the beauty of mathematics can appreciated most of this article without reading mathematical formulas and proofs.