Masakazu Naito, Sohtaro Doro, Daisuke Minematsu and Ryohei Miyadera

Kwansei Gakuin, Nishinomiya City JAPAN

e-mail: miyadera1272000@yahoo.co.jp

Kwansei Gakuin, Nishinomiya City JAPAN

e-mail: miyadera1272000@yahoo.co.jp

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.

- Introduction and the traditional Josephus Problem.
- The Josephus Problem in both Directions.
- Linear Josephus Problem.
- The Linear Josephus Problem in Both Directions.
- Mathematical Theory of the Linear Josephus Problem in both Directions.
- APPENDIX