: Introduction and an example.

The Josephus Problem in Both Directions.

Masakazu Naito, Toshiyuki Yamauchi, Daisuke Minematsu and Ryohei Miyadera

Kwansei Gakuin

miyadera1272000@yahoo.co.jp

runners@kwansei.ac.jp

ABSTRACT

In this article we study the Josephus problem in both direction. In this variant of the Josephus problem two numbers are to be eliminated at the same time, but two processes of elimination go for different directions. Suppose that there are $n$-numbers and every $k$-th numbers are to be eliminated. We denote the number that remains by $JI(n,k)$.
At a glance this Josephus problem looks like a simple puzzle and nothing more than a good example for computer programming, but the sequence $\{ JI(n,k), n = 1,2,3,...\}$ presents interesting self-similarity of graph and the self-similarity of the sequence when each term is divided by a certain number.
We have presented the self-similarities of this Josephus problem in "The Self-Similarity of the Josephus problem and its Variants, Visual Mathematics Volume 11, No. 2, 2009", but we have not proved the existence of the self-similarity of the sequence.
In this article we prove it using the recursive relations when $k = 2$.

• Introduction and an example.

• Recursive relations and some formulas for JI(n).

• An interesting fact about the Josephus Problem in Both Directions.

• References

: Introduction and an example.