These variants have interesting recursive relations and beautiful self-similarities of the graphs.

The authors have studied variants of the Josephus Problem, and have published our result in [4], and our article is going to be published in [6].

In this article the authors are going to present new results of our research on the variants of the Josephus Problem.

With a proper computer program it becomes very easy to study this variant of the Josephus Problem. Please read the appendix of this article for the programs of the Josephus Problem.

First we are going to study the traditional Josephus Problem.

When , then the function has very simple recursive relations.

Since , by Theorem1.1 we can calculate for any natural number .

The horizontal coordinate is for the number of numbers involved in the game, and the vertical coordinate is for the last number. For example, by we have the point in the graph.

As you can see, the graph of the function is very simple. It is interesting to compare this to the graphs in the following sections.