Josephus Problem Under Various Mod n.

Abstract

We are going to study the Josephus Problem and its variants under various moduli in this article. In the Josephus Problem we can find two kinds of self-similarity. One is the self-similarity of graphs of the Josephus Problem. Another is the self-similarity of the sequence of the Josephus Problem. In this article we are going to study the sequence of the Josephus Problem and its similarity. The authors are the first persons who are going to study this kind of similarity of the Josephus Problem. Let $n$ be a natural number. We put $n$ numbers in a circle, and we are going to remove every second number. Let $J2(n)$ be the last number that remains. This is the traditional Josephus Problem. The list {$J2(n)$, n = 1,2,...,20 } is {1, 1, 3, 1, 3, 5, 7, 1, 3, 5, 7, 9, 11, 13, 15, 1, 3, 5, 7, 9}. When this sequence is reduced $mod \ 4$, then we have {1, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1 }.
Next we are going to study a variant of the Josephus Problem in which two numbers are to be eliminated at the same time, and let $JS(n)$ be the last number that remains. If the sequence {$JS(2n)$, n = 1, 2, ...63 } is reduced $mod \ 2$, then we have {1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0 }. The pattern that exists in the sequence is obvious if you look at the sequence carefully.
In this way we get interesting patterns of sequences for the Josephus Problem and its variants under various moduli. The authors were the first people who began to study these problems under various moduli. The authors have discovered many interesting facts, and they are going to present them in this article. They have also studied the graphs that are made from these problems. They are going to present a program of Java applet.

Mathematical Games