## Abstract

We are going to study the Josephus Problem and its variants under various moduli in this article. Let n be a natural number. We put n numbers in a circle, and we are going to remove every second number. Let J(n) be the last number that remains. This is the traditional Josephus Problem. The list { J(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 J2(n) be the last number that remains. If the sequence { J2(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.

## Faculty Sponsor

Ryohie Miyadera

## Recommended Citation

Yamauchi, Toshiyuki; Inoue, Takahumi; and Tatsumi, Soh
(2009)
"Josephus Problem Under Various Moduli,"
*Rose-Hulman Undergraduate Mathematics Journal*: Vol. 10:
Iss.
1, Article 10.

Available at:
https://scholar.rose-hulman.edu/rhumj/vol10/iss1/10