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#### Article Title

Josephus Problem Under Various Moduli

#### 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.

#### Author Bio

Toshiyuki Yamauchi is now a student at Kwansei Gakuin University and he majors in physics. He wants to be a scientist.Yamauchi loves Apple computer, and he respects Steve Jobs. He is going to attend the Intel International Science and Engineering Fair representing Japan.

Takahumi Inoue is now a student at Keio University and he majors in Policy Study. He wants to be a politician. He is going to attend the Intel International Science and Engineering Fair representing Japan.

Soh Tatsumi is a student at Kwansei Gakuin High School. He loves computer programming and reading.

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