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The Linear Josephus Problem

This is a variant of the Josephus problem. There are

players in a line. Beginning with the first player, move from left to right, removing every

player. Change the direction at the end of the line and again remove every

player. Denote by

the position of the last player remaining.

You can see how players are removed with the "demo" button. With the "graph" button you can see the graphs of

for

and

. You can also see the combination of all the graphs for

Contributed by: Hiroshi Matsui, Toshiyuki Yamauchi, Daisuke Minematsu, Soh Tatsumi, Masakazu Naito, Takafumi Inoue, Takuto Hieda, and Ryohei Miyadera.

Show Initialization Code

(402 lines omitted)

Fix the number

and choose "graph" to see the graph of the list

. For

the graph is very simple and beautiful. See the first snapshot for the case of

. For other values of

the graphs are quite complicated. See the second snapshot for

Combining all the graphs for

with different colors gives a very beautiful graph. The picture in the third snapshot is for

As far as the authors know, the first paper on the linear Josephus problem is C. Groer, "Mathematics of Survival," American Mathematical Monthly, 110, 2003.

For research on the variants of the Josephus problem, please see the following two papers written by the authors:

R. Miyadera, S. Hashiba, and D. Minematsu, "How High School Students Can Discover Original Ideas of Mathematics Using Mathematica," Mathematica in Education and Research, 11(3), 2006.

H. Matsui, T. Yamauchi, S. Tatsumi, T. Inoue, M. Naito, and R. Miyadera, "Interesting Variants of the Josephus Problem," Kokyuroku (to be published).

Josephus Problem (Wolfram MathWorld)

Josephus Problem (The Wolfram Demonstrations Project)

The Josephus Problem in Both Directions (The Wolfram Demonstrations Project)

"The Linear Josephus Problem" from The Wolfram Demonstrations Project
http://demonstrations.wolfram.com/TheLinearJosephusProblem/

Contributed by: Hiroshi Matsui, Toshiyuki Yamauchi, Daisuke Minematsu, Soh Tatsumi, Masakazu Naito, Takafumi Inoue, Takuto Hieda, and Ryohei Miyadera.

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