Block Josephus Problem: When the reality is more cruel than the old story

Abstract

In the Josephus Problem, there are $n$ people numbered from $0$ to $n-1$ around a circle and proceeding around the circle every second person is executed until no one survives. Determining where to stand on the circle to be the last survivor is called the Josephus Problem. In this paper, we present a generalized version of the Josephus Problem and study cases where multiple executions occur at each iteration. Especially, we focus on the Block Josephus problem where the number of skips and the number of executions are the same. In particular, we present nonrecursive formulas for the initial positions of survivors in the Block Josephus Problem.

Keywords

• Discrete Mathematics
• Josephus problem
• function iterations

References

1. [1] K. Burde, Das Problem der Abzählreime und Zahlentwicklungen mit gebrochenen Basen, J. Number Theory 26 (2), 192–209, 1987.
2. [2] L. Halbeisen and N. Hungerbühler, The Josephus Problem, Journal de Thèorie des Nombres de Bordeaux, 9, 303–318, 1997.
3. [3] F. Jakóbczyk, On the Generalized Josephus Problem, Glasg. Math. J. 14, 168–173, 1973.
4. [4] T. F. Josephus, The Jewish War (translated by William Whiston).
5. [5] A.M. Odlyzko and H.S. Wilf, Functional iteration and the Josephus problem, Glasg. Math. J. 33 (2), 235–240, 1991.
6. [6] J.W. Park and R. Teixeira, Serial Execution Josephus Problem, Korean J. Math. 26 (1), 1–7, 2018.
7. [7] W.J. Robinson, The Josephus Problem, Math. Gaz. 44 (347), 47–52, 1960.
8. [8] F. Ruskey and A. Williams, The Feline Josephus Problem, Theory Comput. Syst. 50 (1), 20–34, 2012.

9. [9] A. Shams-Baragh, Formulation of the Extended Josephus Problem, in: National Computer Conference, December 2002.
10. [10] S. Sharma, R. Tripathi, S. Bagai, R. Saini and N. Sharma, Extension of the Josephus Problem with Varying Elimination Steps, DU Journal of Undergraduate Research and Innovation 1 (3), 211–218, 2015.
11. [11] R. Teixeira and J.W. Park, Mathematical Explanation and Generalization of Penn and Teller’s Love Ritual Magic Trick, Journal of Magic Research 8, 21–32, 2017.
12. [12] N. Thèrialut, Generalizations of the Josephus Problem, Util. Math. 58, 161–173, 2000.
13. [13] D. Woodhouse, The Extended Josephus Problem, Rev. Mat. Hisp.-Amer. 33 (4), 207– 218, 1973.