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Randomisation in the Josephus problem

Faustin Adiceam, Steven Robertson, Victor Shirandami and Ioannis Tsokanos

Vol. 13 (2024), No. 3, 277–298
Abstract

The Josephus problem is a well-studied elimination problem consisting of determining the position of the survivor after repeated applications of a deterministic rule removing one person at a time from a given group.

A natural probabilistic variant of this process is introduced in this paper. More precisely, in this variant, the survivor is determined after performing a succession of Bernoulli trials with parameter p designating each time at which person is removed. When the number of participants tends to infinity, the main result characterises the limit distribution of the position of the survivor with an increasing degree of precision as the parameter approaches the unbiased case p = 1 2. Then, the convergence rate to the position of the survivor is obtained in the form of a central limit theorem.

A number of other variants of the suggested probabilistic elimination process are also considered. They each admit a specific limit behaviour which, in most cases, is stated in the form of an open problem.

To Angela James

Keywords
Josephus, elimination process
Mathematical Subject Classification
Primary: 05A16, 11Z05, 60C05
Secondary: 60F05
Supplementary material

Appendix: Numerical simulations

Milestones
Received: 27 March 2024
Revised: 21 August 2024
Accepted: 7 September 2024
Published: 23 September 2024
Authors
Faustin Adiceam
Laboratoire d’Analyse et de Mathématiques Appliquées (LAMA)
Université Paris-Est Créteil
Créteil
France
Steven Robertson
Department of Mathematics
The University of Manchester
Manchester
United Kingdom
Victor Shirandami
Department of Mathematics
The University of Manchester
Manchester
United Kingdom
Ioannis Tsokanos
Departmento de Mathemática, Instituto de Biociências, Letras e Ciências Exatas
Universidade Estadual Paulista
São José do Rio Preto
Brasil