The feline Josephus problem

Abstract

In the classic Josephus problem, elements 1, 2,.. ., n are placed in order around a circle and a skip value κ is chosen. The problem proceeds in n rounds, where each round consists of traveling around the circle from the current position, and selecting the κth remaining element to be eliminated from the circle. After n rounds, every element is eliminated. Special attention is given to the last surviving element, denote it by j. We generalize this popular problem by introducing a uniform number of lives ℓ, so that elements are not eliminated until they have been selected for the ℓth time. We prove two main results: 1) When n and κ are fixed, then j is constant for all values of & larger than the nth Fibonacci number. In other words, the last surviving element stabilizes with respect to increasing the number of lives. 2) When n and j are fixed, then there exists a value of κ that allows j to be the last survivor simultaneously for all values of ℓ. In other words, certain skip values ensure that a given position is the last survivor, regardless of the number of lives. For the first result we give an algorithm for determining j (and the entire sequence of selections) that uses O(n 2) arithmetic operations. © 2010 Springer-Verlag Berlin Heidelberg.