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 k is chosen. The problem proceeds in n rounds, where each round consists of traveling around the circle from the current position, and selecting the kth 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 k are fixed, then j is constant for all values of ℓ 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 k 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.
- Publication:
-
Lecture Notes in Computer Science
- Pub Date:
- 2010
- DOI:
- 10.1007/978-3-642-13122-6_33
- Bibcode:
- 2010LNCS.6099..343R
- Keywords:
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- Josephus problem;
- Fibonacci number;
- Chinese remainder theorem;
- Bertrand's postulate;
- number theory;
- algorithm