Vol. 7, No. 6, 2014

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Fibonacci Nim and a full characterization of winning moves

Cody Allen and Vadim Ponomarenko

Vol. 7 (2014), No. 6, 807–822
Abstract

In this paper we will fully characterize all types of winning moves in the “take-away” game of Fibonacci Nim. We prove the known winning algorithm as a corollary of the general winning algorithm and then show that no other winning algorithms exist. As a by-product of our investigation of the game, we will develop useful properties of Fibonacci numbers. We conclude with an exploration of the probability that unskilled player may beat a skilled player and show that as the number of tokens increase, this probability goes to zero exponentially.

Keywords
Fibonacci Nim, take away games, dynamic Nim, combinatorial games, Fibonacci
Mathematical Subject Classification 2010
Primary: 91A46
Secondary: 11A63
Milestones
Received: 23 December 2013
Revised: 8 January 2014
Accepted: 24 January 2014
Published: 20 October 2014

Communicated by Scott T. Chapman
Authors
Cody Allen
Department of Mathematics and Statistics
San Diego State University
5500 Campanile Drive
San Diego, CA 92182-7720
United States
Vadim Ponomarenko
Department of Mathematics and Statistics
San Diego State University
5500 Campanile Drive
San Diego, CA 92182-7720
United States