Involve: A Journal of Mathematics

  • Involve
  • Volume 7, Number 6 (2014), 807-822.

Fibonacci Nim and a full characterization of winning moves

Cody Allen and Vadim Ponomarenko

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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.

Article information

Involve, Volume 7, Number 6 (2014), 807-822.

Received: 23 December 2013
Revised: 8 January 2014
Accepted: 24 January 2014
First available in Project Euclid: 20 December 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 91A46: Combinatorial games
Secondary: 11A63: Radix representation; digital problems {For metric results, see 11K16}

Fibonacci Nim take away games dynamic Nim combinatorial games Fibonacci


Allen, Cody; Ponomarenko, Vadim. Fibonacci Nim and a full characterization of winning moves. Involve 7 (2014), no. 6, 807--822. doi:10.2140/involve.2014.7.807.

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