Abstract
Fibonacci nim is a popular impartial combinatorial game, usually played with a single pile of stones: two players alternate in removing no more than twice the previous player’s removal. The game is appealing due to its surprising connections with the Fibonacci numbers and the Zeckendorf representation. In this article, we investigate some properties of a variant played with multiple piles of stones, and solve the 2-pile case. A player chooses one of the piles and plays as in Fibonacci nim, but here the move-size restriction is a global parameter, valid for any pile.
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Notes
Also, note the similarity with the Tribonacci morphism \(x\rightarrow xy,y\rightarrow xz,z\rightarrow x\), which satisfies \(T_n = T_{n-1}T_{n-2}T_{n-3}\), with fixed point \(xyxzxyxxyxzxyx{\underline{y}}xz\ldots \). We have indicated the first letter that differs from our type 1 word in Example 19.
A complete proof of all details of this result would involve many tedious verifications, so we merely provide the main idea and fill in some of the many details.
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Acknowledgements
Part of the work for this paper was completed at the Games at Dal workshop at Dalhousie University in Halifax, Nova Scotia, in August 2015. The first author was supported by the Killam Trusts. We would like to thank the referees for their helpful suggestions.
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Larsson, U., Rubinstein-Salzedo, S. Global Fibonacci nim. Int J Game Theory 47, 595–611 (2018). https://doi.org/10.1007/s00182-017-0574-x
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DOI: https://doi.org/10.1007/s00182-017-0574-x