Fibonacci Nim and a full characterization of winning moves (Q2513730)

Fibonacci Nim is a 2-player combinatorial game. It is a take-away game: two players remove alternatively token from a pile. In this game, a player may remove a positive number of token and at most twice the number of token removed during the previous round by the other player. From that respect, it is an example of dynamic one-pile Nim as studied by \textit{A. Holshouser} et al. [Fibonacci Q. 41, No. 3, 253--262 (2003; Zbl 1093.91013)]. It is a game in normal convention: the player taking the last token wins. The authors first recall classical results about Zeckendorff greedy expansion of integers as sum of non-consecutive Fibonacci numbers. Then, they study winning and losing positions of the game to perform an analysis of the game. The paper ends with a discussion about an upper bound on the probability that an unskilled player may beat a skilled player.