The Reversed Zeckendorf Game

TL;DR

This paper introduces the reversed Zeckendorf game, a new variant starting from the end state of the original game, and demonstrates that Player 1 has a winning strategy for certain values of n, expanding understanding of game strategies.

Contribution

The paper presents the reversed Zeckendorf game, a novel variant, and provides strategies for Player 1 in specific cases, advancing the analysis of Zeckendorf-related games.

Findings

Player 1 has a winning strategy for n=F_{i+1}+F_{i-2}

Solved various modified versions of the game

Extended understanding of strategic play in Zeckendorf games

Abstract

Zeckendorf proved that every natural number $n$ can be expressed uniquely as a sum of non-consecutive Fibonacci numbers, called its Zeckendorf decomposition. Baird-Smith, Epstein, Flint, and Miller created the Zeckendorf game, a two-player game played on partitions of $n$ into Fibonacci numbers which always terminates at a Zeckendorf decomposition, and proved that Player 2 has a winning strategy for $n\geq 3$. Since their proof was non-constructive, other authors have studied the game to find a constructive winning strategy, and lacking success there turned to related problems. For example, Cheigh, Moura, Jeong, Duke, Milgrim, Miller, and Ngamlamai studied minimum and maximum game lengths and randomly played games. We explore a new direction and introduce the reversed Zeckendorf game, which starts at the ending state of the Zeckendorf game and flips all the moves, so the reversed game ends with all pieces in the first bin. We show that Player 1 has a winning strategy for $n = F_{i+1} + F_{i-2}$ and solve various modified games.