Black Hole Zeckendorf Games

TL;DR

This paper explores variants of the Zeckendorf game involving 'black holes' that restrict moves, providing new solutions and strategies for specific configurations and initial placements.

Contribution

It introduces 'black hole' variants of the Zeckendorf game, analyzing gameplay constraints and constructing solutions for particular cases and initial setups.

Findings

Constructive solutions for specific black hole configurations.

Analysis of pre-game initial placements and strategies.

Extension of Zeckendorf game theory with new constraints.

Abstract

Zeckendorf proved that every positive integer can be written as a decomposition of non-adjacent Fibonacci numbers. Baird-Smith, Epstein, Flint, and Miller converted the process of decomposing an integer $n$ into a 2-player game, using the moves of $F_i + F_{i-1} = F_{i+1}$ and $2F_i = F_{i+1} + F_{i-2}$, where $F_i$ is the $ith$ Fibonacci number. They showed non-constructively that for $n \neq 2$, Player 2 has a winning strategy: a constructive solution remains unknown. We expand on this by investigating ``black hole'' variants of this game. The $F_m$ Black Hole Zeckendorf game is played with any $n$ but solely in columns $F_i$ for $i < m$. Gameplay is similar to the original Zeckendorf game, except any piece that would be placed on $F_i$ for $i \geq m$ is locked out in a ``black hole'' and removed from play. With these constraints, we analyze the games with black holes on $F_3$ and $F_4$ and construct a solution for specific configurations, using a non-constructive proof to lead to a constructive one. We also examine a pre-game in which players take turns placing down $n$ pieces in the outermost columns before the decomposition phase, and find constructive solutions for any $n$.