Black Hole Zeckendorf Games

Abstract

Zeckendorf proved a remarkable fact 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 a positive integer into its Zeckendorf decomposition into a 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 $i$thFibonacci number. Players take turns applying these moves, beginning with $n$ pieces in the $F_1$ column. They showed that for $n \neq 2$, Player 2 has a winning strategy, though the proof is non-constructive, and a constructive solution is unknown. We expand on this by investigating "black hole'' variants of this game. The Black Hole Zeckendorf game on $F_m$ 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 parity-stealing based 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 $n$.