Winning Strategies for Generalized Zeckendorf Game

TL;DR

This paper investigates winning strategies in generalized Zeckendorf games based on linear recurrence sequences, providing proofs of strategic dominance for players under various conditions and extending results to multiplayer alliances.

Contribution

It introduces new non-constructive proofs for winning strategies in generalized Zeckendorf games, including multiplayer and alliance scenarios, for various recurrence parameters.

Findings

Player 2 wins for Fibonacci game when n > 2.

Player 1 wins for even c in (c,k)-nacci game when n is large.

No player has a winning strategy for certain parameters with p ≥ c+2.

Abstract

Zeckendorf proved that every positive integer $n$ can be written uniquely as the sum of non-adjacent Fibonacci numbers; a similar result holds for other positive linear recurrence sequences. These legal decompositions can be used to construct a game that starts with a fixed integer $n$, and players take turns using moves relating to a given recurrence relation. The game eventually terminates in a unique legal decomposition, and the player who makes the final move wins. For the Fibonacci game, Player $2$ has the winning strategy for all $n>2$. We give a non-constructive proof that for the two-player $(c, k)$-nacci game, for all $k$ and sufficiently large $n$, Player $1$ has a winning strategy when $c$ is even and Player $2$ has a winning strategy when $c$ is odd. Interestingly, the player with the winning strategy can make a mistake as early as the $c + 1$ turn, in which case the other player gains the winning strategy. Furthermore, we proved that for the $(c, k)$-nacci game with players $p \ge c + 2$, no player has a winning strategy for any $n \ge 3c^2 + 6c + 3$. We find a stricter lower boundary, $n \ge 7$, in the case of the three-player $(1, 2)$-nacci game. Then we extend the result from the multiplayer game to multialliance games, showing which alliance has a winning strategy or when no winning strategy exists for some special cases of multialliance games.