Winning Strategies for Generalized Zeckendorf Game

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.

24 pages, 8 figures