The Zeckendorf Game

TL;DR

This paper introduces a two-player game based on Zeckendorf's theorem, analyzing strategies, bounds on game length, and showing Player 2's winning strategy for all n > 2, despite the proof being non-constructive.

Contribution

It formulates a new combinatorial game related to Zeckendorf decompositions and provides bounds and strategies, including the non-constructive proof of Player 2's winning strategy.

Findings

Game terminates in Zeckendorf decomposition

Upper bound on moves is O(n log n)

Player 2 has a winning strategy for all n > 2

Abstract

Zeckendorf proved that every positive integer $n$ can be written uniquely as the sum of non-adjacent Fibonacci numbers. We use this to create a two-player game. Given a fixed integer $n$ and an initial decomposition of $n = n F_1$, the two players alternate by using moves related to the recurrence relation $F_{n+1} = F_n + F_{n-1}$, and whoever moves last wins. The game always terminates in the Zeckendorf decomposition, though depending on the choice of moves the length of the game and the winner can vary. We find upper and lower bounds on the number of moves possible. The upper bound is on the order of $n\log n$, and the lower bound is sharp at $n-Z(n)$ moves, where $Z(n)$ is the number of terms in the Zeckendorf decomposition of $n$. Notably, Player 2 has the winning strategy for all $n > 2$; interestingly, however, the proof is non-constructive.