Bounds on Zeckendorf Games

Zeckendorf proved that every positive integer \(n\) can be written uniquely as the sum of non-adjacent Fibonacci numbers. We use this decomposition to construct 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; depending on the choice of moves the length of the game and the winner can vary, though for \(n\ge 2\) there is a non-constructive proof that Player 2 has a winning strategy. Initially the lower bound of the length of a game was order \(n\) (and known to be sharp) while the upper bound was of size \(n \log n\). Recent work decreased the upper bound to of size \(n\), but with a larger constant than was conjectured. We improve the upper bound and obtain the sharp bound of \(\frac{\sqrt{5}+3}{2}\ n - IZ(n) - \frac{1+\sqrt{5}}{2}Z(n)\), which is of order \(n\) as \(Z(n)\) is the number of terms in the Zeckendorf decomposition of \(n\) and \(IZ(n)\) is the sum of indices in the Zeckendorf decomposition of \(n\) (which are at most of sizes \(\log n\) and \(\log^2 n\) respectively). We also introduce a greedy algorithm that realizes the upper bound, and show that the longest game on any \(n\) is achieved by applying splitting moves whenever possible.