The Ordered Zeckendorf Game

Abstract

We introduce and analyze the ordered Zeckendorf game, a novel combinatorial two-player game inspired by Zeckendorf's Theorem, which guarantees a unique decomposition of every positive integer as a sum of non-consecutive Fibonacci numbers. Building on the original Zeckendorf game\ -- previously studied in the context of unordered multisets\ -- we impose a new constraint: all moves must respect the order of summands. The result is a richer and more nuanced strategic landscape that significantly alters game dynamics. Unlike the classical version, where Player 2 has a dominant strategy for all $n > 2$, our ordered variant reveals a more balanced and unpredictable structure. In particular, we find that Player 1 wins for nearly all values $n \leq 25$, with a single exception at $n = 18$. This shift in strategic outcomes is driven by our game's key features: adjacency constraints that limit allowable merges and splits to neighboring terms, and the introduction of a switching move that reorders pairs. We prove that the game always terminates in the Zeckendorf decomposition\ -- now in ascending order\ -- by constructing a strictly decreasing monovariant. We further establish bounds on game complexity: the shortest possible game has length exactly $n - Z(n)$, where $Z(n)$ is the number of summands in the Zeckendorf decomposition of $n$, while the longest game exhibits quadratic growth, with $M(n) \sim \frac{n^2}{2}$ as $n \to \infty$. Empirical simulations suggest that random game trajectories exhibit log-normal convergence in their move distributions. Overall, the ordered Zeckendorf game enriches the landscape of number-theoretic games, posing new algorithmic challenges and offering fertile ground for future exploration into strategic complexity, probabilistic behavior, and generalizations to other recurrence relations.

Figures (5)

• Figure 1: Directed graph showing the possible ways to move between game states in Lemma \ref{['lem:higher-repetitions']}.
• Figure 2: State-transition graph of the ordered Zeckendorf Game with lower-order term tracking.
• Figure 3: Frequency graphs of the number of moves in 10,000 simulations of the Zeckendorf Game with random moves when $n = 150$ with the best fit log Gaussian over the data points.
• Figure 4: The distribution of outcomes for 10,000 random games for $n = 150$.
• Figure 5: As $n \to \infty$, the ratio $\dfrac{\frac{n^2}{2}-M(n)}{n\log_\phi(n)}$

Theorems & Definitions (20)

• Definition 1.1: The Two Player Zeckendorf Game
• Definition 1.2: The Two-Player Ordered Zeckendorf Game
• Theorem 1.3: Termination and Final State
• Proposition 1.4: Shortest Game
• Theorem 1.5: Upper Bound on the Maximal Game Length
• Definition 1.6: Long Game Strategy
• Conjecture 1.7: Strategy for Maximum Length
• Definition 1.8: Higher-Index Terms and Lower-Index Terms
• Lemma 1.9: Repetitions in Higher Index Terms
• Theorem 1.10: Lower Bound of Maximal Game Length