Ordered Zeckendorf Game Overview

• Ordered Zeckendorf Game is a two-player combinatorial game that uses Fibonacci-based moves to transform an initial degenerate state into a unique Zeckendorf decomposition.
• The game employs legal moves including merging, splitting, and switching, with a strictly decreasing monovariant ensuring termination and revealing novel winning patterns.
• Analyses demonstrate tight bounds on game lengths, a log-Gaussian distribution of move trajectories, and prompt further exploration into algorithmic complexity and generalized recurrences.

The Ordered Zeckendorf Game is a combinatorial two-player adversarial process grounded in the number-theoretic structure of Zeckendorf's Theorem. The game transforms the unordered multiset dynamics of the classical Zeckendorf Game into a richer, order-sensitive arena, where adjacency constraints and additional move types yield complex strategic phenomena. Each play begins from the degenerate Fibonacci sum $(F_1, \dots, F_1)$ and evolves through a sequence of legal local moves on neighboring terms—which include merges, splits, and switches—culminating in the strictly increasing Zeckendorf decomposition of a given $n$. The study of this ordered variant reveals new patterns in winner determination, game-length extremality, and probabilistic trajectory distributions, and it connects with broader questions in algorithmic combinatorics and integer decomposition games (Bortnovskyi et al., 27 Aug 2025).

1. Formal Definition and Move-Set

Let $F_1 = 1$, $F_2 = 2$, and $F_{k+1} = F_k + F_{k-1}$ for $k \geq 2$. For a fixed $n \in \mathbb{N}$, the initial state is the ordered tuple $S_0 = (F_1,\,F_1,\,\dots,\,F_1)$, with $n$ copies.

On each turn, a player must perform exactly one of the following legal moves on adjacent tuple entries (see (Bortnovskyi et al., 27 Aug 2025), Definition 1.2):

• Merging: If two neighbors satisfy $i_{j+1} = i_j + 1$, then $$(\,F_{i_j},\,F_{i_j+1}\,) \longrightarrow (\,F_{i_j+2}\,)$$.
• Merging Ones: $(F_1, F_1) \longrightarrow (F_2)$.
• Splitting (for $i_j > 2$): $$(\,F_{i_j}, F_{i_j}\,) \longrightarrow (\,F_{i_j-2}, F_{i_j+1}\,)$$.
• Splitting Twos: $(F_2,F_2) \longrightarrow (F_1,F_3)$.
• Switching: If $i_j > i_{j+1}$, swap the terms: $$(F_{i_j},F_{i_{j+1}}) \longrightarrow (F_{i_{j+1}},F_{i_j})$$.

The termination condition requires that the tuple be strictly increasing and admit no further merge or split. Theorem 1.3 (Bortnovskyi et al., 27 Aug 2025) asserts that the terminal state is exactly the ascending Zeckendorf decomposition: the unique sum of non-consecutive Fibonacci numbers.

2. Monovariant and Termination Analysis

Termination and correctness are established by constructing a strictly decreasing, integer-valued monovariant:

$f(S) = \sum_{j=1}^k (k+1-j)\,F_{i_j}$

where $S = (F_{i_1}, F_{i_2}, \dots, F_{i_k})$ is the current tuple ((Bortnovskyi et al., 27 Aug 2025), Theorem 2.1). At each move, $f(S)$ decreases by at least one. Specifically:

• Merges and splits decrease $f$ by at least one Fibonacci unit.
• Switches strictly reduce $f$ whenever applied.

The initial value is $f(S_0) = n(n+1)/2$, while the minimum possible is $n$ (sum of the Zeckendorf decomposition summands). Since $f$ cannot decrease indefinitely, the process always terminates, guaranteeing arrival at the unique decomposition (otherwise a merge or split would still be available).

3. Win/Loss Structure for Small $n$

The strategic implications of the order restriction are substantial. In the unordered Zeckendorf game, Player 2 wins for all $n > 2$ due to a parity-based invariance ([Baird-Smith et al., cited in (Bortnovskyi et al., 27 Aug 2025)]). The ordered game, however, exhibits a different empirical pattern for $n \leq 25$:

$n$	Winner
$1 \leq n \leq 17$	Player 1 (always)
$n = 18$	Player 2 (unique)
$19 \leq n \leq 25$	Player 1 (always)

These results are obtained via exhaustive game-tree search under optimal play ((Bortnovskyi et al., 27 Aug 2025), Section 4.2). The need for ordered moves and the additional switch operation annihilate the classical parity argument and rebalance the game in favor of Player 1 for nearly all small $n$. This suggests fundamentally different strategic landscapes between the ordered and unordered variants.

4. Extremal Game Lengths

Game-length analysis distinguishes sharply between minimal and maximal trajectories.

• Shortest Game: Each merge reduces the tuple length by one. The minimum number of moves required is exactly

$n - Z(n)$

where $Z(n)$ is the number of terms in Zeckendorf's decomposition of $n$ ((Bortnovskyi et al., 27 Aug 2025), Proposition 3.1).

• Longest Game: Via Theorem 2.1 and subsequent refined constructions:

$M(n) \leq \frac{n(n-1)}{2}$

and, via explicit "Long Game Strategy" (Definition 3.2),

$$M(n) \geq \frac{n^2}{2} - n \log_\varphi n + o(n \log n)$$

where $\varphi = \frac{1+\sqrt{5}}{2}$ is the golden ratio ((Bortnovskyi et al., 27 Aug 2025), Theorems 3.4, 3.6, Corollary 3.7).

The asymptotic form is $M(n) \sim n^2/2$ for large $n$.

5. Distribution of Random Trajectories

Empirical simulation of $10^4$ random games for $n=150$ reveals that the length distribution of random trajectories is well approximated by a log-Gaussian shape ((Bortnovskyi et al., 27 Aug 2025), Section 4.1, Figures 4.1–4.2). The histograms show that the probability mass for wins is roughly evenly split between players. Conjecture 4.3 posits:

As $n \to \infty$, under uniformly random play, the distribution of game lengths converges to a log-Gaussian, and each player has equal winning probability.

This behavior reflects underlying multiplicative effects in move sequences and the high combinatorial complexity of allowed transitions.

6. Generalizations and Algorithmic Directions

Several avenues for extension and complexity analysis are identified ((Bortnovskyi et al., 27 Aug 2025), Section 5):

• General Recurrence Relations: The replacement of Fibonacci numbers with $k$-bonacci or arbitrary linear recurrences requires adaptation of the allowed merges and splits. Whether the ordered game's termination and bounds extend to this setting is an open question ([Boldyriew et al., cited in (Bortnovskyi et al., 27 Aug 2025)]).
• Computational Complexity: The game-tree's exponential growth renders winner determination (minimax search) demanding, raising the prospect of PSPACE-completeness, as is typical for generalized placement games.
• Probabilistic Limit Theorems: Deriving the conjectured log-Gaussian law for move-lengths rigorously is a nontrivial open problem, likely to involve advanced probabilistic tools (branching processes, martingale CLTs).
• Algebraic Invariants: Refinements of the main monovariant, potentially including inversion counts or gap statistics, might yield finer-grained control over trajectory length distributions and transition dynamics.
• Multiplayer or Coalition Games: Allowing more than two players or incorporating alliances, possibly with order and switching moves, could generate yet deeper combinatorial and algorithmic structures ([Cusenza et al., cited in (Bortnovskyi et al., 27 Aug 2025)]).

These facets point to the Ordered Zeckendorf Game as a rich locus for ongoing research at the intersection of additive number theory, combinatorial game theory, and computational complexity.