Zeckendorf-Based Game

• Zeckendorf-based game is an impartial combinatorial game defined by unique Fibonacci decompositions and rules based on the Fibonacci recurrence.
• It employs move types like combine and split, using a monovariant (term plus index sum) to guarantee termination at a unique Zeckendorf decomposition.
• Rigorous analyses of move complexity, algorithmic strategies, and probabilistic outcomes offer deep insights into impartial game theory and recurrence-based decompositions.

A Zeckendorf-based game is a finite combinatorial game defined by the structure of Zeckendorf decompositions and the Fibonacci recurrence. These games, introduced in (Baird-Smith et al., 2018) and further developed in (Cusenza et al., 2020, Li et al., 2020), and subsequent works, generalize to positive linear recurrence sequences, support rigorous results on move complexity, and serve as archetypal case studies in impartial game theory for arithmetically-motivated move sets. Below is a comprehensive treatment of their setup, termination, extremal and probabilistic properties, and strategic theory.

1. Formal Definition and Rules

Let $F_1 = 1$, $F_2 = 2$, and $F_{k+1} = F_k + F_{k-1}$ for $k \ge 2$. Zeckendorf’s Theorem states every $n \in \mathbb{N}$ admits a unique decomposition as a sum of non-adjacent Fibonacci numbers. The Zeckendorf-based game operationalizes this as a two-player impartial game on multisets of Fibonacci numbers:

• Initial position: For fixed $n$, the game starts with the multiset $\{ F_1^{\, n} \}$ (i.e., $n$ copies of $F_1$).
• Moves: At each turn, a player must perform exactly one of the following (with current multiset $\mathcal{S}$):
  1. Combine consecutive: If $F_{k-1}, F_k \in \mathcal{S}$, replace them with $F_{k+1}$. Formally,

  $\{ F_{k-1}, F_k \} \longrightarrow \{ F_{k+1} \}$

1. Split duplicates: If $\mathcal{S}$ has two copies of $F_k$, replace as follows
   • For $k=1$: $\{ F_1, F_1 \} \to \{ F_2 \}$
   • For $k=2$: $\{ F_2, F_2 \} \to \{ F_1, F_3 \}$
   • For $k \ge 3$: $\{ F_k, F_k \} \to \{ F_{k-2}, F_{k+1} \}$

• Termination: The game ends when no move is possible, i.e., the current multiset is the Zeckendorf decomposition of $n$ (all summands distinct, no two consecutive indices). The last player to move is declared the winner (Cusenza et al., 2020).

2. Termination and Structure

Termination is guaranteed by a monovariant argument:

Let $T$ be the total number of terms (with multiplicity) in the current multiset, and $I$ the sum of their indices. Define $M = T + I$. Under every combine move, $M$ strictly decreases, while under split moves, $M$ is preserved. Since $M$ is bounded below, and only a finite range of multisets of given sum $n$ arise, all plays terminate after finitely many steps (Cusenza et al., 2020, Baird-Smith et al., 2018).

The unique terminal position is the Zeckendorf decomposition of $n$, as all other multisets admit at least one legal move by the recurrence relation. This is a direct combinatorial manifestation of Zeckendorf’s theorem.

3. Extremal Bounds on Game Length

Denote by $L(n)$ the maximal number of moves and by $\ell(n)$ the minimal number of moves in any Zeckendorf-based game on $n$.

• Minimal length:

$\ell(n) = n - Z(n)$

where $Z(n)$ is the number of terms in the Zeckendorf decomposition of $n$ (Li et al., 2020, Cusenza et al., 2020). This is realized by the "Combine Largest" greedy strategy: always combine the highest-index available pair, then combine $1$'s, and only split if forced.

• Sharp upper bound (Cusenza–D’Antonio et al.):

$$L(n) \le \frac{\sqrt{5}+3}{2}\, n - IZ(n) - \frac{1+\sqrt{5}}{2}\, Z(n)$$

where $IZ(n)$ is the sum of indices in the Zeckendorf decomposition. The leading constant is $\varphi^2 = (3+\sqrt{5})/2 \approx 2.618$, and the subtracted terms are of order $O((\log n)^2)$ and $O(\log n)$, respectively (Cusenza et al., 2020). The sharp upper bound is achieved via a split-priority strategy: whenever possible, split; otherwise, combine minimal consecutive indices. The deterministic "Split Smallest" variant achieves $L(n)$ (Li et al., 2020).

• Asymptotic behavior: $Z(n) = \Theta(\log n)$, $IZ(n) = \Theta((\log n)^2)$, so $L(n) = \varphi^2\, n + o(n)$ (Cusenza et al., 2020).
• Interval of attainable values: For every integer $m$ with $n-Z(n) \le m \le L(n)$, there exists a Zeckendorf-based game of length exactly $m$ (Cheigh et al., 2022).

4. Algorithmic and Probabilistic Properties

There is a combinatorial explosion in the number of possible plays:

• The number of distinct shortest games on $n$ is at least $\prod_{k=1}^{\ell-2} \mathrm{Cat}(F_{k})$, where $\mathrm{Cat}(m)$ is the $m$th Catalan number and $F_\ell \le n < F_{\ell+1}$ (Cheigh et al., 2022).

Probabilistic studies establish that:

• Under both the uniform measure (all games equally likely) and the move-uniform measure (at each turn, moves chosen uniformly at random), as $n \to \infty$,

$$\mathbb{P}(\text{Player 1 wins}) \to \tfrac{1}{2}, \quad \mathbb{P}(\text{Player 2 wins}) \to \tfrac{1}{2}$$

More generally, with $Z$ players, each wins with limiting probability $1/Z$ (Cheigh et al., 2022).

• For large $n$, the distribution of game lengths (number of moves) under random play approaches a Gaussian on natural subfamilies defined by move-type partitions, with the full unconditional limit conjectured to be Gaussian (Cheigh et al., 2022).

5. Strategic and Game-Theoretic Analysis

• Two-player classic case: For $n \ge 2$, Player 2 has a non-constructive winning strategy, proven via the standard "strategy-stealing" argument: if Player 1 could force a win, Player 2 could "pretend" to be Player 1 after the first move, yielding a contradiction (Baird-Smith et al., 2018, Cusenza et al., 2020, Li et al., 2020). No explicit constructive strategy is known.
• Deterministic variants: Four deterministic protocols were analyzed (Li et al., 2020):
  • Combine Largest and Split Largest: Realize the sharp lower bound $\ell(n)$.
  • Split Smallest: Attains the (nearly) sharp upper bound $L(n)$, with empirical growth rate close to $\varphi^2\, n$.
  • Combine Smallest: Intermediate complexity, conjectured to be $\sim 1.20647\, n$ moves.
• Generalizations and multiplayer behaviors:
  • For $p \ge 3$ players and $n \ge 5$, no player has a deterministic winning strategy due to robust stealing patterns (Cusenza et al., 2020, Miller et al., 2022).
  • Alliance games: For $t\ge 3$ alliances of $k = t-1$ consecutive players, no team can force a win for sufficiently large $n$ (Cusenza et al., 2020).
  • In two-team splits with $p\ge6$ and one alliance of size $p-2$, the larger team has a winning strategy beyond certain thresholds (Miller et al., 2022, Cusenza et al., 2020).
• Order-constrained variants and reversed games:
  • The ordered Zeckendorf game, restricting moves to ordered summands, gives a fundamentally altered landscape: Player 1 can force a win for almost all $n \le 25$ except $n = 18$, with longest play lengths growing as $\sim \frac{n^2}{2}$ (Bortnovskyi et al., 27 Aug 2025).
  • The reversed Zeckendorf game (starting at the terminal Zeckendorf decomposition and inverting moves) yields infinitely many Player 1 wins, explicitly for $n = F_{i+1} + F_{i-2}$ (Batterman et al., 2023).

6. Generalizations to Other Recurrence Relations

The setup extends to positive linear recurrence sequences (PLRS) and even non-constant recurrences:

• For $(c,k)$-nacci games, the recurrence $G_n = \sum_{i=1}^{k} c\, G_{n-i}$ defines the game mechanics. Winning strategies depend on the parity of $c$: for sufficiently large $n$, Player 2 wins if $c$ odd, Player 1 wins if $c$ even (Miller et al., 2022, Baird-Smith et al., 2018).
• Games built on non-constant recurrences, such as $a_{n+1} = n\, a_n + a_{n-1}$, admit unique decompositions and analogous play structures. For two-player games, both first and second player wins occur, depending on the parity of shortest play (Bołdyriew et al., 2020).

7. Connections and Applications

These games serve as fertile ground for research in combinatorial and number-theoretic games, bridging unique base and recurrent decompositions, impartial game theory, and probabilistic combinatorics.

Key conceptual links include:

• Potential-theoretic monovariants governing termination and bounds
• Deep links between move structure and Zeckendorf-type representation theory
• Rich dynamical systems in deterministic versus randomized play paths
• Alliance and parity-based strategy-stealing phenomena, robust across recurrence classes

Further research is active in extending explicit strategy characterization, analyzing random play, and generalizing to new recurrence and tiling-based decompositions (Cusenza et al., 2020, Miller et al., 2022, Bortnovskyi et al., 27 Aug 2025).