---
title: Zeckendorf-Based Game
url: https://www.emergentmind.com/topics/zeckendorf-based-game
type: topic
---

# Zeckendorf-Based Game

A Zeckendorf-based game is a finite combinatorial game defined by the structure of Zeckendorf decompositions and the Fibonacci recurrence. These games, introduced in [1809.04881] and further developed in [2009.09510], [2006.16457], 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,
      $$
      \{ F_{k-1}, F_k \} \longrightarrow \{ F_{k+1} \}
      $$
  2. **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 [2009.09510].

## 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 [2009.09510], [1809.04881].

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 $n$ [2006.16457], [2009.09510]. 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 [2009.09510]. 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 $L(n)$ [2006.16457].

- **Asymptotic behavior:** $Z(n) = \Theta(\log n)$, $IZ(n) = \Theta((\log n)^2)$, so $L(n) = \varphi^2\, n + o(n)$ [2009.09510].

- **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 $m$ [2210.11038].

## 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 $F_\ell \le n < F_{\ell+1}$ [2210.11038].

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 $1/Z$ [2210.11038].

- 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 [2210.11038].

## 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 [1809.04881], [2009.09510], [2006.16457]. No explicit constructive strategy is known.

- **Deterministic variants:** Four deterministic protocols were analyzed [2006.16457]:
  - **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 [2009.03708], [2211.14973].
  - Alliance games: For $t\ge 3$ alliances of $k = t-1$ consecutive players, no team can force a win for sufficiently large $n$ [2009.03708].
  - In two-team splits with $p\ge6$ and one alliance of size $p-2$, the larger team has a winning strategy beyond certain thresholds [2211.14973], [2009.03708].

- **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 $\sim \frac{n^2}{2}$ [2508.20222].
  - The reversed Zeckendorf game (starting at the terminal Zeckendorf decomposition and inverting moves) yields infinitely many Player 1 wins, explicitly for $n = F_{i+1} + F_{i-2}$ [2309.12748].

## 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 [2211.14973], [1809.04883].
- 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 [2009.12475].

## 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 [2009.09510], [2211.14973], [2508.20222].

Source: https://www.emergentmind.com/topics/zeckendorf-based-game