Reversed Zeckendorf Game Analysis

• The reversed Zeckendorf game is a combinatorial game defined by reversing standard Fibonacci partition moves, ending when all tokens are in the first bin.
• The game’s methodology employs split and combine moves that strictly decrease an index-sum monovariant, ensuring termination and enabling rigorous strategy analysis.
• Explicit Player 1-winning families are established through strategy-stealing and parity-copycat methods, highlighting intricate combinatorial dynamics.

The reversed Zeckendorf game is a two-player combinatorial game defined by applying reversed move operations to partitions of natural numbers into Fibonacci numbers, beginning from a number’s Zeckendorf decomposition and ending when all tokens occupy the first bin. Unlike the original Zeckendorf game, which always terminates at the Zeckendorf decomposition and supports a guaranteed Player 2 win for $n\geq3$, the reversed variant manifests more nuanced win structures, including explicit Player 1-win families for certain forms of %%%%1%%%% (Batterman et al., 2023).

1. Formal Definition and Move Set

Let $(F_1, F_2, F_3, \dots) = (1, 2, 3, 5, 8, \dots)$ denote the Fibonacci sequence with $F_1 = 1$, $F_2 = 2$. Zeckendorf’s theorem asserts that every $n \in \mathbb{N}$ has a unique sum representation as $\sum_{i\ge1} h_i F_i$, with $h_i \in \{0,1\}$ such that $h_i h_{i+1}=0$. This “bin-vector” $(h_1, h_2,\dots)$ encodes the decomposition.

The reversed Zeckendorf game starts with this bin-vector and ends when all chips have been moved into the first bin, i.e., terminal state $(n, 0, 0, \dots)$. Legal moves—which precisely invert the standard Zeckendorf operations—are:

• Split move: For any $i\ge2$ with $h_{i+1}\ge1$, replace one $F_{i+1}$ with $F_i + F_{i-1}$; i.e., $$(h_{i-1}, h_i, h_{i+1}) \mapsto (h_{i-1}+1, h_i+1, h_{i+1}-1)$$. For $i=1$, $F_2 \mapsto 2F_1$.
• Combine move: For any $i\ge3$ with $h_{i-2}\ge1$ and $h_{i+1}\ge1$, combine $F_{i+1} + F_{i-2} \mapsto 2F_{i}$; i.e., $$(h_{i-2}, h_i, h_{i+1}) \mapsto (h_{i-2} -1, h_i+2, h_{i+1}-1)$$. For $i=2$, $F_3 + F_1 \mapsto 2F_2$.

Every move conserves $n$, and each action strictly decreases the total index-sum $I = \sum_i i h_i$, enforcing guaranteed termination.

2. Termination Properties and Monovariants

The reversed Zeckendorf game is always terminating; no infinite play is possible. The monovariant $I = \sum_i i h_i$ strictly decreases after each move. The unique terminal state is $(n, 0, 0, \dots)$, with all tokens in bin 1. This terminating behavior structurally contrasts with the endpoint characterization of the original, “forward” game, which concludes at a Zeckendorf decomposition (Batterman et al., 2023).

3. Winning Strategy for $n=F_{i+1}+F_{i-2}$ and Parity Analysis

A central result is the explicit Player 1-winning strategy for positions where $n = F_{i+1} + F_{i-2}$. Two proofs are outlined:

• Strategy-stealing argument: Any assumed winning strategy for Player 2 from the start position $(0,\dots,0,1,0,\dots,1,0,\dots)$ (one $F_{i+1}$ and one $F_{i-2}$) can be co-opted by Player 1 using the available moves—either a Combine yielding $2F_i$ or a Split creating $F_i+F_{i-1}+F_{i-2}$—and by symmetry, Player 1 can force a win.
• Constructive parity-copycat method: This hinges on the “Even-Heights Copycat” lemma. If every bin height $h_i$ is even, the second player can always mirror the first player's move, preserving evenness. The corollary is that positions with $2F_i$ (i.e., $h_i=2$, others zero) are always second-player wins. Therefore, Player 1's initial move to $2F_i$ guarantees a forced win (Batterman et al., 2023).

4. Variants and Ancillary Results

Multiple variants and further outcomes are rigorously classified:

• Game-length extremes: The shortest possible game is $n-Z(n)$, where $Z(n)$ is the number of terms in the Zeckendorf decomposition. The longest length is bounded by $$\lfloor \phi^2 n - Z_I(n) - 2Z(n) + \phi - 1 \rfloor$$, with $Z_I(n) = \sum_{j}j$ over Zeckendorf terms.
• Random-play statistics: Game lengths are equidistributed modulo $Z$ for any $Z\ge1$ in both reversed and forward games.
• Alternative starting partitions: Starting with arbitrary bin configurations $(a,b,c)$ for $F_1,F_2,F_3$, win/loss outcomes can be classified via parity and the relations between $a$ and $c$ (see Theorem 4.1). Proofs proceed via case-by-case parity induction and forced replies.

A two-phase variant, the “Build-Up 1–2–3 Game,” first partitions $n$ as a sum of 1’s, 2’s, and 3’s with subsequent reversed play. The outcome is: $n$ odd $\implies$ Player 1 wins; $n$ even ($n\neq4$) $\implies$ Player 2 wins.

5. Illustrative Examples

Concrete small-$n$ instances clarify mechanics and typical outcomes:

$n$	Zeckendorf Start Bins	First Move(s)	Outcome
2	$(2,0,0,\dots)$	$F_2 \mapsto 2 F_1$	Player 2 win
5 ($F_4+F_2$)	$(0,1,0,1)$	Combine $F_4+F_2\to2F_3$	Player 2 win
11 ($F_6+F_3$)	$(0,1,0,0,1)$	Combine $F_6+F_3\to2F_5$	Player 1 win

These cases illustrate the reversal of “forward” Zeckendorf strategy logic and the emergence of forced wins dependent on parity structure and form of $n$ (Batterman et al., 2023).

6. Computational Complexity, Broader Implications, and Open Problems

Enumeration of outcomes for all $n \leq 129$ via breadth-first search reveals exponential growth of potential game states, roughly $\exp(O(\sqrt n))$. Current numerics indicate the proportion of Player 1-wins for $n \leq N$ appears to approach $1/\varphi \approx 0.618$.

Open directions include:

• Constructing infinite families of Player 2-win positions.
• Exploring reversed variants in other impartial games (such as Chomp).
• Investigating “stagnant 1” variants, where moves involving $F_1$ are disallowed.
• Seeking direct correspondence between forward and reverse game strategies; notably, the conjecture that the challenge of designing constructive strategies for the forward game mirrors the nontriviality of win/loss patterns in the reversed game.

A plausible implication is that the complexity and richness of the reversed Zeckendorf game offer pathways to deeper combinatorial and algorithmic investigations in the context of Fibonacci number partition games (Batterman et al., 2023).