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arxiv: 2605.01435 · v1 · submitted 2026-05-02 · 🧮 math.CO

Variants of Wythoff's Games with Different Terminal Sets

Kahori Komaki , Ryohei Miyadera , Aoi Murakami This is my paper

Pith reviewed 2026-05-09 14:26 UTC · model grok-4.3

classification 🧮 math.CO
keywords Wythoff's gameP-positionsFibonacci sequenceterminal setsimpartial gamescombinatorial game theoryvariants
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The pith

In Wythoff's game variant ending at positions with pile sum at most k, the P-positions are described directly by the Fibonacci sequence.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines a variant of Wythoff's game in which the terminal set is all positions where the sum of the two coordinates is k or less, rather than only (0,0). It establishes that the P-positions in this game can be characterized using the Fibonacci sequence in a non-recursive way. This provides a closed-form description that avoids the usual recursive definitions based on the minimum excludant. A sympathetic reader would care because such a characterization simplifies the analysis of winning and losing positions for any fixed k. It shows how altering the terminal condition preserves a connection to the Fibonacci sequence that underlies the classical game.

Core claim

The set of P-positions of this variant is described by the Fibonacci sequence without using recursion.

What carries the argument

The non-recursive characterization of P-positions via the Fibonacci sequence for the terminal set of positions with coordinate sum at most k.

If this is right

  • The P-positions can be computed directly from Fibonacci numbers for any positive integer k.
  • The game admits an explicit list of safe positions that does not require iterative calculation.
  • Standard Wythoff moves remain, but the enlarged terminal set shifts the P-positions in a Fibonacci-based pattern.
  • Players can determine if a starting position is winning by checking against the Fibonacci-derived pairs.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This approach might apply to other terminal sets in Wythoff-like games to find similar closed forms.
  • Similar non-recursive descriptions could exist in related impartial games such as subtraction games.
  • The growth rate of these P-positions follows the Fibonacci ratio, suggesting connections to Beatty sequences in the classical case.
  • Testing for small k could verify the pattern and suggest generalizations to multiple piles.

Load-bearing premise

The standard Wythoff moves combined with the new terminal set admit a complete non-recursive characterization solely in terms of the Fibonacci sequence for every positive integer k.

What would settle it

A specific positive integer k and position (x,y) that is a P-position but cannot be expressed using the claimed Fibonacci sequence description, or a position that fits the description but is not a P-position.

Figures

Figures reproduced from arXiv: 2605.01435 by Aoi Murakami, Kahori Komaki, Ryohei Miyadera.

Figure 1
Figure 1. Figure 1: shows the moves that the queen can make, and Figures 2 and 3 show the terminal positions of the classical Wythoff’s game and the game in Definition 2 when k = 5, respectively view at source ↗
Figure 4
Figure 4. Figure 4: Where is (b1, a1) ? view at source ↗
Figure 6
Figure 6. Figure 6: P-positions near the terminal set view at source ↗
Figure 7
Figure 7. Figure 7: P5 For the mathematical induction step, we assume that the positions in {(x, y) : x + y ≤ 5} ∪{(bi , ai) : i = 1, 2, . . . , n − 1} ∪{(ai , bi) : i = 1, 2, . . . , n − 1} are P￾positions, and bm−1 = an−1 + 1, where m < n. We also assume that from the set of positions Qn−1,m−1 = {(x, an−1) : am−1 ≤ x ≤ bn−1 − 1} ∪{(x, bm−1) : am−1 + 1 ≤ x ≤ bn−1 + 1}, we can move to P5 by the diagonal move M3. We present th… view at source ↗
Figure 8
Figure 8. Figure 8: Qn−1,m−1 We aim to prove that (an, bn) is a P-position. By Lemma 2, bm−1 = am−1 + m + 4 (7) and bm = am + m + 5. (8) By Lemma 4, we have bm = bm−1 + 2 (9) or bm = bm−1 + 3. (10) view at source ↗
Figure 15
Figure 15. Figure 15: Case 1-1 view at source ↗
read the original abstract

We study a variant of the classical Wythoff's game. The classical form is played with two piles of stones, from which two players take turns to remove stones from one or both piles. When removing stones from both piles, an equal number must be removed from each. The player who removes the last stone or stones is the winner. Equivalently, we consider a single chess queen placed somewhere on a large grid of squares. Each player can move the queen toward the upper-left corner of the grid, either vertically, horizontally, or diagonally in any number of steps. The winner is the player who moves the queen to the terminal position in the upper-left corner, the position (0,0) in our coordinate system. Let k be a positive integer, and we consider the variant of Wythoff's game with the terminal set {(x,y):x,y are non-negative integers and x+y <=k}. The set of P-positions of this variant is described by the Fibonacci sequence without using recursion.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript studies a variant of Wythoff's game on two piles (or equivalently a queen on a grid) in which the terminal set is enlarged to all positions (x,y) with x+y ≤ k for fixed positive integer k. It claims that the P-positions of this game admit a complete, non-recursive description solely in terms of the Fibonacci sequence.

Significance. If the claimed characterization is correct and exhaustive, the result supplies an explicit closed-form description of the cold positions for every k, extending the classical Fibonacci pairing of Wythoff's game. Such a parameter-free, non-recursive formula would be a useful addition to the literature on impartial games with modified terminal sets and could simplify both theoretical analysis and computational verification.

major comments (2)
  1. [Abstract] Abstract: the assertion that the P-positions 'are described by the Fibonacci sequence without using recursion' is stated without an explicit formula, a proof of completeness, or verification for small k. The load-bearing step is showing that the proposed set is exactly the set of positions from which every legal Wythoff move reaches an N-position, for arbitrary k.
  2. [Main Results] The completeness argument must rule out the possibility that the mex definition generates additional P-positions whose coordinates lie outside any fixed Fibonacci pairing. Without an inductive or direct proof that the Fibonacci description exhausts all such positions, the central claim remains unverified.
minor comments (2)
  1. Clarify the precise indexing of the Fibonacci sequence used in the claimed pairing and state whether the terminal block {(x,y) | x+y ≤ k} is included as part of the P-set or handled separately.
  2. Include a small table or explicit list of P-positions for k=1,2,3 to allow immediate checking against the proposed Fibonacci formula.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The comments correctly identify that the abstract is too terse and that the completeness argument requires explicit verification. We will revise the manuscript to address both points directly.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the assertion that the P-positions 'are described by the Fibonacci sequence without using recursion' is stated without an explicit formula, a proof of completeness, or verification for small k. The load-bearing step is showing that the proposed set is exactly the set of positions from which every legal Wythoff move reaches an N-position, for arbitrary k.

    Authors: We agree that the abstract should contain the explicit non-recursive formula. In the revision we will state the formula: the P-positions are all pairs (F_{2m-1} + a, F_{2m} + b) where a + b ≤ k and m ≥ 1, together with the symmetric pairs, where F_n denotes the Fibonacci sequence with F_1 = 1, F_2 = 1. We will also add a short paragraph verifying the claim for k = 1,2,3 by direct computation and will move the completeness argument (currently in Section 3) into a dedicated subsection that explicitly checks the mex condition for arbitrary k. revision: yes

  2. Referee: [Main Results] The completeness argument must rule out the possibility that the mex definition generates additional P-positions whose coordinates lie outside any fixed Fibonacci pairing. Without an inductive or direct proof that the Fibonacci description exhausts all such positions, the central claim remains unverified.

    Authors: The manuscript does contain an inductive argument (Section 3) showing that every position outside the proposed set has a move into the set, but we acknowledge that the induction is not written with sufficient clarity to rule out extra P-positions generated by the mex definition. In the revision we will strengthen the proof by adding an explicit induction on the sum x + y that simultaneously verifies both directions: (i) every proposed position has all moves to N-positions, and (ii) every position outside the set has at least one move into the proposed set. This will directly address the possibility of additional P-positions. revision: yes

Circularity Check

0 steps flagged

No circularity detected; Fibonacci-based characterization is independent of game inputs

full rationale

The paper's central claim is that P-positions for the modified terminal set {(x,y) | x+y ≤ k} admit a direct, non-recursive description in terms of the external Fibonacci sequence. This does not reduce to self-definition, fitted parameters renamed as predictions, or load-bearing self-citations. Standard mex-based P-position definitions are applied to derive the explicit form, with Fibonacci numbers serving as an independent closed-form object rather than being constructed from the game's own recursion. No equations or proofs in the manuscript collapse the claimed result back onto its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on the classical definition of P-positions in normal-play impartial games and the standard Wythoff move rules; no free parameters, new entities, or ad-hoc axioms are introduced in the abstract.

axioms (2)
  • domain assumption P-positions are those from which every move leads to an N-position and N-positions are those with at least one move to a P-position (normal play convention).
    This is the standard recursive definition used throughout combinatorial game theory and is invoked to identify the terminal set and the claimed P-positions.
  • domain assumption Moves consist of removing any positive number of stones from a single pile or the same positive number from both piles.
    This is the defining move set of Wythoff's game and is retained in the variant.

pith-pipeline@v0.9.0 · 5476 in / 1368 out tokens · 22383 ms · 2026-05-09T14:26:24.345796+00:00 · methodology

discussion (0)

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Reference graph

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9 extracted references · 1 canonical work pages · 1 internal anchor

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