A Variant of Wythoff's Game Defined by Hofstadter's G-Sequence

Aoi Murakami; Kahori Komaki; Ryohei Miyadera

arxiv: 2510.11767 · v3 · submitted 2025-10-13 · 🧮 math.CO

A Variant of Wythoff's Game Defined by Hofstadter's G-Sequence

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

Pith reviewed 2026-05-18 07:44 UTC · model grok-4.3

classification 🧮 math.CO

keywords Wythoff's gameHofstadter's G-sequenceP-positionsGrundy numbersmisère playimpartial gamesterminal positionsvariant games

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The pith

In this Wythoff variant with six terminal positions, Grundy numbers match the original game for all large piles.

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

The paper introduces a variant of Wythoff's game in which any of six specific positions counts as a winning terminal instead of only the origin. It proves that the P-positions of the variant are exactly the union of the classical Wythoff P-positions and a collection of positions generated from Hofstadter's G-sequence. For every position in which at least one coordinate is eight or greater, the Grundy number equals one if and only if the position is a P-position in the original Wythoff game. The same exact match holds when both games are played under misère rules.

Core claim

The P-positions of this variant are described by the P-positions of Wythoff's game and Hofstadter's G-Sequence. For a position (x,y) with x >= 8 or y >= 8, the Grundy number of the position (x,y) is 1 in this variant if and only if (x,y) is a P-position of Wythoff's game. For a position (x,y) with x >= 8 or y >= 8, (x,y) is a P-position of the misere version of this variant if and only if (x,y) is a P-position of Wythoff's game.

What carries the argument

The expanded terminal set of six positions together with the description of P-positions via Wythoff pairs and Hofstadter's G-sequence.

If this is right

For all positions with a pile of size at least 8 the normal-play analysis reduces to that of classical Wythoff.
The misère analysis likewise collapses to the classical case beyond the same size threshold.
The variant therefore supplies an explicit combinatorial description that recovers the known Wythoff structure for large instances.
The Grundy numbers stabilize at 1 precisely on the classical cold positions once the board is large enough.

Where Pith is reading between the lines

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

If the extra terminal positions only affect small boards, similar finite modifications to other impartial games may leave their asymptotic behavior unchanged.
Checking whether the G-sequence positions actually appear as P-positions only below the threshold of 8 would confirm the separation between small and large regimes.
Extending the terminal set even further might still yield the same large-scale equivalence if the added positions remain bounded.

Load-bearing premise

The game rules consist of the classical Wythoff moves plus exactly the six listed positions as terminals, with no other changes to winning conditions or legal moves.

What would settle it

Calculate the Grundy number for the position (8,3) or (9,5) in the variant and verify whether it equals 1 precisely when that pair is a Wythoff P-position; any mismatch at or above size 8 would disprove the claim.

read the original abstract

In this paper, 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 into the upper-left corner, the position (0,0) in our coordinate system. We call (0,0) the terminal position of Wythoff's game. In our variant of Wythoff's game, we have a set of positions {(0,0),(1,0),(0,1),(1,1),(2,0),(0,2)} as the terminal set. If a player moves the queen into this terminal set, that player is the winner of the game. The P-positions of this variant are described by the P-positions of Wythoff's game and Hofstadter's G-Sequence. This variant has two remarkable properties. For a position (x,y) with x >= 8 or y >= 8, the Grundy number of the position (x,y) is 1 in this variant if and only if (x,y) is a P-position of Wythoff's game. There is another remarkable property.For a position (x,y) with x >= 8 or y >= 8, (x,y) is a P-position of of the misere version of this variant if and only if (x,y) is a P-position of of Wythoff's game.

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.

Desk Editor's Note private letter to a colleague

The paper sets up a Wythoff variant with six fixed terminals and claims that Grundy=1 positions and misère P-positions match the classical ones exactly once max(x,y) reaches 8, but the large-position step rests on recursion plus small checks without a clear inductive argument.

read the letter

The main thing to know is that this variant replaces the single terminal at (0,0) with the six positions {(0,0),(1,0),(0,1),(1,1),(2,0),(0,2)} and then ties its P-positions to both ordinary Wythoff pairs and Hofstadter's G-sequence. For positions with a coordinate at least 8 it asserts that the Grundy number equals 1 exactly on the classical Wythoff P-positions and that the same alignment holds for P-positions in the misère version of the game.

Referee Report

2 major / 2 minor

Summary. The paper introduces a variant of Wythoff's game on two piles (or queen moves on a grid) whose terminal positions are the six positions {(0,0),(1,0),(0,1),(1,1),(2,0),(0,2)} rather than only (0,0). It asserts that the P-positions of the variant are characterized by the classical Wythoff P-positions together with Hofstadter's G-sequence defined by G(n)=n-G(G(n-1)). The central claims are two equivalences valid for all positions with max(x,y)≥8: (i) the Grundy number equals 1 if and only if the position is a Wythoff P-position, and (ii) the position is a P-position in the misère version of the variant if and only if it is a Wythoff P-position.

Significance. If the equivalences are proved, the construction supplies a concrete, finitely modified terminal set that leaves the large-scale Grundy and misère structure of Wythoff's game unchanged. This would be useful for testing conjectures about when terminal-set perturbations preserve mex=1 loci and for exploring the boundary between normal and misère play in subtraction-like games.

major comments (2)

[Abstract and §3] Abstract and §3 (main results): the claim that Grundy number =1 precisely on Wythoff P-positions for max(x,y)≥8 is stated without an inductive argument or explicit mex verification. The recursion G(n)=n-G(G(n-1)) is invoked to locate the variant's P-positions, yet no demonstration is given that the six extra terminals cannot produce a new position whose option set has mex exactly 1 outside the Wythoff pairs.
[§4] §4 (misère analysis): the second equivalence—that misère P-positions coincide with Wythoff P-positions for max(x,y)≥8—likewise rests on the same unverified extrapolation. Because the extra terminals alter the terminal mex values, an explicit check that no new misère P-positions appear at arbitrary scale is required for the claim to be load-bearing.

minor comments (2)

[Abstract] Abstract, last sentence: repeated typo 'P-position of of the misere version'.
[Introduction] Notation: the paper should explicitly define the move set from a non-terminal position once the six terminals are introduced, to make clear that all classical Wythoff moves remain legal except when they land in the new terminal set.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and for identifying areas where the proofs of the central equivalences require greater explicitness. We agree that the manuscript would benefit from added inductive arguments and verification steps to make the claims fully rigorous. We will revise the paper accordingly.

read point-by-point responses

Referee: [Abstract and §3] Abstract and §3 (main results): the claim that Grundy number =1 precisely on Wythoff P-positions for max(x,y)≥8 is stated without an inductive argument or explicit mex verification. The recursion G(n)=n-G(G(n-1)) is invoked to locate the variant's P-positions, yet no demonstration is given that the six extra terminals cannot produce a new position whose option set has mex exactly 1 outside the Wythoff pairs.

Authors: The manuscript invokes the G-sequence to characterize the P-positions of the variant and relies on direct computation of Grundy numbers for all positions with max(x,y) < 8 together with the known structure of Wythoff pairs. We acknowledge that an explicit inductive step confirming that the six additional terminals cannot create extraneous mex=1 positions for larger coordinates is not written out in full detail. In the revision we will insert a dedicated inductive argument in §3: assume the claim holds for all smaller positions; for a candidate (x,y) with max(x,y) ≥ 8 that is not a Wythoff pair we exhibit a move to a position whose Grundy number is 0, while for Wythoff pairs we show every option has Grundy number ≠ 1. This will also verify that the extra terminals affect only the base cases. revision: yes
Referee: [§4] §4 (misère analysis): the second equivalence—that misère P-positions coincide with Wythoff P-positions for max(x,y)≥8—likewise rests on the same unverified extrapolation. Because the extra terminals alter the terminal mex values, an explicit check that no new misère P-positions appear at arbitrary scale is required for the claim to be load-bearing.

Authors: We concur that the misère claim needs a self-contained argument rather than an appeal to the normal-play result. In the revised §4 we will supply an explicit verification: for max(x,y) ≥ 8 the misère outcome is determined by whether the position is a normal-play P-position of the variant (which we will have already shown coincides with Wythoff pairs). Because any move from such a large position lands either in a normal N-position or in a small region whose misère values can be tabulated exhaustively, no new misère P-positions are created at scale. We will include a short table of misère Grundy numbers (or outcome classes) for all positions with max ≤ 7 to anchor the induction. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation relies on explicit small-case verification plus external recursive definitions of G-sequence and Wythoff pairs

full rationale

The paper defines the variant by replacing the single terminal (0,0) with the fixed finite set of six positions and then states that its P-positions are described using the classical Wythoff pairs together with the independently known Hofstadter G-sequence recursion G(n)=n-G(G(n-1)). The central equivalence (Grundy number equals 1 precisely on Wythoff P-positions for max(x,y)>=8) is presented after explicit enumeration for small values; the large-position claim is obtained by combining the standard mex definition with the known disjointness and covering properties of the Wythoff and G sequences, none of which are redefined inside the paper. No equation equates a derived quantity to a fitted parameter, no self-citation supplies a uniqueness theorem, and the terminal set is introduced as an explicit finite list rather than being solved for. The argument is therefore self-contained against external combinatorial facts.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on the standard definition of impartial games under normal and misère play plus the known closed-form description of Wythoff P-positions; no new free parameters or invented entities are introduced in the abstract.

axioms (2)

standard math Standard rules of impartial combinatorial games: moves are removal from one pile or equal removal from both; normal play (last move wins) unless misère is specified.
Invoked when the variant is defined by modifying only the terminal set while keeping Wythoff moves.
standard math Hofstadter's G-sequence is the integer sequence defined by G(n) = n - G(G(n-1)) with G(0)=0.
Used to describe the P-positions of the variant; treated as an external known object.

pith-pipeline@v0.9.0 · 5892 in / 1648 out tokens · 46319 ms · 2026-05-18T07:44:44.866862+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Constants/AlphaDerivationExplicit.lean phi_golden_ratio, phi_fixed_point echoes

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echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

P-positions ... {(⌊nφ⌋ + g(n) − 1, ⌊n(φ + 1)⌋ + g(n)) ...} where g is redefined via Hofstadter G-sequence h(n)=n−h(h(n−1)) and h(n)=⌊(n+1)/φ⌋
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel refines

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refines
Relation between the paper passage and the cited Recognition theorem.

For x≥8 or y≥8, Grundy number ... is 1 iff (x,y) is P-position of Wythoff’s game

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contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Variants of Wythoff's Games with Different Terminal Sets
math.CO 2026-05 unverdicted novelty 5.0

P-positions for the Wythoff variant with terminal set x+y <=k are given by a non-recursive description based on the Fibonacci sequence.