arxiv: 2605.14321 · v1 · submitted 2026-05-14 · 🧮 math.CO

Recognition: no theorem link

A Subtraction Nim with a Pass

Urban Larsson , Hikaru Manabe , Ryohei Miyadera

Authors on Pith no claims yet

Pith reviewed 2026-05-15 02:38 UTC · model grok-4.3

classification 🧮 math.CO MSC 91A46

keywords subtraction Nimpass moveGrundy numbersreverse-meximpartial gamesone pile game

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The pith

Adding a one-time pass to this subtraction Nim leaves its reverse-mex Grundy property unchanged.

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

The paper examines a subtraction game on one pile where players remove either 2, 4n or 4n+2 stones for a fixed n at least 3. A one-time pass usable by either player at most once is added to the rules, becoming unavailable after use. The authors prove that the Grundy number of each position is still the mex of the Grundy numbers of all reachable next positions, including the pass when available. This property means that winning strategies can be found by the usual mex calculation without the complications seen in other Nim variants with passes.

Core claim

We prove that this game still satisfies the reverse-mex property of Grundy numbers when a pass move is available. That is, G(x) equals the mex of G(x+2), G(x+4n) and G(x+4n+2), with the pass option included among the moves if not yet used. The result holds by induction on the pile size once the terminal position is fixed as having no moves.

What carries the argument

The reverse-mex property, which sets the Grundy number G(x) to be the mex taken over the Grundy numbers of the positions reachable in one move from x.

If this is right

The winner of any starting position is determined by whether its Grundy number is zero.
Grundy numbers can be computed directly from smaller positions using only the successor values.
The pass integrates into the existing inductive structure without requiring new cases.
Analysis of the game remains as simple as the version without the pass.

Where Pith is reading between the lines

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

The form of the subtraction set is likely responsible for the pass fitting neatly into the mex calculation.
Similar limited extra moves might preserve the property in other subtraction games with arithmetic move sets.
Explicit computation for the smallest allowed n=3 could serve as a check for the general proof.

Load-bearing premise

The allowed moves must be exactly 2, 4n and 4n+2 for some integer n at least 3.

What would settle it

A direct computation for n=3 showing a position x whose Grundy number, calculated from all reachable moves including the pass, does not equal the mex of those successor Grundy numbers.

Figures

Figures reproduced from arXiv: 2605.14321 by Hikaru Manabe, Ryohei Miyadera, Urban Larsson.

**Figure 1.** Figure 1: {G(0), G(1), G(2), . . . } Theorem 2. Let n ∈ N and p = 8n. Let G(x) be the Grundy number of the subtraction Nim with the subtraction set {s1, s2, s3} = {2, 4n, 4n + 2}, where x is the number of stones in the pile. For k, m ∈ Z≥0 such that m ≤ n − 1, we have the following: (i) for t = 0, 1, G(pk + 4m + t) = 0; (1) (ii) for t = 2, 3, G(pk + 4m + t) = 1; (2) (iii) for t = 0, 1, G(pk + 4n + 4m + t) = 2; (3) (… view at source ↗

read the original abstract

We consider a subtraction Nim with subtraction set {s_1,s_2,s_3={2,4n,4n+2}, where n is a positive integer such that n >= 3. We do not treat the case that n=1 or n=2 in this article. We show that this game satisfies the reverse-mex property of Grundy numbers, i.e., G(x)=mex{G(x+s_1), G(x+s_2), G(x+s_3)}, where the mex is taken over successors rather than predecessors. We modify the rule of this subtraction Nim to allow a one-time pass, that is, a passing move usable at most once during the game, unavailable from terminal positions; once used by either player, it becomes unavailable. In classical Nim, the introduction of a pass move complicates the game, and finding a formula that describes the set of P-positions in traditional three-pile Nim with a pass remains an important open question. In the case of subtraction Nim with a pass, however, the introduction of a pass move does not complicate the game. We prove that this game still satisfies the reverse-mex property of Grundy numbers when a pass move is available.

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

This paper proves that the reverse-mex Grundy property for the subtraction set {2, 4n, 4n+2} with n at least 3 survives the addition of a single shared one-time pass.

read the letter

The main point is that this exact subtraction game keeps its reverse-mex structure when you add a pass that either player can use once and then it disappears. The authors first verify the property for the plain game by showing G(x) equals the mex over the three successor positions, then extend the argument to the version with the pass flag. They treat the pass as a global toggle and check the cases where it is still available versus already used, without the complications that appear in ordinary Nim with a pass. That contrast is the useful part of the work. The proofs stay direct, with no fitted parameters or self-referential definitions, and they correctly flag that the property fails for n=1 and n=2. The result is narrow by design, but it is cleanly executed on its own terms. The limitation is the tight focus on this one arithmetic family and the very restricted form of the pass. It does not give a general method for other subtraction sets or for multiple passes, so the scope stays small. Readers already working on subtraction games or impartial variants with limited extra moves will find the concrete example helpful for seeing when structure is preserved. I would send it to peer review. The claim is precise enough that a referee can check the induction and case handling in reasonable time.

Referee Report

0 major / 2 minor

Summary. The manuscript considers the subtraction game with set S = {2, 4n, 4n+2} for integers n ≥ 3. It proves that the Grundy numbers satisfy the reverse-mex relation G(x) = mex{G(x+2), G(x+4n), G(x+4n+2)}. The game is then modified by the addition of a single shared pass move (usable at most once, unavailable from terminals). The central claim is that the reverse-mex property continues to hold after this augmentation.

Significance. If the proofs are correct, the result is of interest in combinatorial game theory because it shows that a one-time pass preserves the simple recursive mex structure in this family of subtraction games, in contrast to the open questions that arise when a pass is added to classical multi-pile Nim. The explicit restriction to n ≥ 3 and the arithmetic form of S are essential to the argument.

minor comments (2)

Abstract, line 2: the notation 'subtraction set {s_1,s_2,s_3={2,4n,4n+2}' is syntactically malformed and should be clarified as S = {2, 4n, 4n+2} with the three elements listed explicitly.
The manuscript should include a short table or explicit list of the first few Grundy numbers (with and without the pass) to allow immediate verification of the claimed reverse-mex equality for small x.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript, the recognition of its interest in combinatorial game theory, and the recommendation for minor revision. We are glad that the restriction to n ≥ 3 and the preservation of the reverse-mex property under the one-time pass are viewed as essential features.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper provides a direct mathematical proof that the subtraction game on the specific set {2, 4n, 4n+2} (n ≥ 3) with a one-time shared pass move satisfies the stated reverse-mex relation G(x) = mex{G(x+s1), G(x+s2), G(x+s3)}. The reverse-mex property is introduced once as the target relation to be verified, then shown to hold via explicit case analysis or induction on the arithmetic structure of the subtraction set; the pass is incorporated as an extra option that toggles a global flag without altering the core successor definition. No parameter is fitted to data and then renamed as a prediction, no self-citation chain is load-bearing for the central claim, and the derivation does not reduce any output to its own inputs by construction. The explicit exclusion of n=1,2 further indicates the result is not tautological but depends on the chosen arithmetic form.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The analysis rests on the standard definition of Grundy numbers via the mex operator and the impartiality of the game, plus the explicit arithmetic form of the subtraction set for n≥3.

axioms (2)

standard math Grundy number G(x) defined as mex of options
Core definition used to state the reverse-mex property
domain assumption The game remains impartial after adding the one-time pass
Required for Grundy numbers to be well-defined

pith-pipeline@v0.9.0 · 5513 in / 1426 out tokens · 59297 ms · 2026-05-15T02:38:54.294616+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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