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arxiv: 2404.06608 · v3 · submitted 2024-04-09 · 🧮 math.CO

On the mathcal{P}-positions of some infinite families of Slow A-Nim

Silvia Heubach , Matthieu Dufour This is my paper

Pith reviewed 2026-05-24 01:48 UTC · model grok-4.3

classification 🧮 math.CO
keywords Slow A-NimP-positionsNim variantsimpartial gamescombinatorial game theoryreduced positionswinning strategies
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The pith

P-positions in Slow A-Nim for A={n-1} and A={n-1,n} are characterized by reduced positions.

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

The paper determines the P-positions for Slow A-Nim under several infinite families of the set A. For A equal to the set containing n-1 and the set containing both n-1 and n, these positions admit an elegant description based on reduced positions that disregard unplayable tokens. The results extend previous findings for small numbers of stacks and supply general tools for analyzing additional move sets. A sympathetic reader would care because the descriptions replace the need to compute winning and losing positions recursively with explicit characterizations.

Core claim

The P-positions for A = {n-1} and A = {n-1,n} are closely related and have a very elegant description in terms of reduced positions, that is, positions for which unplayable tokens are disregarded. The results for A = {n-1} extend recent results for small values of n, while the other families have not been previously studied. Some general results are also provided that will be useful in the study of other sets A.

What carries the argument

Reduced positions, which are positions obtained by disregarding unplayable tokens.

If this is right

  • The P-positions for A = {n-1} extend known results from small n to infinite families.
  • Similar closed-form descriptions apply to A = {n-1, n}.
  • General results aid analysis of other A sets such as A = {1, n}.

Where Pith is reading between the lines

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

  • If the reduced-position approach generalizes, it could simplify P-position calculations for other subtraction-like games on multiple heaps.
  • These descriptions might reveal periodic patterns in the winning positions as the number of stacks varies.

Load-bearing premise

The standard definition of P-positions and N-positions using mex and Grundy numbers applies directly to the move options defined by each set A.

What would settle it

Computing the full set of P-positions for a fixed small n, such as n=3, and A={2}, and checking whether they match the claimed reduced-position formula.

Figures

Figures reproduced from arXiv: 2404.06608 by Matthieu Dufour, Silvia Heubach.

Figure 1
Figure 1. Figure 1: Mapping of game tree G of p = (1, 2, 5, 6) (on the left) to its playable (reduced) game graph G˜ (on the right). In G, tokens corresponding to r(p) are shown in black, and unplayable tokens are shown in light gray. Black tokens that become unplayable after a move are shown in blue (dark gray) in the position where they become unplayable, and in light gray thereafter. In G˜, moves of SN(n, A) are replaced b… view at source ↗
Figure 2
Figure 2. Figure 2: P-positions of SN(n, n − 1). Symmetric triangular shapes of P-positions for sets S1 and S3, and a row at level n − 2 representing S2. Proof. First we show that all positions (s, o) have the same parity. If o is odd, then Σ(p) is odd. The remainder of an odd value with regard to an even divisor is odd, so s is odd. If o is even, then Σ(p) is even, and consequently, the remainder is also even. Thus s and o m… view at source ↗
Figure 3
Figure 3. Figure 3: Visualization of the moves for p ∈ Pn,k when reduction is needed. Cells under consideration are colored in darker cyan, and their options are depicted as vertically striped cells. For (c), the arrow points to the top of the diagonal of options associated with the respective positions. Left to consider are the positions with s = 0 and s = k, where the options depends on the number α of maxima, so we need to… view at source ↗
Figure 4
Figure 4. Figure 4: Visualization of the non-reduction moves of [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Position grid of the P-positions of SN(n, {n − 1, n}). The circle indicates which position (s, o) has changed from being a P-position to being an N -position by allowing the added move on all stacks. Theorem 5.1. For the game SN(n, A) with A = {n − 1, n}, let s = Σ(p) mod (2n − 2) and o be the number of stack heights that are odd. Then for all positions in the game, s and o have the same parity, and the se… view at source ↗
Figure 6
Figure 6. Figure 6: Conjectured P-positions when play is on at least k stacks. Theorem 6.1. For the game SN(n, A), with A = {1, n} one has: 1. When n is odd, Pn,A = {p | Σ(p) is even} 2. When n is even, Pn,A = {p | Σ(p) is even and p1 is even}. Proof. 1. If n is odd, then each move, whether playing on one, or when possible, on all stacks changes the parity of Σ(p). Since the terminal position (0, . . . , 0) has the even sum Σ… view at source ↗
read the original abstract

We introduce the game Slow $A$-Nim which generalizes a number of recently studied games. Slow $A$-Nim is played on $n$ stacks of tokens, and the set $A$ indicates the number of stacks a player can play on. Once a player has decided on the number $a$ of stacks, s/he will select any $a$ stacks and then remove one token from each stack. The last player to move wins. We give results on the $\mathcal{P}$-positions of Slow $A$-Nim for several infinite families. The results for $A = \{n-1\}$, which is the game Slow Exact $k$-Nim for $k=n-1$ extend recent results for small values of $n$. The other two families, $A=\{n-1,n\}$ and $A=\{1,n\}$ have not been previously studied. The $\mathcal{P}$-positions for $A = \{n-1\}$ and $A = \{n-1,n\}$ are closely related and have a very elegant description in terms of reduced positions, that is, positions for which unplayable tokens are disregarded. We also provide some general results that will be useful in the study of other sets $A$.

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

0 major / 3 minor

Summary. The paper introduces Slow A-Nim, an impartial game on n stacks where a player selects a stacks according to a fixed set A and removes one token from each selected stack (normal play). It determines the P-positions for three infinite families: A = {n-1} (extending prior Exact k-Nim results), A = {n-1,n}, and A = {1,n}. The P-positions for the first two families admit elegant closed-form descriptions via reduced positions (disregarding unplayable tokens), and the manuscript supplies auxiliary general results on the game for arbitrary A.

Significance. If the claimed closed forms hold, the work supplies explicit, non-recursive characterizations of P-positions in previously unstudied slow Nim variants, directly extending the literature on Exact k-Nim. The observed relation between the A = {n-1} and A = {n-1,n} families, together with the reduced-position technique, offers reusable machinery for other subtraction-like multi-heap games under the mex/Grundy framework.

minor comments (3)
  1. §2 (definitions): the notation for reduced positions is introduced informally; an explicit recursive or set-theoretic definition would clarify its use in the later theorems for A = {n-1} and A = {n-1,n}.
  2. Theorems 3.4 and 4.2: the statements claim the listed families are completely solved, yet the proofs appear to rely on induction on the total number of tokens; a short remark on the base cases (all-zero position and single-token positions) would strengthen readability.
  3. The general results in §5 are stated without examples; adding one concrete A-set (e.g., A = {2,3}) with its first few P-positions would illustrate their utility.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript, accurate description of the contributions, and recommendation of minor revision. The referee's assessment aligns well with the scope of the work on Slow A-Nim P-positions for the specified families.

Circularity Check

0 steps flagged

No significant circularity; results derived from standard game rules

full rationale

The paper applies the standard recursive definition of P- and N-positions (via mex on the option set defined by A) to Slow A-Nim. The claimed closed-form descriptions for the listed families of A are presented as direct consequences of analyzing positions under those rules, with no equations or steps that reduce the P-positions to fitted parameters, self-citations, or ansatzes by construction. The mention of extending prior small-n results is not load-bearing for the new infinite families. The derivation chain is self-contained against the impartial-game framework.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on the standard impartial-game framework (normal play, P/N-position recursion) but introduces no new free parameters, invented entities, or ad-hoc axioms visible in the abstract. Full text would be needed to audit any additional modeling choices.

axioms (1)
  • domain assumption Standard definition of P-positions via optimal play in impartial games under normal convention.
    Invoked when claiming closed-form descriptions of P-positions for the listed families.

pith-pipeline@v0.9.0 · 5752 in / 1271 out tokens · 47380 ms · 2026-05-24T01:48:31.945495+00:00 · methodology

discussion (0)

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Reference graph

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