pith. sign in

arxiv: 2605.23213 · v1 · pith:FSXYDO4Xnew · submitted 2026-05-22 · 🧮 math.CO

Generalizing OOOOOOB

Alon Danai , Paul Ellis , Thotsaporn Aek Thanatipanonda This is my paper

Pith reviewed 2026-05-25 04:25 UTC · model grok-4.3

classification 🧮 math.CO
keywords combinatorial game theoryimpartial gamesP-positionsmulti-pile gamesgeneralizationone-or-one-or-bothexperimental enumeration
0
0 comments X

The pith

One generalization of the one-or-one-or-one-of-both game to multiple piles has P-positions following a simple pattern.

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

The paper defines three distinct ways to extend the classic two-pile impartial game one-or-one-or-one-of-both to any number of piles and then classifies the resulting P-positions. In the first extension a closed pattern appears that directly identifies every losing position. The other two extensions produce only partial classifications, each obtained by running exhaustive searches that either hold the total number of piles fixed or hold the number of tokens per pile fixed. A reader would care because an explicit description of P-positions supplies an immediate test for whether the first player can force a win in any concrete starting position.

Core claim

In the first of the three multi-pile generalizations the P-positions obey a simple explicit pattern. In the remaining two generalizations only partial solutions are obtained, each found by computational enumeration that either limits the number of piles or limits the number of tokens per pile.

What carries the argument

P-positions, the terminal losing positions from which every legal move reaches a winning position for the opponent.

If this is right

  • The first generalization supplies an immediate, non-recursive rule that decides the winner for every starting position.
  • The second generalization yields conjectural formulas for P-positions once the number of piles is held constant.
  • The third generalization yields conjectural formulas for P-positions once the token count per pile is held constant.
  • Both partial approaches produce concrete lists that can be checked against larger instances to test consistency.

Where Pith is reading between the lines

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

  • If the observed patterns survive larger checks, they may allow direct computation of Grundy numbers without enumerating every sub-position.
  • The same two experimental restrictions could be applied to other multi-pile subtraction or Nim-like games to obtain partial classifications.
  • Failure of the patterns at moderate sizes would indicate that a different theoretical tool, rather than further computation, is required.

Load-bearing premise

The regularities observed when the number of piles or tokens is small continue unchanged for every larger size.

What would settle it

Any single starting position with more piles or more tokens per pile whose membership in the P-set contradicts the pattern conjectured from the small cases.

Figures

Figures reproduced from arXiv: 2605.23213 by Alon Danai, Paul Ellis, Thotsaporn Aek Thanatipanonda.

Figure 2.1
Figure 2.1. Figure 2.1: All positions of Version B with 4 tokens, along with some relevant options. Proof. Refer to [PITH_FULL_IMAGE:figures/full_fig_p003_2_1.png] view at source ↗
Figure 2.2
Figure 2.2. Figure 2.2: All positions of Version B with 5 tokens and at least one pile of size 1, along with some relevant options. Proof. Our base case is any position with 5 piles where at least one of the piles is 1. Note that these are organized by the size of the second smallest pile in [PITH_FULL_IMAGE:figures/full_fig_p004_2_2.png] view at source ↗
Figure 3.1
Figure 3.1. Figure 3.1: Positions of Version C with small piles. In each grid, the number in the top-left corner is a3, the row is a2, and the column is a1. Exceptions to the general case are highlighted. The position (4, 4, 2) is marked with P and its options are marked with •. Proof [PITH_FULL_IMAGE:figures/full_fig_p007_3_1.png] view at source ↗
Figure 3.2
Figure 3.2. Figure 3.2: Positions of Version C with k piles of size 1 and a single pile of size n. The case k = 2n is boxed . Note that k = 2n implies that both k + n = 0 mod 3 and k + 2n = 0 mod 4. Proof. We refer the reader to [PITH_FULL_IMAGE:figures/full_fig_p008_3_2.png] view at source ↗
read the original abstract

We present three versions of the classic two-pile game \textsc{one-or-one-or-one-of-both} generalized to the multi-pile context. In each case, we explore the resulting $\mathcal{P}$-positions. In the first version, there is a simple pattern. In the other two versions, we find partial solutions in each case through two experimental routes. First by limiting the number of piles, then by limiting the number of tokens per pile.

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

1 major / 0 minor

Summary. The manuscript generalizes the two-pile impartial game one-or-one-or-one-of-both to three multi-pile variants and studies the resulting P-positions. It reports a simple pattern for P-positions in the first variant and obtains partial solutions for the other two variants by computationally restricting either the number of piles or the number of tokens per pile.

Significance. A rigorously established simple pattern for P-positions in the first variant would supply an explicit, closed-form description of winning positions in a multi-pile normal-play game, which is a useful contribution to combinatorial game theory. The experimental partial solutions for the remaining variants provide initial data that may motivate subsequent analytic work, although their bounded scope restricts broader applicability.

major comments (1)
  1. [Abstract] Abstract: The assertion of a 'simple pattern' for the P-positions of the first variant is stated without any reference to a general proof, inductive argument, or verification procedure that applies to arbitrary numbers of piles and tokens. The abstract indicates that the other two variants rely on computational searches limited to small pile or token counts; if the same experimental route underpins the first variant, the reported pattern risks being an artifact of the bounded regime, a common occurrence in impartial games where mex-regularities break beyond small instances.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed feedback on the abstract. We address the single major comment below and will revise the manuscript to improve clarity.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The assertion of a 'simple pattern' for the P-positions of the first variant is stated without any reference to a general proof, inductive argument, or verification procedure that applies to arbitrary numbers of piles and tokens. The abstract indicates that the other two variants rely on computational searches limited to small pile or token counts; if the same experimental route underpins the first variant, the reported pattern risks being an artifact of the bounded regime, a common occurrence in impartial games where mex-regularities break beyond small instances.

    Authors: The abstract is intentionally brief. The manuscript body distinguishes the first variant by presenting an explicit, general pattern for its P-positions that is not obtained via the same bounded experimental searches used for the other two variants. We will revise the abstract to state that this pattern holds for arbitrary numbers of piles and tokens (with the supporting argument or verification procedure referenced to the relevant section). This change will make the distinction explicit and address the risk of misinterpretation as a bounded-regime artifact. revision: yes

Circularity Check

0 steps flagged

No circularity: pattern claim for first version stands apart from experimental partial solutions.

full rationale

The manuscript distinguishes the first version (simple pattern for P-positions) from the other two (partial solutions obtained only via bounded computational searches). No equations, fitted parameters, self-citations, or ansatzes are referenced that would reduce the claimed pattern to its own inputs by construction. The derivation for the first version is presented as independent of the experimental routes used elsewhere, satisfying the requirement for a self-contained result.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no information on free parameters, background axioms, or new entities.

pith-pipeline@v0.9.0 · 5594 in / 966 out tokens · 24143 ms · 2026-05-25T04:25:41.656063+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

5 extracted references · 5 canonical work pages

  1. [1]

    Coskey, P

    S. Coskey, P. Ellis, J. Wood,Five Fabulous Activities For Your Math Circle, Natural Math, 2022

    work page 2022

  2. [2]

    Danai, P

    A. Danai, P. Ellis, T. Thanatipanonda, Two-Dimensional Subtraction-Transfer Games, preprint

    work page

  3. [3]

    Fomkin, S

    D. Fomkin, S. Genkin, I. Itenberg,Mathematical Circles (Russian Experience), American Mathematical Society, 1996

    work page 1996

  4. [4]

    Larsson, I

    U. Larsson, I. Saha, M. Yokoo, Subtraction games in more than one dimension,Theoretical Computer Science,1016, 2024

    work page 2024

  5. [5]

    Morisawa, Winning strategy and periodicity ofm-pile divisor Nim.Int J Game Theory55, 21 (2026).https://doi.org/10.1007/ s00182-026-00991-5 8

    T. Morisawa, Winning strategy and periodicity ofm-pile divisor Nim.Int J Game Theory55, 21 (2026).https://doi.org/10.1007/ s00182-026-00991-5 8

    work page 2026