pith. sign in

arxiv: 2604.02587 · v1 · submitted 2026-04-02 · 🧮 math.CO

The Invariance Reduction Process -- a New Tool to Solve Circular Nim and Related Games

Balaji R. Kadam , Matthieu Dufour , Silvia Heubach This is my paper

Pith reviewed 2026-05-13 20:13 UTC · model grok-4.3

classification 🧮 math.CO
keywords Invariance Reduction Processinvariant vectorsCircular NimPath NimP-positionsSet NimSimplicial Nim
0
0 comments X

The pith

The Invariance Reduction Process simplifies finding P-positions in Circular Nim and Path Nim by reducing games to smaller solved subgames.

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

The paper introduces invariant vectors and develops the Invariance Reduction Process to simplify solving for P-positions in Nim-like games. The process reduces positions via invariance and then uses zero and merge reductions to reach smaller solved sub-games. It is applied to the family of Path Nim games where play is allowed on at least half the stacks, as well as Circular Nim games with n=7 k=3 and n=8 k=3. This approach allows easier proofs about moves to and from P-positions without requiring full background in simplicial complexes.

Core claim

We introduce the notion of invariant vectors of a game and develop the Invariance Reduction Process, which first uses reduction of positions via invariance and then zero and merge reductions of games to arrive at smaller, solved sub-games for closed subspaces of the positions. This process makes it much easier to prove that there are moves from N-positions to P-positions, and can also be used in some cases to show that there are no moves between P-positions. We apply the Invariance Reduction Process to derive results on the P-positions of the family of Path Nim games where play is allowed on at least half the stacks, as well as for the Circular Nim games CN(n,k) with n=7, k=3 and n=8,k=3.

What carries the argument

Invariant vectors, used within the Invariance Reduction Process to reduce game positions and arrive at smaller sub-games.

If this is right

  • Results on the structure of the P-positions in Path Nim games with sufficient playable stacks follow from invariant vectors.
  • The P-positions for CN(7,3) and CN(8,3) are determined via the reduction process.
  • Invariant vectors provide a description of P-positions in Set Nim SN(n,A) without simplicial complex notation.
  • Invariant vectors have wider applicability compared to circuits for describing P-positions.

Where Pith is reading between the lines

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

  • The method might extend to other game families with simplicial complex rule sets to derive similar P-position structures.
  • Applying the process to larger values of n and k in Circular Nim could reveal general formulas for P-positions.
  • Comparison with standard mex and grundy number calculations could show computational advantages for these reductions.

Load-bearing premise

The rule sets of the games form a simplicial complex.

What would settle it

Computing the actual P-positions for CN(8,3) by other means and finding a mismatch with the structures derived from invariant vectors would falsify the claim.

Figures

Figures reproduced from arXiv: 2604.02587 by Balaji R. Kadam, Matthieu Dufour, Silvia Heubach.

Figure 1
Figure 1. Figure 1: Using invariant vectors to find an option [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Visualization of the Invariance Reduction Process. Boxed numbers reference the [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Quantities involved in the move to p ′ ∈ Pn,k in case 3 of the proof of Theorem 4.1. We now turn our attention to the game H, defined in Theorem 4.2. The game H is a sub-game of both CN(7, 3) and CN(9, 4), so Theorem 4.2 is a crucial piece in the derivation of the P-positions of both games. We give the solution for CN(7, 3) in Section 5. While we also have the result for the P-positions of CN(9, 4), our cu… view at source ↗
Figure 4
Figure 4. Figure 4: The steps of the Invariance Reduction Process for a position [PITH_FULL_IMAGE:figures/full_fig_p018_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Generic labeling of positions of CN(8, 3). and assume without loss of generality that a ≤ b. We will express stack heights on the two squares (p0, p2, p4, p6) and (p1, p3, p5, p7) in relationship to their respective square minimum, that is, p = (a, b + x1, a + x2, b + x3, a + x4, b + x5, a + x6, b + x7). As a result, the vectors z1 = (1, 0, 1, 0, 1, 0, 1, 0) and z2 = (0, 1, 0, 1, 0, 1, 0, 1) are invariant … view at source ↗
read the original abstract

We introduce the notion of invariant vectors of a game and develop the Invariance Reduction Process, which first uses reduction of positions via invariance and then zero and merge reductions of games to arrive at smaller, solved sub-games for closed subspaces of the positions. This process makes it much easier to prove that there are moves from N-positions to P-positions, and can also be used in some cases to show that there are no moves between P-positions. This process is suitable for all variations of the game Nim whose rule sets form a simplicial complex. We rephrase Simplicial Nim as Set Nim SN($n,A$) and derive results on the structure of the P-positions in terms of invariant vectors, without needing the background and notation of simplicial complexes. We also show that invariant vectors differ from the circuits used to describe the P-positions in Simplicial Nim and that invariant vectors have wider applicability compared to circuits. We apply the Invariance Reduction Process to derive results on the P-positions of the family of Path Nim games where play is allowed on at least half the stacks, as well as for the Circular Nim games CN($n,k$) with $n=7, k=3$ and $n=8,k=3$.

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 introduces the notion of invariant vectors for impartial games and develops the Invariance Reduction Process, which combines invariance-based position reduction with zero and merge reductions to obtain smaller solved sub-games. The process is claimed to apply to Nim variants whose move rules form a simplicial complex. The paper rephrases Simplicial Nim as Set Nim SN(n,A), derives structural results on P-positions in terms of invariant vectors, shows that these vectors differ from the circuits used in prior simplicial-complex treatments, and applies the process to obtain explicit P-position descriptions for the family of Path Nim games in which moves are allowed on at least half the stacks as well as for the specific Circular Nim instances CN(7,3) and CN(8,3).

Significance. If the modeling and reduction steps are valid, the work supplies a concrete algorithmic-style tool that can simplify proofs of P-position structure for a broad class of Nim-like games on simplicial complexes. The concrete results for Path Nim and the two Circular Nim cases constitute falsifiable, checkable claims that could be used by other researchers working on subtraction or heap games.

major comments (2)
  1. [Sections describing CN(n,k) and the application of the Invariance Reduction Process] The central applications to CN(7,3) and CN(8,3) rest on the assertion that the allowed move sets form simplicial complexes, yet the manuscript supplies no explicit verification that the collection of permitted subsets is downward-closed and closed under pairwise intersections. Without this check, the subsequent invariance-vector construction and the zero/merge reductions cannot be guaranteed to preserve the game value, rendering the stated P-position characterizations conditional.
  2. [Path Nim section] For the Path Nim family (moves on at least half the stacks), the paper must demonstrate that the chosen invariant vectors are indeed invariant under every legal move; the current exposition leaves open whether the half-stack threshold introduces additional constraints that could invalidate the reduction steps used to reach the solved sub-games.
minor comments (2)
  1. [Introduction and Set Nim section] The rephrasing of Simplicial Nim as Set Nim SN(n,A) is notationally helpful, but the definition of an invariant vector should be stated once, early, with a single consistent symbol rather than being reintroduced in each application section.
  2. [Comparison with circuits] A short table or diagram comparing the invariant vectors obtained here with the circuits of the earlier simplicial-complex literature would make the claimed wider applicability immediately visible to readers.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major point below and will revise the manuscript to incorporate the requested verifications.

read point-by-point responses

  1. Referee: [Sections describing CN(n,k) and the application of the Invariance Reduction Process] The central applications to CN(7,3) and CN(8,3) rest on the assertion that the allowed move sets form simplicial complexes, yet the manuscript supplies no explicit verification that the collection of permitted subsets is downward-closed and closed under pairwise intersections. Without this check, the subsequent invariance-vector construction and the zero/merge reductions cannot be guaranteed to preserve the game value, rendering the stated P-position characterizations conditional.

    Authors: We agree that explicit verification is required. In the revised manuscript we will insert a short subsection that directly checks the two properties for the move sets of CN(7,3) and CN(8,3): (i) downward closure under inclusion and (ii) closure under pairwise intersections. These checks will be stated in elementary set-theoretic language without assuming prior knowledge of simplicial complexes, thereby confirming that the invariance-vector construction and the zero/merge reductions preserve game values. revision: yes

  2. Referee: [Path Nim section] For the Path Nim family (moves on at least half the stacks), the paper must demonstrate that the chosen invariant vectors are indeed invariant under every legal move; the current exposition leaves open whether the half-stack threshold introduces additional constraints that could invalidate the reduction steps used to reach the solved sub-games.

    Authors: We accept that the invariance of the selected vectors must be shown explicitly for the half-stack threshold. The revision will add a lemma that, for each chosen invariant vector, verifies invariance under every legal move (including those that touch exactly half the stacks). The proof will enumerate the possible move cardinalities and confirm that the vector remains unchanged, thereby validating the subsequent reduction steps. revision: yes

Circularity Check

0 steps flagged

No circularity: new process introduced and applied without self-referential reduction

full rationale

The paper defines invariant vectors and the Invariance Reduction Process from first principles using standard impartial game theory (P-positions, moves), then applies the process to Path Nim and specific CN(n,k) instances. No equations reduce a claimed prediction to a fitted input by construction, no self-citation chain bears the central result, and the simplicial-complex suitability condition is stated as an external modeling assumption rather than derived from the paper's own outputs. The derivation chain remains self-contained against the stated game rules.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim depends on the domain assumption that Nim rule sets form simplicial complexes and on the newly introduced invariant vectors whose existence and utility are asserted rather than derived from prior literature.

axioms (1)
  • domain assumption Rule sets form a simplicial complex
    Explicitly stated as the condition under which the process applies to all Nim variations.
invented entities (1)
  • invariant vectors no independent evidence
    purpose: To enable reduction of positions via invariance before zero and merge reductions
    Newly defined objects whose properties are used to reach smaller solved sub-games; no independent evidence supplied in abstract.

pith-pipeline@v0.9.0 · 5531 in / 1229 out tokens · 40955 ms · 2026-05-13T20:13:44.680391+00:00 · methodology

discussion (0)

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

Lean theorems connected to this paper

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

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
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.

  1. Necklace Games

    math.CO 2026-04 unverdicted novelty 5.0

    NecklaceNim NN(n,k) is solved for infinite families where play is allowed on at least half the stacks.

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages · cited by 1 Pith paper

  1. [1]

    H., Nowakowski, R

    Albert, M. H., Nowakowski, R. J. & Wolfe, D.Lessons in Play(CRC Press, Boca Raton, FL, 2019)

    work page 2019

  2. [2]

    R., Conway, J

    Berlekamp, E. R., Conway, J. H. & Guy, R. K.Winning ways for your mathematical plays. Vol. 1-4Second Ed. (A K Peters, Ltd., Natick, MA, 2001-2004)

    work page 2001

  3. [3]

    Bouton, C. L. Nim, a game with a complete mathematical theory.Ann. of Math. (2) 3,35–39.https://doi.org/10.2307/1967631(1901/02)

    work page doi:10.2307/1967631(1901/02 1901

  4. [4]

    & Heubach, S

    Dufour, M. & Heubach, S. Circular Nim games.Electron. J. Combin.20,Paper 22, 26. https://doi.org/10.37236/2765(2013)

    work page doi:10.37236/2765(2013 2013

  5. [5]

    Dufour, M., Heubach, S. & Vo, A. Circular Nim games CN(7,4).Integers21B,Paper No. A9, 18 (2021)

    work page 2021

  6. [6]

    & Steingr´ ımsson, E

    Ehrenborg, R. & Steingr´ ımsson, E. Playing Nim on a simplicial complex.Electron. J. Combin.3,Research Paper 9, approx. 33.https://doi.org/10.37236/1233(1996)

    work page doi:10.37236/1233(1996 1996

  7. [7]

    Gurvich, V., Heubach, S., Ho, N. B. & Chikin, N. Slowk-Nim.Integers20,Paper No. G3, 19 (2020)

    work page 2020

  8. [8]

    Holladay, J. C. Matrix Nim.Amer. Math. Monthly65,107–109.https://doi.org/ 10.2307/2308886(1958)

    work page doi:10.2307/2308886(1958 1958

  9. [9]

    Winning positions in simplicial Nim.Electron

    Horrocks, D. Winning positions in simplicial Nim.Electron. J. Combin.17,Research Paper 84, 13.https://doi.org/10.37236/356(2010)

    work page doi:10.37236/356(2010 2010

  10. [10]

    Kadam, B. R. & Shaiju, A. J. Variations of the Nim game played on a cube.Integers 25,Paper No. G2, 14 (2025)

    work page 2025

  11. [11]

    Moore, E. H. A generalization of the game called nim.Ann. of Math. (2)11,93–94. https://doi.org/10.2307/1967321(1910)

    work page doi:10.2307/1967321(1910 1910

  12. [12]

    N.Combinatorial game theoryhttps : / / doi

    Siegel, A. N.Combinatorial game theoryhttps : / / doi . org / 10 . 1090 / gsm / 146 (American Mathematical Society, Providence, RI, 2013). 30

    work page 2013