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arxiv: 2604.10364 · v1 · submitted 2026-04-11 · 🧮 math.CO

Necklace Games

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

Pith reviewed 2026-05-10 15:17 UTC · model grok-4.3

classification 🧮 math.CO
keywords NecklaceNimCircularNimPathNimimpartial gamesNim variantscombinatorial game theorygraph gameswinning positions
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The pith

NecklaceNim NN(n,k) is solved for all cases where moves are allowed on at least half the stacks.

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

NecklaceNim is defined as PathNim on a line of stacks with the added option to play at either end vertex. The game models the position that remains in CircularNim once k-2 consecutive stacks have been emptied, so its solution is required before the full circular game can be settled. The paper supplies complete solutions for the infinite families of NN(n,k) in which players may act on at least half the stacks. A reader following the work sees these results as the concrete bridge that turns the circular problem from intractable to partially resolved.

Core claim

The authors solve the infinite families of NN(n,k) when play is allowed on at least half the stacks. NecklaceNim NN(n,k) is introduced as PathNim PN(n,k) with an extra move permitted on the end vertices; it arises exactly as the subgame of CircularNim CN(n,k) after k-2 consecutive stacks are depleted, and the paper determines the outcome for every such family meeting the half-stack condition.

What carries the argument

NecklaceNim NN(n,k), the impartial game on a path of n stacks of height k that additionally allows moves at the two endpoints, which exactly encodes the depleted circular position.

If this is right

  • Winning and losing positions are now known for every NN(n,k) in the solved families.
  • CircularNim CN(n,k) is reduced by these subgames, leaving only the cases with fewer active stacks.
  • Structural patterns appear in the game values once the number of movable stacks reaches or exceeds half the total.
  • The endpoint rule produces distinct mex calculations that deviate from ordinary PathNim in predictable ways.

Where Pith is reading between the lines

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

  • The remaining unsolved families of CircularNim may become reachable once the half-stack results are combined with other depletion analyses.
  • The same endpoint-extension technique could be applied to impartial games on other cycle-like graphs after partial depletion.
  • Formulas derived for the solved families can be checked by exhaustive computation on small instances to confirm the general pattern.

Load-bearing premise

NecklaceNim correctly reproduces the subgame that remains in CircularNim after k-2 consecutive stacks have been depleted.

What would settle it

A direct comparison, for concrete small n and k satisfying the half-stack rule, between the game value computed for NN(n,k) and the game value obtained by playing out the corresponding depleted position inside CN(n,k).

Figures

Figures reproduced from arXiv: 2604.10364 by Balaji R. Kadam, Matthieu Dufour, Silvia Heubach.

Figure 1
Figure 1. Figure 1: Solved games are shown in color. Arrows indicate the games that [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The game NN(2ℓ, ℓ) reduces to NN(2ℓ − 2, ℓ − 1) through zero and merge reduction, illustrated for ℓ = 5. 3.3 Invariance Reduction Process for Structure In [7], the Invariance Reduction Process was introduced with the goal of finding a move from an N -position to a P-position by identifying subspaces that are equivalent to known games, and then using the winning move in the equivalent game to produce a winn… view at source ↗
Figure 3
Figure 3. Figure 3: Game NN(2ℓ, ℓ) reduces to PN(3, 2) through zero and merge re￾duction on the subspace of positions with zeros at a2, . . . , aℓ , shown for ℓ = 5. Equation (2) implies that a1 = Pℓ−1 i=1 bi, which when added to (3) gives A = a1 + Σmin = X ℓ−1 i=1 bi + bℓ = B, so condition (SE) of Sℓ = PG is satisfied. Next we need to check the (ME) condition, namely that m = min(a1, bℓ) = min(s2, . . . , sℓ+1) = s ∗ where s… view at source ↗
Figure 4
Figure 4. Figure 4: Mapping of p˜ to pˆ when ˜bi = 0. Specifically, aˆj =    a˜j for 2 ≤ j < i a˜i + ˜ai+1 for j = i a˜j+1 for i < j ≤ ℓ − 1 ˆbj =    ˜bj for 1 ≤ j < i − 1 ˜bi−1 + ˜bi for j = i − 1 ˜bj+1 for i ≤ j ≤ ℓ − 2. Now we consider the (ME) equation. Let ˆsj = ˆaj + · · · + ˆbj−2 for j = 2, . . . , ℓ. When j ≤ i, then ˆsj contains ˆai but not ˆbi−1, and for j > i, ˆsj contains ˆbi−1 but not ˆai . Thus sˆj = (… view at source ↗
read the original abstract

We define and give results on the game NecklaceNim NN($n$,$k$) which is PathNim PN($n$,$k$) with an additional move allowed on the end vertices. This game arises as a sub-game in the context of solving CircularNim CN($n$,$k$) when $k-2$ consecutive stacks have been depleted, therefore its solution is critical to solving CircularNim. We solve the infinite families of NN($n$,$k$) when play is allowed on at least half the stacks.

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 / 2 minor

Summary. The manuscript defines NecklaceNim NN(n,k) as PathNim PN(n,k) augmented with moves on the end vertices. It arises as the relevant subgame of CircularNim CN(n,k) once k-2 consecutive stacks are emptied. The central result is an explicit solution for the infinite families of NN(n,k) in the regime where moves are permitted on at least half the stacks.

Significance. If the claimed solutions are correct and the subgame equivalence holds, the work supplies concrete P-positions or Grundy-number formulas for these families, which would constitute a measurable advance toward resolving CircularNim and related impartial games on circular or path graphs. The emphasis on infinite families rather than finite cases is a methodological strength.

major comments (1)
  1. [Introduction / modeling section] The modeling claim that NN(n,k) is the exact subgame of CN(n,k) after k-2 consecutive stacks are depleted is load-bearing for the motivation and for any future transfer of results. The manuscript must supply a self-contained argument (with explicit move-set comparison or Grundy-number preservation) establishing this equivalence; without it, the solutions for NN do not automatically apply to CN.
minor comments (2)
  1. Clarify the precise definition of 'play is allowed on at least half the stacks' (e.g., whether it refers to n/2 playable heaps or a move restriction) and state the solved families explicitly in a theorem or table for quick reference.
  2. Add a brief comparison of the new NN results with known values for ordinary Nim or PathNim to highlight the effect of the end-vertex moves.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address the single major comment below.

read point-by-point responses

  1. Referee: [Introduction / modeling section] The modeling claim that NN(n,k) is the exact subgame of CN(n,k) after k-2 consecutive stacks are depleted is load-bearing for the motivation and for any future transfer of results. The manuscript must supply a self-contained argument (with explicit move-set comparison or Grundy-number preservation) establishing this equivalence; without it, the solutions for NN do not automatically apply to CN.

    Authors: We agree that a self-contained argument is required. The original manuscript stated the subgame relation but did not include a full proof. In the revised version we have added a new subsection (Section 2.1) that supplies the missing argument. We first identify the precise position reached in CN(n,k) once k-2 consecutive stacks are emptied and show, by direct enumeration of legal moves, that the remaining options on the circular board coincide exactly with the move set of NN(n,k) (the path moves plus the two extra end-vertex moves). We then prove by induction on total stones that the Grundy numbers of corresponding positions are identical, establishing that the games are equivalent. This revision ensures the claimed solutions for NN transfer directly to the relevant CN positions. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The paper explicitly defines NecklaceNim NN(n,k) as PathNim PN(n,k) augmented by an end-vertex move, states its emergence as the relevant subgame of CircularNim CN(n,k) once k-2 consecutive stacks are emptied, and then derives Grundy numbers or P-positions for the infinite families where moves are permitted on at least half the stacks. These steps rely on direct combinatorial analysis of the stated rules rather than any self-referential fit, ansatz imported via citation, or uniqueness theorem that collapses back to the authors' prior unverified claims. The modeling equivalence is presented as a modeling choice whose validity can be checked externally against the parent game, and the solutions are obtained by standard impartial-game techniques applied to the defined positions. No equation or claim reduces by construction to its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the game definition and the validity of the claimed solutions for the specified families.

axioms (1)
  • standard math Standard impartial game rules and analysis methods apply to NecklaceNim.
    Implicit in any solution of a Nim-like game.
invented entities (1)
  • NecklaceNim game no independent evidence
    purpose: Models a subgame of CircularNim
    Newly defined variant in the paper.

pith-pipeline@v0.9.0 · 5375 in / 1002 out tokens · 51584 ms · 2026-05-10T15:17:52.926869+00:00 · methodology

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Reference graph

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9 extracted references · 9 canonical work pages · 1 internal anchor

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