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Impartial Chess on Integer Partitions

T0 review · 3 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Playing impartial chess on Young diagrams yields a complete losing-position rule for Rook and places Rook and Queen games in two previously empty CGH classification regions.

desk verdict Rook P-positions and SG formulas are real progress; the CGH classification table still needs fuller proofs — and one headline proof has a subscript typo. read the letter →

arxiv 2501.14640 v1 pith:K7MVEJFF submitted 2025-01-24 math.CO

Eric Gottlieb , Matjaž Krnc , Peter Muršič This is my paper · ORCID
classification math.CO MSC 91A4605A17
keywords impartialcombinatorialgameYoungdiagramintegerpartitionSprague-GrundyvaluemisèreplayConway-Gurvich-HoclassificationRookQueen
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper extends Berlekamp's impartial chess games from rectangular boards to Young diagrams of integer partitions. Its central result is a complete description of the losing positions of Rook: a partition λ is a P-position if and only if λ has Dyson rank 0 and the subpartition λ[1,1] has at least r−1 cells in each part, where r is the number of parts. The authors also compute Sprague-Grundy values for King and Rook on special partition families, show that Pawn and Knight reduce to the known game Downright, and prove that misère play is equivalent to normal play on a truncated diagram. Using the CGH classification of normal/misère interplay, they classify all these games and exhibit the first infinite families in two previously unoccupied regions, generalizing Nim and Wythoff, which arise as Rook and Queen on rectangles.

What carries the argument

The central machinery is the moveset formulation of impartial chess on Young diagrams, where a position is a partition λ and a move deletes a subpartition λ[i,j]. The proof structure relies on three pillars: (1) the directed acyclic graph DAG_M(λ) of a position, which supports partition-equivalence and game-equivalence relations; (2) Sprague-Grundy values and misère Grundy values, combined into Conway pairs (SG, G^−) that drive the CGH classification; and (3) the truncation operation, whose key property is that misère P-positions of (M, λ) coincide with normal-play P-positions of the truncated position. For Rook, the P-position proof introduces 'ample' partitions—partitions whose rows and columns each contain a P-position—and shows that a position is losing exactly when it has rank 0, equal top two parts, and an ample interior.

What would settle it

Compute the Conway pairs of (Q, k_{r,c}) for a small generalized staircase not covered by the omitted proof of Theorem 4.11, such as (Q, 2_{3,3}) or (Q, 3_{2,2}), by direct recursive mex calculation; if any subposition is non-swap and has a move to exactly one of the two swap types, the claim that (Q, generalized staircases) is miserable fails.

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Extended reading notes

Core claim

Theorem 3.16 gives the complete normal-play P-position classification for Rook: (R, λ) is a losing position if and only if λ has rank 0 and λ[1,1] ≥ r−1, where r is the number of parts of λ. The CGH theorems place the restrictions (R, S1) and (Q, S2) into the regions 'forced and tame but not miserable' and 'tame but not miserable and returnable but not forced', respectively, with S1 = {[ℓ+1, ℓ^k][i,j] : ℓ ≥ 3, k > ℓ} and S2 = {[5, 4^k][i,j] : k ≥ 7}; the latter region was previously occupied only by a single small artificial game. These are the paper's most consequential assertions; together with Theorems 4.8–4.11 and the Rook characterization, they yield the full CGH table for these families.

Load-bearing premise

The completeness of the classification table depends on several proofs that are stated but not written: Theorems 4.8 and 4.9 claim 'the others are similar', and Theorem 4.11 omits its proof 'in the interest of brevity'; if any of those unshown cases contains a Conway pair outside the listed sets, the table in Figure 7 and Table 1 is wrong.

Editorial extensions

If this is right

  • Rook on a rectangle is game-equivalent to 2-pile Nim and Queen on a rectangle to Wythoff, so all results on Young diagrams strictly generalize these classical games.
  • The P-position rule for Rook gives a direct test for whether any Young-diagram position is a first-player loss, computable from the partition's rank and a single subpartition.
  • The game-tree equivalences mean Pawn and Knight positions can be solved by mapping to Downright via the explicit transformations φ_p and φ_N.
  • Misère versions of Downright, King, Rook, and Queen are normal play on the corner-removed partition λ^−, so misère play is exactly as tractable as normal play for those pieces.
  • The classification table shows Bishop, Pawn on rectangles, Rook on rectangles and generalized staircases, Queen on generalized staircases, and Knight on staircases are pet and forced, while other combinations occupy distinct known or new CGH regions.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the omitted 'similar-case' proofs in Theorems 4.8–4.11 hold, the CGH regions containing Nim, Wythoff, and Downright now have natural infinite families from chess, suggesting the taxonomy is not artificially sparse.
  • The golden-ratio coefficient that governs Queen's P-position density on rectangles might extend to all partitions; the paper's Lemma 3.22 gives a factor-2 bound for general partitions, and finding an infinite family with density approaching the golden-ratio bound would settle whether the constant can be improved.
  • The Rook characterization invites an analogous search for Queen's P-positions on Young diagrams; the paper argues this is likely hard because even the rectangular case is complicated.
  • A natural testable extension is to compute CGH classifications for thick-hook partitions, which the authors identify as a bridge between rectangles and generalized staircases.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The manuscript extends Berlekamp's impartial chess games from rectangular boards to Young diagrams of integer partitions. It defines chess-piece movesets on partitions, proves Sprague-Grundy results for King, gives a P-position characterization for Rook (Theorem 3.16), and provides a Conway-Gurvich-Ho (CGH) classification for restrictions to rectangles, staircases, and generalized staircases. The headline classification results are Theorem 4.12, placing (R,S1) as forced and tame but not miserable, and Theorem 4.13, placing (Q,S2) as tame but not miserable and returnable but not forced, together with Theorem 4.11 for Queen on rectangles and generalized staircases. The paper claims these occupy previously unknown or sparsely occupied CGH regions.

Significance. If the classification results are correct, the paper makes a solid contribution to combinatorial game theory: it connects impartial chess to partition theory, provides a complete P-position characterization for Rook on all Young diagrams (Theorem 3.16), generalizes Nim and Wythoff equivalences, and supplies new infinite families in CGH regions that were previously empty or nearly empty. The self-contained proofs of Theorem 3.16, Lemma 3.23, and the King lemmas are coherent and appear correct. The main caveat is that the CGH table depends on several proofs that are omitted or compressed (Theorem 4.11) or contain an indexing error (Theorem 4.12), so the headline classifications are not yet fully verified.

major comments (3)
  1. [Theorem 4.12] In the proof of Theorem 4.12, the displayed subpartition identity (⟨ℓ+1,ℓ^k⟩)[k−ℓ,ℓ] = ⟨ℓ^{ℓ+1}⟩ is false: by Definition 2.1, subtracting ℓ from each of the remaining ℓ+1 rows gives all entries 0, so the subpartition is empty for every k>ℓ≥3. The subsequent structural assertions about this subposition (swap positions in the last two columns, and (0,0)- and (1,1)-positions elsewhere) depend on the subpartition being the rectangle ℓ+1,ℓ, which would correspond to the index [k−ℓ,0]. Because this identification is the anchor for showing that all non-swap positions are (t,t)-positions, Theorem 4.12 is not established by the written argument. The authors should correct the index or supply a valid subpartition identity and re-verify the tameness claim.
  2. [Theorem 4.11] Theorem 4.11, which classifies (Q, rectangles) and (Q, generalized staircases) as miserable but not pet and returnable but not forced, is stated with its proof omitted 'in the interest of brevity.' This result is load-bearing for the completeness of Figure 7 and Table 1, and the preceding paragraph indicates that the rectangle case requires separate treatment for r=c=2, r=2,c=3, and r,c≥3. Without a proof or a reference to a complete proof, the CGH classification is conditional. Please provide the full proof or move the statement to a conjecture section.
  3. [Theorem 4.13] The proof of Theorem 4.13 asserts, without derivation, that (Q,⟨4^7⟩) contains only swap and (t,t)-positions with a specific column structure, and that (Q,⟨5,4^k⟩[0,4]) is a (0,1)-position. These assertions are the basis for concluding that every position in the family is (t,t) for t≥2, and hence for the tameness claim. Because (Q,S2) is one of the two headline new-region claims, the local computation should be shown or replaced by a reproducible argument (e.g., a small table of Conway pairs for ⟨4^7⟩).
minor comments (4)
  1. [Definition 3.1] In the definition of Queen, the moveset is written as Q = (1,1)^+ ∪ (1,0)^+ ∪ (1,1)^+, with (1,1)^+ repeated and (0,1)^+ missing; the later text in Section 3.6 gives the correct set Q = (1,0)^+ ∪ (0,1)^+ ∪ (1,1)^+. Please correct the definition.
  2. [Theorem 4.9] Theorem 4.9 states that p, (N, staircases), (K, staircases), (p, generalized staircases), and (D, generalized staircases) are pet and returnable but not forced, but the proof only treats (D, generalized staircases) and says 'the others are similar.' Since the movesets and the witness partitions differ for each family, a few more details or a general criterion would improve verifiability.
  3. [Proposition 3.7] The proof of Proposition 3.7 asserts that the relevant disjunctive sums are in P without spelling out the mirroring strategy; a sentence describing the pairing of moves would make the argument easier to check.
  4. [Theorem 4.10] In the proof of Theorem 4.10, the sentence 'in addition they may they include both swap positions' contains a typo, and the maximality argument for the generalized-staircase extension is compressed; please clarify.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the new classifications are derived from definitions and independent prior results, not reduced to the paper's own inputs.

full rationale

I found no circular step. The paper contains no fitted parameters, no prediction of data, and no uniqueness theorem imported from the authors. The only overlapping-author citation is [3], used for LCTR/Downright and truncation; it is prior independent published work, parameter-free, and not a restatement of the new rook/queen CGH classifications, so it does not create circularity. The load-bearing results are derived from moveset definitions, Sprague-Grundy and misere recursions, and (for rectangles) known Nim/Wythoff equivalences. Theorems 3.16, 4.8, 4.9, 4.10, 4.12, and 4.13 are supported by direct local arguments rather than by citing this paper's own conclusions. The possible indexing slip in Theorem 4.12 (the subpartition written [k - l, l] is empty under Definition 2.1, while the intended rectangle seems to be [k - l, 0]) is a correctness concern, not circularity. Likewise, the omitted proof of Theorem 4.11 and the 'others are similar' clauses in Theorems 4.8 and 4.9 are completeness gaps that weaken the written verification of Table 1, but they do not make any claim equivalent to its input by construction. Therefore the circularity score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No fitted parameters or invented entities appear: the paper is pure mathematics. The main external inputs are standard Sprague-Grundy theory, Young lattice combinatorics, and prior published classifications, including the authors' own LCTR work, which is independent of the new claims.

assumptions (5)
  • standard math Finite impartial game theory: Sprague-Grundy values computed by mex satisfy SGG(p) = mex({SGG(p') : p to p'}), and the Sprague-Grundy theorem holds for sums of games.
    Section 2.2; foundational for all P/N and Sprague-Grundy statements.
  • standard math Young lattice and subpartition identities: partitions are ordered componentwise, and Equation (1) states that (lambda[i1,j1])[i2,j2] = lambda[i1+i2,j1+j2].
    Section 2.1; used in every moveset computation and in induction arguments.
  • domain assumption Truncation theorem and Downright facts from Gottlieb, Krnc, and Muršič [3], reformulated here as Theorem 4.4, relate misère P-positions to truncM(lambda).
    Section 4.1; same-author prior published result used as an input for the misère analysis.
  • domain assumption Conway-Gurvich-Ho classification properties: pet, tame, miserable, domestic, forced, and returnable, together with the known classifications of Nim, Subtraction, Wythoff, LCTR, and Downright.
    Section 4.2, Definitions 4.6 and surrounding paragraphs; used as external benchmarks for Figure 7 and Table 1.
  • domain assumption Rook on rectangles has the same game tree as two-pile Nim, and Queen on rectangles has the same game tree as Wythoff's game.
    Observation 3.15 and Section 3.6, citing Berlekamp, Nivasch, and Wythoff; used as baselines for the new generalizations.

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Pith. "Pith review of Impartial Chess on Integer Partitions." pith.science (2026). https://pith.science/paper/K7MVEJFF

@misc{pith2026250114640,
  author       = {Pith},
  title        = {Pith review of: Impartial Chess on Integer Partitions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/K7MVEJFF}},
  note         = {Machine review of arXiv:2501.14640}
}
abstract

Berlekamp proposed a class of impartial combinatorial games based on the moves of chess pieces on rectangular boards. We generalize impartial chess games by playing them on Young diagrams and obtain results about winning and losing positions and Sprague-Grundy values for all chess pieces. We classify these games, and their restrictions to sets of partitions known as rectangles, staircases, and general staircases, according to the approach of Conway, later extended by Gurvich and Ho. The games $\rm {R\small OOK}$ and $\rm{Q\small UEEN}$ restricted to rectangles are known to have the same game tree as $2$-pile $\rm N{\small IM}$ and $\rm W{\small YTHOFF}$, respectively, so our work generalizes these well-known games.

Figures

Figures reproduced from arXiv: 2501.14640 by the authors.

Figure 1
Figure 1. Thus, the jth column of λ contains λ ′ j cells. The Young diagram of JK has no cells and JK′ = JK [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Partitions corresponding to 6 , 6,4, 6,4 and 2 3,2 . where λ − i = ⎧⎪⎪ ⎨ ⎪⎪⎩ λi − 1 if i = r or λi > λi+1, λi otherwise for any i ∈ [r]. For any i, j such that λ[i, j] > J1K, we have λ[i, j] − = λ − [i, j]. For a nonempty partition λ = Jλ1, . . . , λrK and a positive integer j ≤ λ1 we define the jth column of λ to be the set of pairs (i, j) for which the subpartitions {λ[i−1, j −1] ∣ i ∈ Z >0 } are well-defined. The… view at source ↗
Figure 3
Figure 3. Visual representation of Sprague-Grundy values ( [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: Partition equivalence depends on equality, not is [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: Knight and Pawn on λ = J142 , 13, 124 , 9, 5, 3K, shaded as and , are game￾equivalent to Downright on ϕN(λ) = J6, 5, 4, 2, 1K and ϕp(λ) = J14, 13, 11, 9, 8, 7, 6, 2K, resp. 3.4. King In the video [12], Berlekamp establishes the following result. We give a written proof…
Figure 6
Figure 6. Figure 6: The partition S 3 J3,1K . 15 [PITH_FULL_IMAGE:figures/full_fig_p015_6.png]
Figure 7
Figure 7. Figure 7: The regions of the CGH classification, together wit [PITH_FULL_IMAGE:figures/full_fig_p020_7.png]

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Nim on Integer Partitions and Hyperrectangles

    math.CO 2025-06 conditional novelty 7.0 of 10

    Exact Sprague-Grundy formulas are proven for two new impartial games, PNim on Young diagrams and RNim on hyperrectangles, with a full description of partitions of value one.

Reference graph

Works this paper leans on

18 extracted references · 15 canonical work pages · cited by 1 Pith paper

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