Unique Winning Opening Move in Three-Row Chomp
Pith reviewed 2026-05-25 03:41 UTC · model grok-4.3
The pith
Every 3 by n Chomp board has exactly one winning opening move.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
We prove that every 3 × n Chomp rectangle has exactly one winning opening move. The proof is carried out in the two-variable recurrence for the function f(q,r) whose values encode the P-positions, using a rightmost-hole principle that separates the diagonal values from the starts of constant rows and yields a partition of the positive integers into the two possible types of winning opening moves.
What carries the argument
The two-variable recurrence introduced for f(q,r) encoding P-positions, together with the rightmost-hole principle that forces intermediate values into the sets when a gap appears.
If this is right
- The uniqueness of the winning move holds for arbitrarily large n.
- The positive integers are partitioned into two types corresponding to the two possible winning move positions.
- The rightmost-hole principle separates diagonal values from constant-row starts in the position sets.
- Computations for n up to 100 are confirmed and extended to all n.
Where Pith is reading between the lines
- Similar uniqueness might be investigated in Chomp with more rows or different shapes.
- The recurrence could be used to compute the specific winning move for any given n efficiently.
- If the recurrence holds, it classifies all P-positions across three-row boards without further case analysis.
Load-bearing premise
The two-variable recurrence correctly identifies all P-positions in three-row Chomp and the rightmost-hole principle applies to separate the relevant sets.
What would settle it
A computation or counterexample showing some 3 by n board has either zero or more than one winning opening move would falsify the claim.
read the original abstract
Chomp was introduced by Gale in 1974 \cite{Gale1974}. In the same paper, Gale reported that the $3\times n$ games had been completely analyzed for $n\le 100$, with a unique winning first move in every case, and asked whether winning first moves are unique in general. Although the general uniqueness statement is false \cite[Section~7.1]{BrouwerEtAl2005}, we prove that the three-row uniqueness phenomenon suggested by Gale's computations holds for all $n$: every $3\times n$ Chomp rectangle has exactly one winning opening move. This settles the three-row case of Gale's 52-year-old first-move uniqueness question. The proof is carried out in the two-variable recurrence introduced by Brouwer, Horv\'ath, Moln\'ar-S\'aska, and Szab\'o \cite{BrouwerEtAl2005} for the function $f(q,r)$ whose values encode the $P$-positions. The main local ingredient is a rightmost-hole principle: if a value $p$ is absent from the set $C(q,r)$ but belongs to all corresponding sets $C(t,r)$ for $q<t<p$, then all intermediate values $q+1,\ldots,p-1$ are forced to belong to $C(q,r)$. This separates the diagonal values from the starts of constant rows, and yields a partition of the positive integers into the two possible types of winning opening moves. It also identifies the row of the unique opening move: no first-row opening move is winning; the second-row and third-row cases are precisely the two complementary Chomp sequences A029900 and A029901.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper proves that every 3×n Chomp rectangle has exactly one winning opening move. It works in the two-variable recurrence of Brouwer–Horváth–Molnár-Sáska–Szabó that encodes the P-positions of three-row Chomp, introduces a rightmost-hole principle on the sets C(q,r), and uses that principle to partition the positive integers into the two admissible types of winning first moves (diagonal versus constant-row).
Significance. If the argument is correct, the result settles the three-row case of Gale’s 1974 first-move uniqueness question. The manuscript gives explicit credit to the prior recurrence and isolates the new local principle that separates the two families of moves; this is a concrete advance for the three-row family even though the general uniqueness statement is already known to be false.
major comments (1)
- [Rightmost-hole principle paragraph] Rightmost-hole principle (the paragraph immediately following the statement of the recurrence): the implication “if p ∉ C(q,r) yet p ∈ C(t,r) for all q < t < p, then q+1,…,p−1 ∈ C(q,r)” is asserted without an explicit inductive verification that the recurrence cannot insert an extra hole at the boundary t = p. Because the uniqueness claim is obtained precisely by showing that this principle forces every integer into exactly one of the two families, a self-contained proof of the principle (or a machine-checked check for small r) is required before the separation argument can be accepted.
minor comments (2)
- Notation: the sets C(q,r) are introduced without an explicit sentence reminding the reader that they are the column indices of the P-positions for fixed r; a one-line definition would improve readability.
- The manuscript cites the Brouwer et al. recurrence but does not restate its exact two-variable form; including the recurrence as Equation (1) would make the subsequent argument self-contained.
Simulated Author's Rebuttal
We thank the referee for the careful reading and constructive suggestion regarding the rightmost-hole principle. We agree that making the verification explicit will strengthen the manuscript and will add a self-contained inductive proof in the revised version.
read point-by-point responses
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Referee: [Rightmost-hole principle paragraph] Rightmost-hole principle (the paragraph immediately following the statement of the recurrence): the implication “if p ∉ C(q,r) yet p ∈ C(t,r) for all q < t < p, then q+1,…,p−1 ∈ C(q,r)” is asserted without an explicit inductive verification that the recurrence cannot insert an extra hole at the boundary t = p. Because the uniqueness claim is obtained precisely by showing that this principle forces every integer into exactly one of the two families, a self-contained proof of the principle (or a machine-checked check for small r) is required before the separation argument can be accepted.
Authors: We agree that the manuscript would benefit from an explicit inductive argument establishing the rightmost-hole principle directly from the recurrence. In the revision we will insert a new subsection immediately after the recurrence statement that proves the claimed implication by induction on p, using the two-variable recurrence to show that no extra hole can appear at the boundary t = p. The argument relies only on the already-established properties of the sets C(q,r) and does not require machine verification for small r, although we are happy to add a short computational check for r ≤ 10 as an appendix if the referee prefers. revision: yes
Circularity Check
No circularity; derivation uses external recurrence plus independently stated local principle
full rationale
The paper defines C(q,r) via the two-variable recurrence of Brouwer et al. (distinct authors) and introduces the rightmost-hole principle as an additional local ingredient that is applied to separate diagonal and constant-row winning moves. No step reduces a claimed prediction or uniqueness result to a fitted parameter, self-definition, or self-citation chain; the central partition of positive integers follows from the recurrence plus the principle without the result being presupposed by construction. The argument is therefore self-contained against external benchmarks.
Axiom & Free-Parameter Ledger
axioms (2)
- domain assumption The two-variable recurrence of Brouwer et al. encodes the P-positions of three-row Chomp
- ad hoc to paper The rightmost-hole principle holds and forces the required membership in the sets C(q,r)
discussion (0)
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