REVIEW 3 minor
An Achievement Game on a Cycle
T0 review · 0 major / 3 minor · reviewed 2026-05-24 · grok-4.3
Pith's one-line read The paper computes the exact optimal number of Maker's adjacent pairs in the vertex-claiming game on cycle C_n for every n.
desk verdict This paper settles the exact value of the adjacency game on cycles for every n with explicit strategies and a mod-3 case split. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The vertex-claiming game on C_n in which Maker's payoff equals the number of edges with both endpoints taken by her.
What would settle it
An exhaustive computation of the optimal score for a small fixed n such as n=7 that differs from the paper's claimed value.
Extended reading notes
Core claim
In the achievement game played on the vertices of the cycle C_n with Breaker moving first, the number of adjacent pairs both claimed by Maker under optimal play by both players is determined exactly for every n.
Load-bearing premise
Optimal play by both Maker and Breaker on C_n produces a well-defined value expressible by a simple formula for every n.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper analyzes a Maker-Breaker achievement game on the cycle C_n (Breaker moves first) in which Maker seeks to maximize the number of her adjacent vertex pairs while Breaker seeks to minimize it. The central claim is that the exact value of this number under optimal play is determined for every n, via explicit strategies for both players together with a case analysis that partitions on n mod 3 and accounts for Breaker's opening move; matching upper and lower bounds are established.
Significance. The result supplies a complete, exact determination of the optimal score for every n, resolving the question posed by Dowden et al. The proof is constructive (explicit strategies) and relies only on the finite, deterministic nature of the game; the case analysis yields a simple closed-form expression. This constitutes a self-contained contribution to positional games on graphs.
minor comments (3)
- §2, Definition 1: the notation for the score function could be introduced earlier to avoid forward references when the strategies are described.
- Figure 1: the labeling of vertices in the cycle diagram is not aligned with the case n ≡ 1 (mod 3) discussed immediately below; a small adjustment would improve readability.
- §4, final paragraph: the sentence summarizing the three cases repeats a phrase already used in the introduction; a single consolidated statement would suffice.
Simulated Author's Rebuttal
We thank the referee for their positive report and recommendation to accept the manuscript. The referee's summary accurately captures the main contribution.
Circularity Check
No significant circularity; derivation is self-contained combinatorial analysis
full rationale
The paper computes the exact value of the game via explicit strategies for Maker and Breaker, with a case analysis on n mod 3 and Breaker's first move to obtain matching upper and lower bounds. No parameters are fitted, no self-citations are load-bearing, and no equations reduce to prior results by construction. The finite perfect-information game admits an exact value by standard game theory, and the manuscript derives it directly without circular reduction.
Assumptions & free parameters
Cite this review
Pith. "Pith review of An Achievement Game on a Cycle." pith.science (2026). https://pith.science/paper/6KZQ3UTK
@misc{pith2026190711152,
author = {Pith},
title = {Pith review of: An Achievement Game on a Cycle},
year = {2026},
howpublished = {\url{https://pith.science/paper/6KZQ3UTK}},
note = {Machine review of arXiv:1907.11152}
}
abstract
Consider the following game played by Maker and Breaker on the vertices of the cycle $C_{n}$, with first move given to Breaker. The aim of Maker is to maximise the number of adjacent pairs of vertices that are both claimed by her, and the aim of Breaker is to minimise this number. The aim of this paper is to find this number exactly for all $n$ when both players play optimally, answering a related question of Dowden, Kang, Mikala\v{c}ki and Stojakovi\'{c}.
Reviewed May 24, 2026 · model on record in the stance chip above.
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