Mis\`ere Cricket Pitch

Ethan J. Saunders; Richard J. Nowakowski

REVIEW 4 major objections 4 minor 11 references

Mis\`ere Cricket Pitch

T0 review · 4 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Two numbers $M(\alpha)$ and $M(\beta)$ extracted from the bumps on either side of the roller decide the misère outcome of every reduced Cricket Pitch board.

desk verdict Lemma 2 is false—the paper's main reduction fails, so the claimed classification of all single Cricket Pitch positions is not established, though the reduced-position theorems may survive. read the letter →

arxiv 2411.17518 v1 pith:4AY2LWRE submitted 2024-11-26 math.CO

Richard J. Nowakowski , Ethan J. Saunders This is my paper · ORCID

classification math.CO MSC 91A46

keywords misèreplayCricketPitchBlockinggamescombinatorialgametheoryoutcomeclassificationdisjunctivesumsoption-closed

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 treats the misère version of the combinatorial game Cricket Pitch, in which a roller moves across a row of integer bumps and reduces each bump it crosses by one, with the player unable to move losing. The main claim is complete: for any single board, the winner can be read from two numbers, one attached to the left side of the roller and one to the right. The first theorem handles positions where one side is empty or has a simple outcome; the second theorem handles the remaining case, reducing every outcome to a comparison of $M(\alpha)$ and $M(\beta)$, the critical odd bumps on the two sides. For sums of one-bump positions, the paper proves reductions valid in the larger class of Blocking games and shows the outcome is decided by counting one-sided components. The point is that misère games normally lose algebraic structure, yet this game keeps enough structure for a full outcome classification of these positions.

What carries the argument

The load-bearing object is the marker function $M(\gamma)$, defined on each side of the roller, together with the reduction Lemma 2 (trailing odd bumps are removable) that brings arbitrary boards into reduced form. The proof of Theorem 2 also relies on the basic strategy: on each turn the moving player aims to roll just past their own marker bump, or to the very end when that marker is 1, forcing the opponent to be the first to exhaust their critical odd bump. Lemma 3 supplies three identities inside the Blocking universe — $e\circledcirc=0$ for even $e$, $d\circledcirc=1\circledcirc$ for odd $d$, and $\circledcirc 1+1\circledcirc=0$ — which are what turn sums of one-bump positions into a simple count.

What would settle it

Compute, by exhaustive game-tree search, the misère outcome of every reduced board with bumps of size at most 5 and total side length at most 8, and compare with the prediction of Theorem 2; one mismatch refutes the classification. A more targeted test looks for any board $\alpha\circledcirc\beta,(2d+1)$ whose outcome differs from $\alpha\circledcirc\beta$, which would directly contradict Lemma 2.

Watch

Extended reading notes

Core claim

The central discovery is a two-number classification. A side sequence $\gamma$ has a marker $M(\gamma)$: look from the roller outward and find the first bump that is odd and no larger than every bump before it; $M(\gamma)$ is that bump's size, and $\infty$ if no such bump exists. After removing trailing odd bumps (Corollary 1), every position becomes reduced, with both sides ending in even bumps. Theorem 2 states that for a reduced position $\alpha\circledcirc\beta$ with both sides nonempty, $o(G)=L$, $R$, $N$, or $P$ according as $M(\alpha)<M(\beta)$, $M(\alpha)>M(\beta)$, $M(\alpha)=M(\beta)<\infty$, or $M(\alpha)=M(\beta)=\infty$. The proof is driven by a 'basic strategy' in which each player pushes the roller just past the critical bump on their own side, and the first player whose critical bump gets exhausted loses the race. Theorem 3 then adds reductions in the Blocking universe — even one-bumps vanish, odd one-bumps act like a single bump of size one, and an opposite pair cancels — so a sum of one-bump positions is decided by comparing the numbers of Left-win and Right-win components.

Load-bearing premise

The classification rests on Lemma 2, which says a trailing odd bump on the right can be removed without changing the winner because Right never profits from rolling all the way to the end; if that strategy claim ever fails, the reduction to even-ended 'reduced' positions would not cover every board.

Editorial extensions

If this is right

Every single Cricket Pitch board, not just the examples, can be resolved by computing $M$ on each side after deleting odd tails; the computation is linear in the number of bumps.
The normal-play trick of subtracting 2 from every bump has no misère analogue, but the new odd-tail removal gives a misère-specific reduction that preserves outcomes.
Sums of one-bump positions are completely solved by counting Left-win versus Right-win components, with equality of counts giving a next-player win.
The reductions $e\circledcirc=0$ and $d\circledcirc=1\circledcirc$ are proved for the whole Blocking universe, so they are available for outcome questions in other Blocking games.
The worked examples exhibit positions with no additive inverse (zugzwang positions), showing why full game values still require a theory of Blocking values that does not yet exist.

Reading between the lines

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

One testable extension is that a similar marker race decides the misère outcome of other 'roller' games in the Blocking universe, provided the board is linear and moves are one-directional.
The paper leaves open general disjunctive sums; a plausible conjecture is that the full pair $(M(\alpha),M(\beta))$, not just their order, will be needed once Blocking values are defined, since equal markers can still give non-zero positions.
Because Lemma 3 is value-free, one can try to strengthen it to an equivalence of values inside a future Blocking value theory; if that strengthening holds, the outcome classification for sums would extend to one-bump positions with arbitrary coefficients.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Dear Colleague,

The headline: Lemma 2 is false, so the paper's central claim—that Theorems 1 and 2 resolve every single Cricket Pitch position—is not established. The counterexample is small: G = ⊚2,3 (left empty, right bumps 2 then 3). Lemma 2 removes the trailing odd 3 and would give o(G) = o(⊚2). But o(⊚2) = N, while a direct misère analysis gives o(⊚2,3) = L. In ⊚2,3, Left to move has no moves and wins. If Right starts, moving over one bump yields 1⊚3, and moving over both yields 2,1⊚; in both lines Left can force a win (I traced the full game tree). So the reduction underpinning Corollary 1 and the passage to reduced positions is not merely terse—it is wrong.

What the paper does well: the problem is well motivated, the reduced-position Theorems 1 and 2 may still be correct (the counterexample does not touch them directly), and the Blocking-game reductions in Lemma 3 are stated in a general setting and the arguments there look plausible. Part (2) of Lemma 3 depends on Part (1), which is fine.

Soft spots beyond the false lemma: Theorem 3 has a typo in the middle case (ℓ = s should be ℓ = r), and the proof swaps the labels for 1⊚ and ⊚1 (1⊚ is Right-win, ⊚1 is Left-win). The proof of Lemma 2 is a sketch that hides the error. These are secondary; the primary problem is the false reduction.

Who this is for: readers working on misère CGT might still find the Blocking-game reductions and the reduced-position analysis useful, but the paper as submitted is not reliable. If the authors fix Lemma 2—perhaps by restricting to reduced positions or finding a correct reduction—there is a salvageable paper here.

My recommendation: this should not go to peer review in its current form. Send the counterexample back to the authors and ask them to rework the reduction. A revised version that fixes Lemma 2 would deserve a serious referee. As it stands, I would not cite it.

Referee Report

4 major / 4 minor

Summary. The paper studies the impartial? actually partizan? Cricket Pitch game under the misère winning convention. It defines left and right sides of the roller, introduces outcome classes, and claims a complete classification of single-component positions: Theorems 1 and 2 give the misère outcome of every single Cricket Pitch position after a reduction that removes trailing odd bumps (Lemma 2 and Corollary 1) and then compares an invariant M on the two sides. For sums, Lemma 3 gives reductions claimed to hold in the Blocking universe, and Theorem 3 gives the outcome of sums of one-bump positions. The proofs of the reduced-position classification and the blocking reductions are the main technical content of the paper.

Significance. If correct, the paper would provide the first complete misère outcome classification for a nontrivial Blocking game and would show how Blocking-universe reductions simplify disjunctive sums. The definition of the M-invariant and the strategy in Theorem 2 are plausible for reduced positions, and the paper makes a good-faith attempt to give explicit winning strategies. However, the central reduction lemma is false, so the advertised completeness of the single-position classification is not achieved. The partial results about reduced positions and the Blocking reductions may be salvageable, but they do not support the abstract's claim as stated.

major comments (4)

[Section 2, Lemma 2 and Corollary 1] Lemma 2 is false. Consider G = ⊚2,3, i.e. α empty, β=(2), and odd tail 3; β is non-empty, so the hypotheses are satisfied. Direct misère analysis gives o(G)=L: Left, moving first, has no move and wins; if Right moves first over one bump, the forced line is 1⊚3 → ⊚3 → 2⊚ → ⊚1 → ⊚, after which Left has no move and wins; if Right moves over both bumps, Left wins by rolling over both to ⊚1, and Right's only move leaves Left with no moves. Hence o(G)=L. But Lemma 2 would give o(G)=o(⊚2)=N, since ⊚2 is a next-player win. Thus 'odd tails are removable' is invalid, and Theorem 2, which applies only to reduced positions, does not cover all single positions as claimed in the abstract.
[Section 2, proof of Lemma 2] The proof of Lemma 2 is not a proof even apart from the counterexample. The statement that if Right moves the roller to the end, then Left can regard the game as being (2d)⊚, and after 2d moves it is Left to move in 0⊚, does not account for the other bumps of β that have been rolled over and now appear on the left side, affecting all subsequent moves. The assertion that a winning Right 'never plays to the end' is unsupported. A rigorous proof of the claimed outcome equality is missing.
[Section 3, Theorem 3] The statement of Theorem 3 is defective. The case 'N if ℓ = s' uses the undefined symbol s, presumably r. More substantively, the proof mislabels the components: it says that ⊚1, 1⊚, and 0 are 'Right, Left, and Next wins respectively', and then sets ℓ to count 1⊚ components and r to count ⊚1 components. By the game rules, 1⊚ is an R-position and ⊚1 is an L-position, so the two assignments are inconsistent. The theorem's conclusion cannot be checked as written.
[Section 2.1, proof of Theorem 2] The proof for cases 1–3 handles Right moving first only under the assumption aℓ < br; the equality case aℓ = br, which is needed for part 3 (M(α)=M(β)<∞), is not treated. A symmetry argument may supply the missing case, but it is not stated, so the proof of part 3 is incomplete.

minor comments (4)

[Introduction] There are typographical errors, e.g. 'a an unenviable task' and 'the i-th bump' with a missing article.
[Section 3, Theorem 3] The symbol s in 'N if ℓ = s' should be r; as written it is undefined.
[Section 1.1] The examples such as '2 n⊚ = 0' and '1⊚ + ⊚1 = 0' use notation that is not defined until later; a brief definition at first use would improve readability.
[Section 3, proof of Lemma 3] The proof of part (3) says 'The other cases are similar and are left to the reader', which leaves a substantial part of the equivalence proof unverified; more details would be needed for a complete proof.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the misere outcomes are derived from the game rules by case analysis and induction, with no fitted parameters and no load-bearing self-citation.

full rationale

The paper's central claims, Theorems 1 and 2 for single positions and Theorem 3 for sums of one-bump positions, are proved from the rules of Cricket Pitch rather than assumed from the target conclusions. The definitions of reduced positions, M(alpha) and M(beta), are stated independently and then used to classify outcomes; no parameter is fitted to the outcome data being predicted. Lemma 1 is proved by induction on the bump sequence, Lemma 2 is argued directly from the game's move structure, and Lemma 3 is proved within the paper for the Blocking universe before being applied to Cricket Pitch. The only self-citation, [10] for the 2011 normal-play analysis, is contextual: it identifies the game and states that normal play was solved, but the misere derivation does not rely on that paper's results. The Conjugate Conjecture and Blocking-universe context are background, not load-bearing assumptions. The proof of Lemma 2 is terse and may be a correctness risk, as the skeptic notes, but a disputed or false lemma is not circularity: the lemma is not defined in terms of the theorem it supports, and no cited result from the same authors is invoked to force the outcome. Because the derivation chain is self-contained against the rules and standard CGT conventions, the circularity score is 0.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

No free parameters or invented entities; this is pure combinatorial game theory. The main results rest on the game rules, the Blocking universe definition (introduced without citation), and standard CGT background.

assumptions (3)

domain assumption The game rules and misère convention (last player to move loses)
The analysis is specific to this rule set; a different termination convention would change outcomes.
domain assumption The Blocking universe definition and its closure properties
Lemma 3 is proven within the blocking universe, B; if the class definition is not coherent, the reduction fails.
standard math Standard combinatorial game theory: outcome classes, disjunctive sum, equivalence
Standard background from Conway and Winning Ways.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Mis\`ere Cricket Pitch." pith.science (2026). https://pith.science/paper/4AY2LWRE

@misc{pith2026241117518,
  author       = {Pith},
  title        = {Pith review of: Mis\`ere Cricket Pitch},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4AY2LWRE}},
  note         = {Machine review of arXiv:2411.17518}
}

abstract

Mis\`ere games in general have little algebraic structure, but if the games under consideration have properties then some algebraic structure re-appears. In 2023, the class of Blocking games was identified. Mis\`ere Cricket Pitch was suggested as a problem at the Games-at-Dal-2023 Workshop, and is the first game in this class to be studied. Normal play Cricket Pitch was analyzed in 2011. The game involves flattening `bumps' with a roller. The main reduction of normal play, reducing every bump by $2$, is not applicable in mis\`ere play. In this paper, we find the outcomes of single (linear) component \textsc{cricket pitch} positions, where the proof is based on first considering the bumps to the left and to the right of the roller separately. We also give reductions, true in Blocking games in general, of positions that occur in simple positions of Cricket Pitch. These allow us to find the outcomes of the disjunctive sum of single bump positions. At the time of writing, it was not possible to find game values since the relevant theory for Blocking games does not exist.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

11 extracted references · 11 canonical work pages

[1]

https://www.thegma.org.uk/learning/resources/cricket-pitc h-maintenance- playing-season-maintenance–acessed July 28, 2024

work page 2024
[2]

Albert, R

M. Albert, R. J. Nowakowski, and D. Wolfe, Lessons in Play: An Introduction to Combinatorial Game Theory, Second Edition. AK Peters/CRC Press, 2019

work page 2019
[3]

Winning Ways , Academic Press, London, 1982

Berlekamp, E., Conway, J., Guy, R. Winning Ways , Academic Press, London, 1982

work page 1982
[4]

On Numbers and Games , Academic Press, 1976

Conway, J. On Numbers and Games , Academic Press, 1976

work page 1976
[5]

Dicots, and a taxonomic ranking for mis` ere games

Dorbec, P., Renault, G., Siegel, A., Sopena. E. “Dicots, and a taxonomic ranking for mis` ere games”, The Seventh European Conference on Combinatorics, Graph Theory and Applications , Volume 16 of the series CRM Series, 371–374, 2013

work page 2013
[6]

Invertible elements of the dicot mis` ere universe

Fisher, M., Nowakowski, R. J., Santos, C. “Invertible elements of the dicot mis` ere universe”,Integers, 21(2), #G06, 2022

work page 2022
[7]

Absolute combinato rial game theory

Larsson, U., Nowakowski, R. J., Santos, C. “Absolute combinato rial game theory”, in S. Huntemann and U. Larrson (Eds.) Games of No Chance 6 , MSRI Publ., 2024, (to appear)

work page 2024
[8]

Progress on mis` ere dead ends: game comparison, canonical form, and conj ugate inverses

Larsson, U., Milley, R., Nowakowski, R. J., Renault, G., Santos, C. “ Progress on mis` ere dead ends: game comparison, canonical form, and conj ugate inverses”, in S. Huntemann and U. Larrson (Eds.) Games of No Chance 6 , MSRI Publ., 2024 (to appear)

work page 2024

Show all 11 references

[9]

Milley, R., Partizan Kayles and mis` ere invertibility, Integers, 15, 2015, Paper No. G3, 14

2015
[10]

Option-closed games

R. J. Nowakowski and P. Ottaway, “Option-closed games”, Contributions to Discrete Mathematics, vol. 6, no. 1, pp. 142–153, 2011

2011
[11]

Combinatorial Game Theory , American Mathematical Society, Providence, Rhode Island, 2013

Siegel, A. Combinatorial Game Theory , American Mathematical Society, Providence, Rhode Island, 2013

2013

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

https://www.thegma.org.uk/learning/resources/cricket-pitc h-maintenance- playing-season-maintenance–acessed July 28, 2024

work page 2024

[2] [2]

Albert, R

M. Albert, R. J. Nowakowski, and D. Wolfe, Lessons in Play: An Introduction to Combinatorial Game Theory, Second Edition. AK Peters/CRC Press, 2019

work page 2019

[3] [3]

Winning Ways , Academic Press, London, 1982

Berlekamp, E., Conway, J., Guy, R. Winning Ways , Academic Press, London, 1982

work page 1982

[4] [4]

On Numbers and Games , Academic Press, 1976

Conway, J. On Numbers and Games , Academic Press, 1976

work page 1976

[5] [5]

Dicots, and a taxonomic ranking for mis` ere games

Dorbec, P., Renault, G., Siegel, A., Sopena. E. “Dicots, and a taxonomic ranking for mis` ere games”, The Seventh European Conference on Combinatorics, Graph Theory and Applications , Volume 16 of the series CRM Series, 371–374, 2013

work page 2013

[6] [6]

Invertible elements of the dicot mis` ere universe

Fisher, M., Nowakowski, R. J., Santos, C. “Invertible elements of the dicot mis` ere universe”,Integers, 21(2), #G06, 2022

work page 2022

[7] [7]

Absolute combinato rial game theory

Larsson, U., Nowakowski, R. J., Santos, C. “Absolute combinato rial game theory”, in S. Huntemann and U. Larrson (Eds.) Games of No Chance 6 , MSRI Publ., 2024, (to appear)

work page 2024

[8] [8]

Progress on mis` ere dead ends: game comparison, canonical form, and conj ugate inverses

Larsson, U., Milley, R., Nowakowski, R. J., Renault, G., Santos, C. “ Progress on mis` ere dead ends: game comparison, canonical form, and conj ugate inverses”, in S. Huntemann and U. Larrson (Eds.) Games of No Chance 6 , MSRI Publ., 2024 (to appear)

work page 2024