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REVIEW 3 major objections 5 minor 27 references

Advances in losing

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Pith's one-line read Misère impartial games are solved by computing a finite commutative monoid, the indistinguishability quotient; Pascal's Beans and Guiles are analyzed completely.

desk verdict Useful survey with two credible-but-unproven new quotient analyses; the real gap is missing derivations or public software, not the alleged congruence issue. read the letter →

arxiv math/0603027 v1 pith:ROAC4GTV submitted 2006-03-01 math.CO

Thane E. Plambeck This is my paper · ORCID
classification math.CO MSC 91A46
keywords misèreplayimpartialcombinatorialgamesindistinguishabilityquotientSprague–Grundytheorycommutativemonoidoctalpretendingfunctionswild
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 argues that the Sprague–Grundy theory of normal-play impartial games has a working generalization to misère play in which the last player to move loses: the indistinguishability quotient. Positions that behave alike in every context are identified, and the result is a commutative monoid, called the indistinguishability quotient, together with a pretending function that labels each starting position. The concrete demonstrations are complete misère analyses of two 'wild' games—Pascal's Beans, introduced here, and the octal game Guiles—where finite quotients of order 12 and 42 determine the outcome of every position. If the construction is sound, misère analysis shifts from a case-by-case morass to a computer-assisted algebraic computation, with normal play appearing as the special case in which the quotient is just the nim-addition group.

What carries the argument

The central object is the indistinguishability quotient $Q(\Gamma)=A/\rho$ for a fixed impartial game $\Gamma$: the quotient of the position set by the relation 'interchangeable in every sum,' with commutative monoid structure given by game addition. The argument rides on the claimed congruence property (Equation (1)) and on the pretending function $\Phi$, which sends positions to their quotient elements; the paper cites a theorem that these pretending functions are provably periodic once they stabilize, which turns a finite computation into a complete analysis.

What would settle it

For Guiles, extend the single-heap equivalence table beyond the printed preperiod—through heap size 90 or 100—and compare with the claimed period-10 pattern; any mismatch would refute the analysis. For Pascal's Beans, test the congruence property directly by searching two- and three-bean sums for a pair $G,H$ with $G$ and $H$ in the same quotient class but $G+X$ and $H+X$ in different outcome classes for some $X$.

Watch

Extended reading notes

Core claim

The paper's central assertion is that the indistinguishability relation on the positions of a fixed impartial game is a congruence. Specifically, for $A$ the set of positions closed under addition and taking options, $G \rho H$ whenever $G+X$ and $H+X$ have the same outcome for every $X \in A$, and the paper states (Equation (1)) that $G \rho H$ implies $(G+X) \rho (H+X)$. Consequently the quotient $Q=A/\rho$ is a commutative monoid with addition defined by $\rho_G + \rho_H = \rho_{G+H}$, and the natural map $\Phi: G \mapsto \rho_G$, called the pretending function, labels every position by a monoid element. In normal play the quotient is an elementary abelian 2-group under nim addition, so the construction contains Sprague–Grundy theory as a special case; in misère play it yields finite monoids for games that previously resisted analysis. The paper reports complete misère analyses of Pascal's Beans (monoid $\langle a,b,c \mid a^2=1, c^2=1, b^3=b^2c\rangle$ of order 12, with P-position types $\{a,b^2,ac\}$) and of Guiles, the octal game 0.15, with a 42-element quotient whose single-heap pretending function is eventually periodic of length 10.

Load-bearing premise

The load-bearing premise is that two positions that behave alike in every context still behave alike after the same extra position is attached to both; if that fails, the quotient the whole analysis is built on is not well defined.

Editorial extensions

If this is right

  • Complete misère analyses of Pascal's Beans and Guiles follow: multiplying monoid elements and checking membership in the listed P-position types decides the outcome of any sum of positions.
  • Normal-play Sprague–Grundy theory is a special case: its quotients are direct products of $\mathbb{Z}_2$'s and the pretending function is the familiar nim-value.
  • Wild misère games that resisted the classical genus theory become computable, and the paper reports hundreds of octal games solved in this way, including seventeen of the twenty-one wild quaternary games listed as open bounties.
  • The construction also reformulates the classical tame-game genus theory, producing the small 'tame quotients' $T_1$, $T_2$, ..., from the same monoid machinery.
  • Not every misère game is finite: Dawson's Chess appears to require an infinite quotient, so the construction has a genuine boundary rather than being a universal cure.

Reading between the lines

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

  • If the periodicity theorem behind pretending functions holds in the stated generality, then outcome determination for any misère impartial game with finite quotient is computationally easy—essentially a table lookup after multiplying finitely many monoid elements—which would extend a normal-play-style tractability dichotomy to misère play.
  • The classification problem posed in the paper suggests the monoids arising as misère quotients form an extremely sparse class among commutative semigroups; a complete list at each order could serve as a taxonomy of misère games analogous to the role of nim-values in normal play.
  • The open 'misère mex mystery' could be approached as a property of the pretending function's eventual periodicity: if partial quotients stabilize, then the stabilization itself may encode a recursive rule for computing the next pretending value, effectively a misère analogue of the mex rule.
  • The small coin-sliding example suggests the construction applies beyond heap games to any finite impartial ruleset with a chosen starting position; a testable extension would be to run the computation on other small directed graphs and see whether the quotient order remains bounded.
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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 / 5 minor

Summary. The paper is a survey of the indistinguishability-quotient approach to misere impartial combinatorial games, based on a 2005 Banff lecture. Sections 1-2 frame the method as a generalization of Sprague-Grundy theory to misere play. Sections 3-4 present, as the paper's main new results, "complete misere analyses" of two wild games: Pascal's Beans (introduced here) and Guiles (octal game 0.15). For each, the paper displays a finite commutative monoid presentation, a partition into P- and N-position types, and a single-position pretending function, and it illustrates how to compute outcomes by multiplying monoid elements. Section 5 defines the indistinguishability quotient and pretending function; Sections 6-7 relate them to misere canonical forms and to genus/tame-game theory; Section 8 lists further quotients; Section 9 describes MisereSolver and partial-quotient computation; Section 10 lists open problems; and an appendix reviews Conway's genus theory.

Significance. If the claimed complete analyses are correct, the paper is a valuable demonstration that wild misere games can be solved by finite quotient computation, and the survey is useful for its careful exposition of the quotient construction, tame quotients, genus theory, and a collection of open problems. The paper also credits MisereSolver and gives an instructive example of partial-quotient instability in Section 9.4. However, the paper contains no proofs of its central new claims: the monoid presentations, P-position sets, and pretending functions for Pascal's Beans and Guiles are asserted from MisereSolver output and unpublished sources, so the new results are not independently checkable from the manuscript. I agree with the stress-test note that the congruence step in Eq. (1) is not the weak point; the weak point is the unsupported completeness of the two quotient computations. The framework itself is credible and consistent with the cited literature, and the appendix's genus-theory summary appears sound.

major comments (3)
  1. [§3.2, §4.2] The paper's central new results are the "complete misere analyses" of Pascal's Beans and Guiles, but neither analysis is proved. In §3.2 the order-12 monoid M, its presentation, the P/N partition, and the pretending function in Figure 4 are introduced with only the remark that "assiduous readers might enjoy verifying" the reduction to canonical words; no argument shows that this monoid is the indistinguishability quotient of the game or that the displayed map is the true pretending function for every position. In §4.2 the order-42 quotient Q, the preperiod-66/period-10 single-heap sequence of Figure 6, and the P-position list are attributed to "Aaron Siegel [PS] found" using MisereSolver, with [PS] listed as "in preparation." These assertions carry the entire weight of the claimed new complete analyses, and the later example 4+58+68+78 = d^2 inherits the same gap: the computation is correct only if the quotient presentation and the heap values are correct.
  2. [§9.4] The completeness of the quotient computations is not established by the displayed MisereSolver output. Section 9.4 explicitly demonstrates that a partial quotient can stabilize temporarily and then change when larger heap sizes are considered: in 0.123, 4+4 is indistinguishable from 6 in the heap-6 partial quotient but distinguishable in the full game. For this reason, the assertions in §3.2 and §4.2 that the Pascal's Beans and Guiles quotient computations are complete require a proof of stabilization, or a bound beyond which no new distinguishing positions can appear. No such proof or bound is provided, and the paper does not state a correctness theorem for MisereSolver's stopping rule.
  3. [References [AS2005], [PS], [P2]] The main new results depend on sources that are not available for verification: [AS2005] is a private communication, [PS] is in preparation, and the central construction is credited to [P2], also by the author. This is not by itself evidence of a logical error, but it prevents a referee from checking the two flagship "complete analyses" from the manuscript alone. The author should either include a detailed derivation/verification of the two quotients and pretending functions, or clearly label them as announced results from [PS] and state that no proof is included here.
minor comments (5)
  1. [Global] The paper is presented as a survey but introduces new games and results (Pascal's Beans in §3; the Guiles analysis in §4) without flagging them as unproved announcements; the introduction should state explicitly which results are new and which are surveyed.
  2. [§10.1.1, Figures 17-18] There is a figure-numbering inconsistency: the text refers to "Figure 18" for the growth of Dawson's Chess partial quotients, but the corresponding caption is numbered Figure 17, and Figure 18 is used for the conjectured quotient counts in §10.2.
  3. [§7.1.2, §11.2] The typesetting of the brace notation (for example, "\bracehtipupleft/bracehtipdownright" in the T_n presentation and in the genus-symbol discussion) is garbled in the manuscript; these formulas need to be cleaned up before publication.
  4. [Throughout] There are numerous spelling and typographical slips (e.g., "intractible," "com binatorial," "difference," "concide," "manscript," "abounding") that should be corrected.
  5. [§6.1.6] Equation (5) is written as "/BE + /BE ?ρ /BD", which is not a standard assertion; it should be phrased as a question or as a proposed relation "/BE + /BE ρ /BD" with an explicit qualifier.

Circularity Check

1 steps flagged · score 4.0 of 10

Central new analyses are deferred to coauthored in-preparation work and private software; no equation-level reduction, but the load-bearing derivation chain is self-citational.

  1. self citation load bearing [Section 4.2 (Guiles: misère play) and references [PS], [AS2005]]
    "Using his recently-developed Java-language computer program MisereSolver, Aaron Siegel [PS] found that the misere indistinguishability quotient Q of misere Guiles is a (commutative) monoid of order 42. ... References: [PS] Thane E. Plambeck & Aaron Siegel, “The Φ-values of various games”, in preparation."

    The claimed complete misère analysis of Guiles is not derived in this paper: the order-42 quotient Q, the periodic pretending function in Figure 6, and the P-position set are all attributed to [PS], a coauthored paper listed as 'in preparation', and to the private program MisereSolver [AS2005]. The later worked 'prediction' that 4+58+68+78 is a P-position is valid only conditional on that unshown quotient presentation and on Figure 6 being the true quotient map for all heap sizes. Thus the flagship new result is load-bearing on a self-citation/private-communication chain rather than on a proof contained in the paper. This is not a definitional equation-level circularity, but the central derivation chain reduces to an unverifiable self-citation in the sense of pattern 3.

full rationale

The paper is not circular by construction. The indistinguishability relation ρ is defined by outcome equivalence and Eq. (1), G ρ H ⇒ (G+X) ρ (H+X), follows directly from that definition together with closure of A under addition, so the congruence step is not a smuggled assumption. The normal-play discussion explicitly says the quotient recasts Sprague-Grundy theory and that 'nothing new' is learned, so it is not a renamed-result deception. The Pascal's Beans and Guiles examples are conditional algebraic computations: if the stated monoid presentations and pretending functions are correct, the monoid product calculation correctly determines outcomes. The genuine problem is evidentiary and self-citational: the 'complete misère analyses' for the two showcased games rest on (i) a one-line invitation for the reader to verify only word-reduction identities in the monoid presentation, not that Figure 4 is the quotient map, and (ii) [PS], a coauthored in-preparation paper, plus the private program [AS2005]. Section 9.3 treats stabilization of partial quotients as discovery of the complete quotient without proving that stabilization criteria imply completeness for all heap sizes; that theorem is inherited from [P2]. None of this is an internal definitional identity or a fitted parameter renamed as a prediction, so a score of 6 or higher would overstate the circularity. However, the load-bearing support for the central new claims is a self-citation/private-communication chain, justifying score 4 rather than 0-2.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters or invented entities: the paper contains no fitted numerical constants, and the monoid presentations are computed outputs rather than tuned parameters. The axioms list the unproved or deferred premises on which the example analyses depend, chiefly the congruence property and the correctness of MisereSolver's outputs.

assumptions (4)
  • domain assumption Indistinguishability rho is a congruence on the position set A.
    Section 5, Eq (1); asserted and attributed to [P2]. If rho were not compatible with addition, the quotient Q = A/rho would not be well-defined.
  • ad hoc to paper The monoid presentations and pretending functions output by MisereSolver are correct.
    Sections 3.2, 4.2, 9.3; outputs asserted without proofs or public code, with MisereSolver cited as private communication [AS2005].
  • ad hoc to paper Periodicity of pretending functions for the example games, including Pascal's Beans and Guiles.
    Section 5 says 'provably periodic [P2]'; Section 4.2 asserts period 10 with preperiod 66 for Guiles; no proof or reproduction is provided.
  • standard math Redei's Theorem: every finitely generated commutative monoid is finitely presentable.
    Invoked in Section 10.1.1 to assert the existence of a presentation for Dawson's Chess partial quotients.

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Pith. "Pith review of Advances in losing." pith.science (2026). https://pith.science/paper/ROAC4GTV

@misc{pith2026math0603027,
  author       = {Pith},
  title        = {Pith review of: Advances in losing},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ROAC4GTV}},
  note         = {Machine review of arXiv:math/0603027}
}
read the original abstract

We survey recent developments in the theory of impartial combinatorial games in misere play, focusing on how the Sprague-Grundy theory of normal-play impartial games generalizes to misere play via the indistinguishability quotient construction.

Figures

Figures reproduced from arXiv: math/0603027 by the authors.

Figure 1
Figure 1. The Pascal’s Beans board. boundary position of the triangle. The game ends when all beans have reached the boundary. 3.1. Normal play. In normal play of Pascal’s Beans, the last player to make a legal move is declared the winner of the game [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. The pattern of single-bean nim-values in normal play of Pascal’s Beans. Each interior value is the minimal excludant (or mex) of the two nim values immediately above it. The underlined entries form the first three rows of an infinite subtriangle whose rows alternate between ∗0 and ∗1. 3.2. Misere play. In misere play of Pascal’s Beans, the last player to make a move is declared the loser of the game. Is it possible … view at source ↗
Figure 3
Figure 3. Addition for normal play of Pascal’s Beans. the values to be identified with particular positions of the triangle and the desired misere addition are given by a particular twelve-element commutative monoid M, the misere indistinguishability quotient7 of Pascal’s Beans. The monoid M has an identity 1 and is presentable using three generators and relations: M = h a, b, c | a 2 = 1, c2 = 1, b3 = b 2 c i. Assiduous read… view at source ↗
Figures from the paper (20 more)
Figure 4
Figure 4. Figure 4: shows the identification of positions of the triangle with elements of M. 1 1 1 1 a 1 1 b b 1 1 a b 2 a 1 1 b c c b 1 1 a b2 b 2 b 2 a 1 1 b c ab2 ab2 c b 1 1 a b2 b 2 b 2 b 2 b 2 a 1 . . . . . . . . [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]
Figure 5
Figure 5. Figure 5: Nim values for normal play 0.15 4.2. Misere play. Using his recently-developed Java-language computer program MisereSolver, Aaron Siegel [PS] found that the misere indistinguishability quotient Q of misere Guiles is a (commutative) monoid of order 42. It has the presen…
Figure 6
Figure 6. Figure 6: Misere equivalences for Guiles. Knowledge of the monoid presentation Q, its partition into N- and P-position types, and the single-heap equivalences in [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]
Figure 7
Figure 7. Figure 7: Canonical forms for misere games born by day 4. true that for misere games G, H with different canonical forms that there must be a game X such that G + X and H + X have different outcomes, such an X might possibly never occur in play of the fixed game Γ that we’ve cho…
Figure 8
Figure 8. Figure 8: The first and second tame quotients T1 is called the first tame quotient. It represents the misere play of She-Loves￾Me, She-Loves-Me-Not. In T1, misere P-positions are represented by the monoid (in fact, group) element a, and N-positions by 1. T2, the second tame quot…
Figure 9
Figure 9. Figure 9: The misere impartial game theorist’s coat of arms, or the Cayley graph of T2. Arrows have been drawn to show the action of the generators a (the doubled rungs of the ladder) and b (the southwest-to-northeast-oriented arrows) on T2. See also [PITH_FULL_IMAGE:figures/fu…
Figure 10
Figure 10. Figure 10: When misere Nim is played with heaps of size 1 and 2 only, the resulting misere indistinguishability quotient is the tame six-element monoid T2. For more on genus symbols and tameness, see section 7. See also [PITH_FULL_IMAGE:figures/full_fig_p014_10.png]
Figure 11
Figure 11. Figure 11: The pretending function for misere play of 0.75. 1 2 3 4 5 6 7 8 0+ a 1 a b 1 a 1 ab 8+ a c a b 1 ac 1 ab 16+ a c a b 1 ac 1 ab [PITH_FULL_IMAGE:figures/full_fig_p015_11.png]
Figure 12
Figure 12. Figure 12: The pretending function for misere play of 0.34. 8. More wild quotients 8.1. The commutative monoid R8. The smallest wild misere indistinguishabil￾ity quotient R8 has eight elements, and is unique up to isomorphism [S1] amongst misere quotients with eight elements. It…
Figure 13
Figure 13. Figure 13: The pretending function for misere play of 0.71. 8.2.2. 0.71. The game 0.71 has a misere quotient of order 36 with the presentation Q0.71 = h a, b, c, d | a 2 = 1, b4 = b 2 , b2 c = c, c4 = ac3 , c3 d = c 3 , d2 = 1 i. The P-positions are {a, b2 , bc, c2 , ac3 , ad, b…
Figure 14
Figure 14. Figure 14: Iterative calculation of misere partial quotients differs in a fundamental way from normal play nim-sequence calculation because sums at larger heap sizes (for example, 8+9) may distin￾guish between positions that previously were indistinguishable at earlier partial q…
Figure 15
Figure 15. Figure 15: Misere coin-sliding on a directed heptagon with two additional edges. An arbitrary number of coins are placed at the vertices, and two players take turns sliding a single coin along a single directed edge. Play ends when the final coin reaches the topmost (sink) node …
Figure 16
Figure 16. Figure 16: Assignment of single-coin positions in the heptagon game to misere quotients elements. 10. Outlook At the time of this writing (December 2005), the indistinguishability quotient construction is only one year old. Several aspects of the theory are ripe for further deve…
Figure 17
Figure 17. Figure 17: Is 0.07 infinite at heap 34? Since Redei’s Theorem (see [P2] for discussion and additional references) asserts that a finitely generated commutative monoid is always finitely presentable, the object being sought in [PITH_FULL_IMAGE:figures/full_fig_p020_17.png]
Figure 18
Figure 18. Figure 18: Conjectured number of nonisomorphic misere quo￾tients at small orders. Evidently misere quotients are far from general commutative semigroups—by comparison, the number of nonisomorphic commutative semigroups at orders 4, 6, and 8 are already 58, 2143, and 221805, resp…
Figure 19
Figure 19. Figure 19: The twenty-one wild four-digit quaternary games (with first wild genus value & corresponding heap size) The author offered a bounty of $25 dollars/game to the first person to exhibit the misere indistinguishability quotient and pretending function of the games in the …
Figure 20
Figure 20. Figure 20: Correspondence between normal play nim positions and tame genera. The value of the genus theory lies in the following theorem (cf [ONAG], Theorem 73) [PITH_FULL_IMAGE:figures/full_fig_p023_20.png]
Figure 21
Figure 21. Figure 21: Normal play nim values of 0.123 (2) Two counters may be removed from a heap, but only if it has more than two counters; and (3) One counter may be removed only if it is the only counter in that heap. 11.3.1. Normal play of 0.123. The nim sequence of 0.1239 is periodic…
Figure 22
Figure 22. Figure 22: G*-values of 0.123 There are some tame genus symbols in [PITH_FULL_IMAGE:figures/full_fig_p025_22.png]
Figure 23
Figure 23. Figure 23: Some genus values of games hi + hj in 0.123. 11.3.4. Multiple heaps. We cannot immediately determine the outcome class of multiple-heap 0.123 positions using [PITH_FULL_IMAGE:figures/full_fig_p026_23.png]

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