Is the misère quotient of Dawson's Kayles ($\cdot07$) infinite at heap size 34? (and exhibit $D_{34}$ if so)
Statement
Under the misère convention (the last player to move LOSES), the misère quotient of an impartial game is the commutative monoid $\mathcal{Q}=\mathcal{A}/{\equiv}$ of positions modulo misère-indistinguishability. Dawson's Kayles is the octal game $\cdot07$; its misère quotient computed through heap size 33 is finite, of order 638. Is the misère quotient infinite at heap size 34? If it is infinite then, by Rédei's theorem, it is nonetheless a finitely presented commutative monoid $D_{34}$: exhibit an explicit monoid presentation of $D_{34}$ (and then $D_{35},D_{36},\dots$) and explain the mechanism. Related deliverables: given a set of games $A$, give an algorithm deciding whether the misère quotient of $A$ is infinite; and (harder) an algorithm to compute a presentation when it is infinite.
Acceptance. FULLY RESOLVES (the headline): a rigorous determination of whether the misère quotient of $\cdot07$ is finite or infinite at heap 34, with a certificate -- if finite, its multiplication table (re-derivable from the known positions); if infinite, an explicit Rédei presentation of $D_{34}$ verified by re-deriving the known heap-34 positions. ADVANCES (independent, increasing hardness): a verified presentation of $D_{34}$ (then $D_{35},D_{36},\dots$) with a structural explanation; an algorithm (with proof) deciding whether an arbitrary game set's misère quotient is infinite; an algorithm computing a presentation in the infinite case. A claimed presentation is machine-checkable by re-deriving the tabulated positions; a numeric guess of '$\infty$' without a presentation or proof does NOT qualify.
Background
Problem A15 (old number 13), 'Misère quaternary and octal games' (Plambeck-Siegel question 1), in R. J. Nowakowski, 'Unsolved problems in combinatorial games' (Games of No Chance 5, MSRI Publ. 70, 2017, pp. 136-137), states verbatim: 'The misère quotient of .07 (Dawson's Kayles) has order 638 at heap size 33. Is it infinite at heap size 34?' along with the $D_{34}$-presentation and the two algorithmic riders. Dawson's original (misère) problem dates to 1935 and is called 'perhaps the oldest open problem in CGT.' The machinery is T. Plambeck and A. Siegel, 'Misère quotients for impartial games' (J. Combin. Theory Ser. A 115 (2008) 593-622 = arXiv:math/0609825); data and the MisereSolver software are at miseregames.org. Frontier: the partial misère quotients $Q_h(\cdot07)$ are computed and finite through $h=33$ ($|Q_{33}|=638$, P-portion 109); at $h=34$ the quotient is recorded as '$\infty(?)$' -- believed to blow up but neither rigorously established nor accompanied by a Rédei presentation $D_{34}$; the two algorithmic questions (deciding infiniteness; presenting the infinite quotient) are wide open. The full misère solution of ordinary Kayles ($\cdot77$) is known. Plambeck offers a USD 500 prize for a complete misère analysis of Dawson's Chess ($\cdot137$, alias Dawson's Kayles $\cdot07$). Vetted open as of 2026-07-06 -- with an honest recency caveat: the strongest on-point 'still open' anchors are 2008 (Plambeck-Siegel), 2013 (Siegel's book), and 2017 (GONC5); misère-quotient research has gone quiet since, and no post-2008 paper computes $D_{34}$, so a stealth resolution is unlikely but the affirmative signal is 2008-2017, not 2024+.
References
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