Is the octal game Treblecross ($\cdot007$) eventually periodic, or are its nim-values unbounded (will $2048$ ever be reached)?
Statement
Treblecross is the impartial octal game $\cdot007$: one-dimensional tic-tac-toe on a strip of cells -- players alternately place X's (never on an occupied cell) and a player wins by completing three consecutive X's (normal play, so completing a triple is the last move). As an octal game, a component of $n$ empty cells has its moves given by the code digit $d_3=7$. Its Sprague-Grundy nim-sequence $(\mathcal{G}(n))$ has been computed to very large $n$ with no period detected. Is the nim-sequence eventually periodic? Concrete sub-question (Nowakowski): the nim-values exhibit rare large spikes -- will a nim-value $\ge2048$ ever be attained, i.e. are the nim-values unbounded (which would make the sequence aperiodic)?
Acceptance. FULLY RESOLVES (disjunction): EITHER a certified eventual period -- an explicit (preperiod, period, saltus) certificate for the nim-sequence of $\cdot007$ verified past the recurrence horizon by the standard octal periodicity criterion (finite, re-runnable), settling Guy's conjecture affirmatively for this game; OR a proof that the nim-values are unbounded (equivalently that some nim-value $\ge2048$ is attained and they keep growing) or otherwise aperiodic -- the first proven aperiodic finite octal game. ADVANCES: a rigorous proof that a nim-value $\ge2048$ is attained (settling Nowakowski's concrete sub-question) without yet deciding periodicity; or a substantially extended, independently re-runnable computation together with a verified structural constraint on the rare-value set. Merely extending the computation without reaching 2048 or a period does NOT qualify.
Background
Problem A2 (old number 2), 'octal games,' in R. J. Nowakowski, 'Unsolved problems in combinatorial games' (Games of No Chance 5, MSRI Publ. 70, 2017, p. 128), calls $\cdot007$ 'perhaps the most notorious and deserving of attention,' noting Flammenkamp had pushed it to $2^{25}$ nim-values (largest $\mathcal{G}(6193903)=1401$) and asking 'Will 2048 ever be reached?' UPDATED FRONTIER: Achim Flammenkamp's live octal table (http://wwwhomes.uni-bielefeld.de/achim/octal.html, last updated 2021-05-05) now reaches heap size $n=2^{28}$ ($\approx2.68\times10^8$) with largest nim-value now $\mathcal{G}=1689$ (near index $2.49\times10^8$) -- larger than the 2017 value of 1401 but STILL below 2048 -- and its period/preperiod columns are blank (Flammenkamp's convention for unsolved). So the '2048' sub-question is still open and the frontier moved $2^{25}\to2^{28}$ with no period emerging. The 2026 preprint arXiv:2604.16759, 'A Weak Solution of Inverse Treblecross' (Liang & Li, Apr 2026), treats a DIFFERENT, avoidance-style game (completing three in a row is forbidden) and only classifies its P-positions; it makes no claim about the normal-play nim-sequence periodicity of $\cdot007$. Systemic point (Grossman): no finite octal game has ever been proven aperiodic, so both outcomes remain genuinely possible -- a period may yet be found (finitely certifiable) or $\cdot007$ may be the first proven-aperiodic octal, a landmark counterexample to Guy's conjecture that every finite octal game is ultimately periodic. Under the 'sparse space' phenomenon, if the rare large nim-values die out the sequence must become periodic; they have not died out. Vetted open as of 2026-07-06 (frontier per Flammenkamp's live table, 2021-05-05).
References
Investigations · 0
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