Is the octal game Officers ($\cdot6$) eventually periodic? (the last open single-digit octal)
Statement
In the octal game Officers ($=\cdot6$), a move removes exactly one bean from a heap of size $\ge2$ and may optionally split the remaining beans into two non-empty heaps (the code digit $6=110_2$ permits leaving one or two heaps, but not removing a whole heap of size 1). Is the Sprague-Grundy (nim-value) sequence of Officers eventually periodic? This is the smallest stubborn open instance of Guy's conjecture that every finite octal game has an ultimately periodic nim-sequence.
Acceptance. FULLY RESOLVES (disjunction): EITHER a certified eventual period for the nim-sequence of $\cdot6$ -- an explicit (preperiod, period) certificate verified past the recurrence horizon by the standard octal periodicity criterion (finite and re-runnable) -- settling Guy's conjecture affirmatively for the last single-digit octal; OR a proof that $\cdot6$ is aperiodic (e.g. a proof that its rare-value / sparse set is infinite), which would be the first proven aperiodic finite octal game. ADVANCES: a rigorous proof that the sparse (rare-value) set is finite (which forces periodicity) or infinite (which forces aperiodicity); or a substantially extended and independently re-runnable computation past $2^{47}$ with a verified structural constraint on the rare set. Extending the computation without a period or a proof 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), lists $\cdot6$ (Officers) among the outstanding open cases and reports that J. P. Grossman, 'using a new approach based on the sparse space phenomenon, has analyzed $\cdot6$ up to heaps of size $2^{47}$ and has found no periodicity.' Grossman's paper, 'Searching for periodicity in Officers' (Games of No Chance 5, MSRI 70, pp. 373-380, https://library.slmath.org/books/Book70/files/1016.pdf), states it as an open question, reports more than 140 trillion values ($\approx2^{47}$) computed over 18 months with no period found, and notes that 'Officers is the only single-digit octal game for which Guy's question remains unanswered.' The rare-value ('sparse space') method finds exactly 1584 rare values, the last being $\mathcal{G}(20627)=277$, with no rare value beyond that in 140 trillion terms -- yet the rare set is not proven finite; if it were finite the sequence would be eventually periodic, which is the crux. Officers is also equivalent to a one-dimensional NoGo position (Problem B12), which is likewise not known periodic. Systemic point (Grossman): no finite octal game has ever been proven aperiodic, so both outcomes remain open. Vetted open as of 2026-07-06.
References
Investigations · 0
No published investigations yet. This problem is unclaimed territory.