Outcome DFAs for all octal games

Determine whether every impartial or partizan octal game admits an outcome deterministic finite automaton.

Background

The paper develops an automatic-solving framework for partizan octal games and successfully verifies outcome DFAs for most of the two-digit games examined computationally. However, 258 of the 2,556 tested games overflowed the state or memory limits, and the authors formulate the universal existence of outcome DFAs as a partizan analogue of Guy’s conjecture. They note that, in the impartial case, this would establish eventual periodicity of outcomes but would not imply periodicity of Sprague–Grundy values.

References

Do all impartial or partizan octal games admit an outcome DFA?

— A Finite Automaton Approach to Combinatorial Games  (2608.13273 - Liang, 13 Aug 2026) in Section 3, subsection “Other Results”

Besides the games mentioned in this paper, what other games can be solved using finite-automaton methods?

— A Finite Automaton Approach to Combinatorial Games  (2608.13273 - Liang, 13 Aug 2026) in Section 5, subsection “More Games”

Can this method be generalized to determine the outcomes or SG values of hexadecimal games exhibiting arithmetic periodicity (or other special periodicities)?

— A Finite Automaton Approach to Combinatorial Games  (2608.13273 - Liang, 13 Aug 2026) in Section 5, subsection “More Games”

Can our approach be generalized to stronger automata?

— A Finite Automaton Approach to Combinatorial Games  (2608.13273 - Liang, 13 Aug 2026) in Section 5, subsection “Stronger Automata”

For a given class of automata (especially FAs), is there a rigorous method to prove that the outcomes of a given game cannot be determined by automata in that class?

— A Finite Automaton Approach to Combinatorial Games  (2608.13273 - Liang, 13 Aug 2026) in Section 5, subsection “Stronger Automata”

Can the algorithm be improved so that it can automatically find and solve weak solutions in the above situation?

— A Finite Automaton Approach to Combinatorial Games  (2608.13273 - Liang, 13 Aug 2026) in Section 5, subsection “Weak Solutions”

Does there exist a two-dimensional generalization of our approach?

— A Finite Automaton Approach to Combinatorial Games  (2608.13273 - Liang, 13 Aug 2026) in Section 5, subsection “Two-Dimensional Generalizations”