8 problems

Equal-period conjecture. If an octal game is misere periodic, then its normal play nim sequence is periodic, and the normal and misere periods are equal.

Let the single-theater game have escalation set $S=\{1,\dots,m\}$. Its misère quotient is the commutative monoid obtained by identifying positions that are interchangeable in every…

Let $G$ and $H$ be games in a misère universe, and let $G+H=0$ denote that their disjunctive sum is equivalent to zero in that universe. The Conjugate Conjecture. If … then $H$ is…

A universe is a class of combinatorial games closed under the relevant operations of misère play, and a universe is closed under conjugation when $G$ belonging to the universe impl…

Conjugation conjecture. Being closed under conjugation should prevent non-conjugal invertibility: if $G+H\equiv 0\pmod{\mathcal{U}}$, then $H\equiv\overline{G}\pmod{\mathcal{U}}$.…

Milley's conjecture. If $G+H\equiv 0$ modulo $\mathcal{U}$ and $\mathcal{U}$ is closed under conjugation, then

Replacement conjecture. One can replace $*$ by $\xi$ in any position having $*$ as an option without changing the resulting misère monoid. This would provide a substitution princip…

Let $\alpha$, $\beta$, and $\gamma$ be impartial positions. An arrow $\delta\to\varepsilon$ exists when Left moving second can win $\varepsilon+\delta^{\circ}$, or under a similar…