8 problems

Guy's conjecture. Every finite octal game has an eventually periodic Grundy sequence.

An octal game is a game played with tokens divided into heaps, where a move either removes some or all tokens in one heap, or removes some but not all tokens from a heap and divide…

Caterpillar Grundy-value conjecture. For all $n\geq 4$, there exists a caterpillar $C$ such that

Unbounded-tree Grundy-value conjecture. For all $n\geq 4$, there exists a tree $T$ such that

Sequential-join period conjecture. If $\mathcal{G}_s(n+_{\ell}m)$ eventually has period $p$, then $\mathcal{G}_s(n\rhd m)$ also eventually has period $p$.

Selective-sum period conjecture. If $\mathcal{G}_s(n+_{\ell}m)$ eventually has period $p$, then $\mathcal{G}_s(n\triangledown m)$ also eventually has period $p$.

Eventual-periodicity conjecture. The scoring-play Sprague–Grundy function should be eventually periodic, and for finite octal games its period should be the same as under the disju…

Let $O=(n_1n_2\dots n_t,p_1p_2\dots p_t)$ and $P=(m_1m_2\dots m_l,q_1q_2\dots q_l)$ be two finite taking-no-breaking octal games. Assume that at least one $n_s$ is neither $0$ nor…