4 problems

Consider the divide-and-residue game, and let ${\mathcal{SG}}(n)$ denote the nim-value of a heap of size $n$. Divide-and-residue conjecture. Every nim-value occurs for at least one…

Consider the divide-and-residue game, and let ${\mathcal{SG}}(n)$ be the nim-value of a heap of size $n$. Finite-occurrence conjecture. Every fixed nim-value occurs for only finite…

For the Euclidean-division ruleset, let ${\mathcal{SG}}(n)$ denote the nim-value of a heap of size $n$. Vanishing relative nim-values conjecture. The relative nim-values satisfy ……

Gale–Neyman conjecture.