12 problems

Let $k=2m+1$ with $m\geq1$, and consider an Inverse $k$-cross game with empty-board state $[a]$. Let $\mathcal{G}([a])$ denote its Sprague–Grundy value, and let $\oplus$ denote nim…

Let $[a X b]$ be a single-piece Inverse Treblecross state, with $a$ and $b$ denoting the lengths on either side of the piece, and let $\mathcal{G}$ denote its Sprague–Grundy value.…

Let $R$ be the regular set of board states, let a state have at most one piece, and let $\mathcal{G}$ denote its Sprague–Grundy value. Completeness conjecture. The regular set $R$…

Let $\mathbb{SG}_{\mathrm{King}}(\mathcal{S}_{r,r}^{k})$ denote the Sprague–Grundy value of the impartial chess game with the king move rule on the generalized staircase determined…

i-Mark conjecture. A necessary condition for $\operatorname{SG}(n)=2$ is that one of the following holds: $n=sd$, $n$ is a term of $(a_n)$, or $n$ is a term of one of the sequences…

Consider the array of Sprague–Grundy values for the single-quadrant queens-in-exile problem, and let $x$ be the generating-function variable for a fixed column. Column generating-f…

Diagonal uniqueness conjecture. For all nonnegative integers $a$ and $g$, there exists a unique integer $b$ such that

Let $a\geq 0$, $m\geq 1$, and $k\geq 1$. Then the sequence … is ultimately periodic with period $p=2m$. This conjectures periodicity of the Sprague–Grundy values for positions whos…

Let $\mathcal{G}_{\mathcal{E}}(a,b)$ denote the Sprague–Grundy value of the position $(a,b)$ in E-Wythoff. E-Wythoff row-value conjecture. For integers $a\geq 4$ and…

Let $\mathcal{G}_{\mathcal{E}}(a,b)$ denote the Sprague–Grundy value of the position $(a,b)$ in E-Wythoff. E-Wythoff diagonal conjecture. For integers $r\geq 0$ and $a\geq 2r$, … T…

Fix a nonnegative integer $a$, and let $\{\mathcal{G}_{\mathcal{R}}(a,i)\}_{i\geq 0}$ be the sequence of Sprague–Grundy values along the row with first coordinate $a$. A sequence…

Let $a$ and $b$ be integers with $4\leq a\leq b$, and let $\mathcal{G}_{\mathcal{R}}(a,b)$ denote the Sprague–Grundy value of the position $(a,b)$ in R-Wythoff. R-Wythoff bounds co…