16 problems

Consider Arc-Kayles played on subdivided stars with three paths, one of which has size $1$. Fix the length of the second path and vary the length of the third path, obtaining a seq…

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…

Spectrum conjecture. The spectrum of nim-numbers for both building games is the set of nonnegative integers.

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…

Let the single-quadrant queens-in-exile problem be viewed as an impartial game, with each board cell assigned its Sprague–Grundy value, and arrange these values in an array indexed…

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 $A=(a_1,a_2,\ldots,a_k)$ with $k\geq 3$ and suppose $a_k>a_{k+1}+1$. If there exists $i\leq k-2$ such that $a_{i+1}-a_i\neq a_{i+2}-a_{i+1}$, then the Sprague–Grundy sequence ……

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…