102 problems

Let $A$ and $B$ be poset games, with $B=2^n$ for some $n\in\mathbb{N}_0$, and suppose that $B$ is weakly-canonical. Let $A\otimes B$ be their lexicographic product, whose underlyin…

Triple-coincidence conjecture. For every integer $n\geq4$, no three distinct pieces in $\mathcal{P}'$ share the same strength on the $n\times n$ board.

Consider a position with $k\geq 3$ piles of tokens, and let … Assume that every pile has more than $m$ tokens. Parity-pattern conjecture. The position is a $\mathcal{P}$-position p…

Consider a position of Version C with $a_i$ piles of size $i$, for $1\leq i\leq n$, where … Exclude the position with $1,1,2,2,3,3,3$ as its piles, equivalently…

A rectair is the rectair family used in CRIM, and $\mathcal G$ denotes its Sprague–Grundy value. Near-square rectair conjecture. For $r\geq7$, … In addition, … All other rectairs h…

For positive integers $i,s,k$, let $\llbracket i+(k-1)s,\ldots,i+s,i\rrbracket$ denote the partition whose parts form a decreasing arithmetic progression with initial part…

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…

Let $K_{m,n}$ be the complete bipartite graph with part sizes $m$ and $n$, and let $P_{\textsc{Col},K_{m,n}}(1)$ denote the number of positions in Col (equivalently, by the biparti…

Shifted Nim-sum conjecture. The shifted nim-sum condition characterizes exactly the $\mathcal{P}$-positions for this family of chocolate-bar games. The claim was suggested by calcu…

Parity conjecture for trees. The Grundy value of a tree is zero if and only if its size is even.

A position is a triple $(x,y,z)$ of nonnegative integers, and its Grundy number is the Sprague–Grundy value of that position; write $x\oplus y\oplus z$ for the bitwise nim-sum. For…

Eventual parity conjecture. The outcome class of the position is determined solely by the parities of $a_1,a_2,\ldots,a_{n-1}$. The source reports preliminary calculations for smal…

For each integer $n\ge 3$, consider the unbiased Maker–Breaker game on the edge set of $K_n$: in each round Maker claims one previously unclaimed edge, and then Breaker claims $q$…

Given a Nim position $\mathbf{x}=(x_1,\ldots,x_n)\in\mathbb{Z}_{\ge 0}^n$ together with a pass-status $\varepsilon\in\{0,1\}$, where $\varepsilon=1$ means that one pass remains ava…

Let $\mathcal{C}$ be a cut set satisfying … and suppose that $\mathcal{C}\neq\{1,2\}$. This is Family C. Family C arithmetic-periodicity conjecture. The nim-sequence for every game…

Let $p$ be the number of spots in the starting position of a game of Sprouts, and let its nimber be the associated Sprague–Grundy value. Nimber conjecture. The nimber for the start…

Period conjecture. The Nim value function is periodic with a period equal to one of $a+b$, $b+c$, or $c+a$.

The Bachet's game with lottery moves model is governed by the assumptions preceding the main theorem, including condition … is violated. The condition is used technically in the pr…

Eventual period-two conjecture. There exists an integer $N$ such that

Let $\lambda$ be a losing partition and let $\Gamma_{r,c}$ be a losing hook. Write … when the stack operation $\lhd$ is defined. Losing-hook stack conjecture. If…

A thick hook is the thick-hook family of partitions used in CRIM, and a meld is the partition operation defined in the paper. A losing position is a $\mathcal P$-position. Thick-ho…

A partition is a Young diagram, and its Conway pair is the pair of game-theoretic invariants used in the Conway–Gurvich–Ho classification. An even-by-even hook is a hook with an ev…

A rectair is the rectair family used in CRIM, and $\mathcal G$ denotes its Sprague–Grundy value. Let $r$ and $k$ be integers with $0\leq k<r$. Square-rectair conjecture. … The conj…

A partition is a Young diagram, and its rank is the relevant rank statistic on the partition used in the CRIM analysis; a position is a $\mathcal P$-position when the next player l…

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…