101 problems

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…

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…

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 $p$ 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…

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…

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

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…

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 $\mathscr{A}$ be a closed set of games with the evil twin property. A pair $(\mathscr{A},\mathscr{K})$, where $\mathscr{K}\subseteq\mathscr{A}$ is a kernel, is evilly normal if…

Let $Q_n$ be the discrete $n$-dimensional hypercube, and let a $k$-player profile be a tuple of $k$ chosen vertices of $Q_n$. A profile is in equilibrium if no player can strictly…

Let $S=\{s_3,s_2,s_1\}$ with $s_3<s_2<s_1$. Let $o_{\mathrm{zs}}$ denote the zero-sum scoring-play outcome and let the antagonistic self-interest game use AvA tie-breaking. Three-a…

Let $S=\{s_3,s_2,s_1\}$ with $s_3<s_2<s_1$, and suppose the largest subtraction is additive: … Let a friendly/antagonistic discrepancy mean a difference between the FvF and AvA sel…

Let $S=\{s_3,s_2,s_1\}$ with $s_3<s_2<s_1$, and let a friendly/antagonistic discrepancy mean a difference between the FvF and AvA self-interest games. Dominant three-action inherit…

Let $S=\{s_2,s_1\}$ with $s_2<s_1<2s_2$, and suppose … for every $k\in\mathbb{N}$. Let FvF and AvA denote the symmetric friendly and antagonistic self-interest games. First-discrep…

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…