165 problems

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…

A Sprague–Grundy value is the nimber assigned to a position in an impartial combinatorial game. Welter's game is an impartial game whose positions can be represented by integer par…

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…

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

Let $S$ be a subtraction set, and call an integer an expansion of $S$ if adjoining it to $S$ does not change the nim-sequence. If $p$ is the period length, call $S$ non-expandable…

Let $S$ be a general three-move subtraction ruleset, and let its maximum entry be $m$. Althöfer–Bültermann's conjecture. The period length is bounded by a quadratic polynomial in t…

Let $S=\{a,b,c\}$ be a three-move subtraction set with $a<b<c$, and let the nim-value sequence have an eventual period length. The set is additive when $c=a+b$. Flammenkamp–Ward co…

A Sprouts game begins with $n$ spots, and an $n$-spot game denotes a game with that initial position. Sprouts conjecture. Each $n$-spot game is winning for the first player if and…

A pawn endgame on an $m\times n$ chessboard is a position in the generalized pawns game whose outcome is represented by a combinatorial-game-theoretic Nimber $*k$. The construction…

Consider the truncated-support subtraction-game positions and the options described immediately before the conjecture: the source identifies a specific collection of dominated opti…

Let $G=\mathcal{T}_\tau(n;a,b)$ with $a<b$, and let $R_{\widehat\beta}=\widehat\beta(a+\tau+1)$, where $\widehat\beta=\left\lfloor\frac{b-a}{b-a-\tau}\right\rfloor$. The atomic wei…

A partition $\lambda$ is parity self-conjugate if $\lambda_k\equiv\lambda'_k\pmod 2$ for every $k$, where $\lambda'$ is the conjugate partition. An odd self-conjugate hook is a sel…

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…

The families $S_n$ and $\mathrm{PS}_n$ are rectair families, and $SG$ denotes their Sprague–Grundy value. Rectair-family Sprague–Grundy conjecture. For $S_n$, … … For…

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…

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…

White-region non-periodicity conjecture. The Nim value function for $W_n$ is not purely periodic with any period, and each $W_n$ is split into infinitely many parts.

Ward's period conjecture. The Nim value function has a period equal to one of $a+b$, $b+c$, or $c+a$.

Fallow-cell conjecture. Fallow cells are the only cells where both players ever want to play; moreover, the unique taut positions guaranteed by Theorem are the fallow positions wit…

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…

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…

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…