123 problems

Let $D(3,n)$ denote the dynamical system governing the relevant tripod Nim arrays, and consider its periodic orbits. Period-divisor conjecture. Every periodic orbit of $D(3,n)$ has…

Let $P_k$ and $P_m$ be paths on $k$ and $m$ vertices, respectively, and let $G=P_k\mathbin{\square}P_m$ denote their Cartesian product. For a graph $G$, write ${\mathbb D}(G)$ for…

Let $n$ be the number of stacks, let $A=k,k+1,\ldots,n$, and consider the game $\textsc{SN}{\left(n,A\right)}$. For a position ${\mathbf p}$, let … and let $o$ be the number of sta…

Classification conjecture. The P-positions in ${\rm SCN}(6,4)$ are classified into one of the following categories:

Consider the three-element subtraction Nim setup above, including the eventual period $p$ and the associated distance function $dist(w)$. The reverse-mex condition. If…

Invariance Reduction Process conjecture. Every non-trivial sub-game of $G$ that arises through the Invariance Reduction Process also admits invariant vectors.

Consider subtraction Nim with subtraction set $\{s_1,s_2,s_3\}$ and one pass. Let $\mathcal{G}(x,0)$ denote the Grundy number without a pass and $\mathcal{G}(x,1)$ the Grundy numbe…

The reverse-mex characterization. The subtraction Nim has the reverse-$mex$ property if and only if, for every $\mathcal{P}$-position $w$ with $dist(w)=2m$ for some…

Let $C_n$ be the cycle graph on vertices $v_1,\ldots,v_n$, where $n\geq 4$ and each $v_i$ is adjacent to $v_{i+1}$ with indices taken modulo $n$. Let $\operatorname{TER}(C_n)$ deno…

Let $G$ be a graph with a weight function $w\colon V(G)\to\mathbb{R}_{\geq 0}$ whose codomain is $\{0,1\}$. Let $\mathcal{C}_\text{odd}$ be the family of corona products of an odd…

Modulo-three conjecture. The sequence $1^n$ is removable if and only if

Let $A$ be a finite set, and consider the subset take-away game in which players alternately choose proper, non-empty subsets of $A$, with no chosen set containing a set chosen ear…

Let $A$ be a finite set. In subset take-away, two players alternately choose proper, non-empty subsets of $A$, with no chosen set containing a set chosen earlier; a player unable t…

Let $K_n$ be the complete graph on $n$ vertices, let $\mathrm{Ramsey}(n,n)$ be the symmetric binary Ramsey number, and let $K_{\mathrm{Ramsey}(n,n)}$ be the complete graph on that…

Let $K_k$ and $K_n$ be complete graphs, let $E^r$ and $E^g$ be precolored red and green edge sets, and let $\mathrm{Ramsey}(n,n)$ denote the classic symmetric binary Ramsey number.…

Let $K_n$ be the complete graph on $n$ vertices, let $A$ be the achievement graph, and let $E^r$ and $E^g$ be the precolored red and green edge sets. Achievement-game tractability…

Let $G$ be a graph and let $A$ be the achievement graph, with no precolored red or green edges, denoted by $(G,A,\{\},\{\})$. Unrestricted graph Ramsey-game conjecture. Graph Ramse…

Let the single-theater game have escalation set $S=\{1,\dots,m\}$. Its misère quotient is the commutative monoid obtained by identifying positions that are interchangeable in every…

Let $q$ be a prime power. In the strong $(m,n)_q$-Tic-Tac-Toe game on the affine space $4\mathbb{F}_q^m4$, let $T(n,q)$ denote the smallest dimension $m$ for which the first player…

A position in $4 \times n$ Chomp is a quadruple $(a,b,c,d)$ of non-negative integers with $a\geq b\geq c\geq d\geq 0$; a P-position is a position from which the previous player has…

Let $P$ be the set of players and let $\mathcal{K}$ be a coalition structure, with core stability meaning that no subset $S \subseteq P$ can deviate to form a new coalition and mak…

Classification conjecture. The only CircularNim games that admit invariant vectors are

Large-order reverse-strategy conjecture. For every $k\geq 3$, there exists $n_0(k)$ such that the reverse strategy is a winning strategy for Player II on $S_n$ for all…

First-player difference conjecture. For all graphs $G$,

Main horizontal-period conjecture. Under these assumptions,