30 problems

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

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…

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 $C_c(a,b)$ denote the quantity associated with the tripod Nim array parameterized by $c$, and say that the numbers from $0$ through $m$ form a band when the corresponding band…

Let $x$ be a position in the exact slow game $\operatorname{NIM}(n,n-1)$ played by $\ell$ players in a fixed cyclic order. A move reduces $n-1$ positive entries by $1$ and leaves o…

Conjecture on NP positions. NP-positions are exactly the positions satisfying $M(x)\in\{0,3\}^*$ and $\xi_3(x)>x_5$.

Characterization conjecture. The displayed inequalities characterize the $m$-critical positions of NIM$(n,k)$.

Let ${\sc Nim}^1_{n,=k}$ be the impartial game on nonnegative integer vectors in which a move decreases exactly $k$ coordinates by one. A position from which the previous player ca…

One-quantum-move Nim solvability conjecture. For every such Z\ and \phi\, Z^{\mathrm{symBudgeted}[1]\mathcal Q(\phi,2)}\ is polynomial-time solvable. The same is conjectured…

Reachable poly-wide Demi-Quantum Nim conjecture. Reachable poly-wide \ruleset{\mathrm{demi\text{-}Quantum Nim}\ remains \cclass{PSPACE}\-hard to play optimally.

Constant-wide Demi-Quantum Nim conjecture. Constant-wide \ruleset{Demi-Quantum Nim}\ is \cclass{NP}\-hard to play optimally.

Quantum Avoid True conjecture. Determining the outcome class of \ruleset{Quantum Avoid True}\ with a classical start is \cclass{PSPACE}\-complete. Consequently, determining the…

Extremal Windsor-move conjecture. There exist, not necessarily distinct, games $G_1$ and $G_2$ with $N(G_1)=N(G_2)=N$ such that

Candy Nim pile-splitting conjecture. There exist such $j$ and $a_1,\ldots,a_j$ for which

Fractal strategy conjecture. Some version of the fractal strategy is optimal for games in standard form.

Fractal strategy conjecture. The fractal strategy is optimal for games in standard form.

Parity Heuristic conjecture. PH holds for graphs with property S and an odd number of edges, for complete bipartite graphs $K_{3,q}$ for any $q$, for graphs of girth at least $5$,…

Support-bound conjecture. If $\mathcal{G}(0,a,b)=g\geq 2$, then

Non-periodicity conjecture. Rows of Table 1 are not ultimately periodic.

Bounded-row conjecture. Given $a\geq 1$, the nim-sequence $(\mathcal{G}(0,a,n))_{n\geq a}$ is bounded.

Boundedness conjecture. The rows and diagonals in Table 1 seem to be bounded but not ultimately periodic.

Let $x^*_0=1$ and let $x^*_1$ be even with $2^{k-1}<x^*_1<2^k$ for some $k\in\mathbb{Z}_{\geq}$. Define the defect…

Let $x=(x_0,x_1,x_2)$ be a position satisfying, for some $k\in\mathbb{Z}_{\geq}$, $2^{k-1}<x_1<x_0+x_1<2^k$. Let $u(x)=x_0+x_1+x_2$ be the upper bound. The eventual upper-bound con…

Let $x=(x_0,x_1,x_2)$ be a position satisfying, for some $k\in\mathbb{Z}_{\geq}$, $2^{k-1}<x_1<x_0+x_1<2^k$, and assume $x_1\leq x_2$. Let $u(x)=x_0+x_1+x_2$ be the upper bound. Th…

Let $x=(x_0,x_1,x_2)$ be a position with $x_1\leq x_2$, and let $u(x)=x_0+x_1+x_2$ denote the upper bound for its Sprague–Grundy value. The power-of-two interval conjecture. If…