25 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 $n$ 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:

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…

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

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.

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…

Let $x=(x_0,x_1,x_2)$ be a position with $x_0\in\mathbb{Z}_{\geq}$ and $x_2\geq x_1=2^k$ for some $k\in\mathbb{Z}_{\geq}$. Write $\ell(x)=x_0+(x_1\mathbin{\oplus}x_2)$ for the lowe…

Let $\operatorname{BN}(2n,\ell)$ denote the Building Nim game with parameters $2n$ and $\ell$, and let $\text{P1}$ and $\text{P2}$ be its two players. Winning-strategy conjecture.…

Let $k,l\in\mathbb{N}$ with $k<l$, and consider $(k,l)$-Maharaja Nim, obtained from Wythoff Nim by adjoining moves of the form $(k,l)$ and $(l,k)$. An upper P-position is a losing…

Let $m$ be any positive integer, let $F$ be a set of positions in $m$-heap Nim with labeled piles, and let $S\subseteq\mathbb{R}^m$ be open. For each $k$, let $\pi(S,k)$ denote the…

Let $F$ be a forbidden set defining an instance of CIS-Nim, and let $S ameqsubseteqmathbb{R}^3$ be open. For each $k$, let $pi(S,k)$ denote the number of $P$-positions of the form…