217 problems

Guo–Miller rational strategy conjecture. Every lattice game has a rational strategy.

Let ${}_0(n)$ denote the initial state of the Bergman Game, and let $\mathcal{L}_n$ be the distribution of the length of a random game played from ${}_0(n)$, where at each state an…

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 winning strategy conjecture. For $a\geq b\geq 3$, the first player has a winning strategy.

Consider Arc-Kayles played on subdivided stars with three paths, one of which has size $1$. Fix the length of the second path and vary the length of the third path, obtaining a seq…

Duchêne–Rigo conjecture. There exists an invariant game having

Longest-game conjecture. The longest game on any $n$ is achieved by applying splitting moves whenever possible: first adding $1$'s, then splitting from smallest to largest, and fin…

Guy's conjecture. Every finite octal game has an eventually periodic Grundy sequence.

Affine stratification conjecture. Every squarefree lattice game possesses an affine stratification.

Equal-period conjecture. If an octal game is misere periodic, then its normal play nim sequence is periodic, and the normal and misere periods are equal.

Reducibility conjecture. The sequence $A$ is reducible to $11$ and therefore is not removable.

Non-removability conjecture. The sequence $11110011111011001$ is not removable.

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 $P$ denote the set of ${\cal P}$-positions in single-hop Peg Duotaire. A language is context-free if it is generated by a context-free grammar. Single-hop context-freeness conj…

A position in single-hop Peg Duotaire has a nim-value, namely the Grundy value of the associated impartial game position. Unbounded-nim-value conjecture. There are positions in sin…

Let $P$ and $N$ denote the sets of ${\cal P}$-positions and ${\cal N}$-positions, respectively, in single-hop Peg Duotaire. A language is regular if it is recognized by a finite-st…

A Domineering board of width $w$ is a rectangular board with $w$ rows and arbitrary length, and a polynomial-time strategy is a winning strategy whose moves can be computed in poly…

Let an $n\times n$ torus be a square Domineering board of side length $n$, with Hepzibah and Vera as the two players; ${\rm 1st}$ and ${\rm 2nd}$ denote the first and second player…

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…

In duotaire, an impartial two-player peg-solitaire game, players take turns jumping pegs, and the winner is determined by normal play. The source states that the complexity of this…