14 problems

Let $u_\mathbb{P}(q;n)$ be the counting quasipolynomial for nonattacking configurations of $q$ copies of a two-move rider $\mathbb{P}$ on the dilated unit board, and let…

Let $\mathbb P$ be a piece, and let a twisted Fibonacci spiral be the vertex configuration defined by four selected moves of $\mathbb P$ in the Fibonacci-spiral pattern, together w…

Let $F_q$ be the $q$th Fibonacci number and let $[F_q]=\{1,2,\ldots,F_q\}$. For the inside-out polytope associated with $q$ queens, let $D$ denote its denominator. Queen denominato…

Let a piece have exactly three moves, and consider the vertex configurations arising from its golden parallelogram constructions. Three-move golden-parallelogram conjecture. For a…

Let $F_q$ be the $q$th Fibonacci number, and consider vertex configurations of $q$ queens in the square-board inside-out polytope. Queen maximal-denominator conjecture. The largest…

Let $F_k$ be the Fibonacci numbers, and let $\mathbb Q^{21}$ denote the semiqueen. For $q$ semiqueens, consider all vertices of the associated inside-out polytope. Semiqueen maxima…

Let $F_q$ denote the $q$th Fibonacci number, and consider vertex configurations of the inside-out polytope for $q$ queens. Queen maximal-denominator conjecture. The largest denomin…

Let $\mathbf z$ be a vertex of an inside-out polytope for $q$ two-move riders, with components $z_i\in\mathbb R^2$. A simple trajectory is a primitive trajectory with no equal cons…

Let $\mathscr{A}_{\mathbb P}$ be the inside-out arrangement associated with the piece $\mathbb P$, and let $D([0,1]^{2q},\mathscr{A}_{\mathbb P})$ denote the denominator of the cor…

For a piece $\mathbb P$ with move set $\mathbf{M}$, write … For a positive integer $\lambda$, let $\mathbb P^{\operatorname{lcm}}_\lambda$ be the lcm-maximal piece whose move set c…

Let $\mathbf{M}$ be the move set of a piece $\mathbb P$, with basic moves $(c_r,d_r)$, and set … Let $\gamma_4$ denote the fourth coefficient of the quasipolynomial…

Let $\mathbf{M}$ be the move set of a piece $\mathbb P$, with basic moves $(c_r,d_r)$, and write … Let $\gamma_3$ denote the third coefficient of the quasipolynomial…

Maximal-move-range piece conjecture. Among all pieces with $\|\mathbb P\|\leq\lambda$, $\mathbb P^{\max}_\lambda$ maximizes the period of every coefficient $\gamma_i$, for fixed…

Period and variation conjectures. The following assertions are proposed: