129 problems

Let $a_s(r,n)$ denote the number of two-toned tilings of a $1\times(n+r+s)$ grid with $r$ red squares, where the last $s$ tiles must be white. For all $s,r,n\geq 1$, the conjecture…

Let $T_{n\times m}(s,k)$ denote the number of ways to tile an $n\times m$ rectangle using $s\times s$ squares and $1\times1$ squares, with exactly $k$ squares of size $s\times s$.…

Let $DPL(n,d)$ denote a diameter $d$ perfect Lee code in $\mathbb{Z}^{n}$. For odd $d$, this agrees with a perfect Lee code of radius $(d-1)/2$; for even $d$, it is defined using a…

Let $I^{d}$ denote the $d$-dimensional unit cube, and let $L\subset\mathbb{R}^{d}$. The cube spectral-tiling conjecture. $(I^{d},L)$ is a spectral pair if and only if $(I^{d},L)$ i…

Let $L\subset\mathbb{R}^{d}$. A spectrum is a set $L$ for which there exists a set $\Omega$ such that $(\Omega,L)$ is a spectral pair, and a tiling set is a set $L$ for which there…

For the explicitly defined $q$-weighted Kasteleyn and Kasteleyn-Percus matrices $M(a,b,c;q)$, $M_\rho(a,a,a;q)$, $A_\tau(a,b,b;q)$, $\widetilde{A}_\tau(a,b,b;q)$, and…

Let $a$, $b$, $c$, $x$ and $y$ be arbitrary integers. Consider a hexagon with side lengths $2n+a$, $2n+b$, $2n+c$, $2n+a$, $2n+b$, and $2n+c$, and a horizontal rhombus whose bottom…

Let $N_T(m,n)$ denote the number of tromino tilings of an $m\times n$ rectangle. Consider an extended domino-deficient rectangle with minimal dimensions $m\times n'$, where $n'>n$.…

Middle-regime doubling conjecture. For every $m\ge4$ and every $n$ in this middle regime,

Periodic orbit-closure conjecture. The orbit closure

Nivat's conjecture. Then $c$ is $1$-periodic.

Erdős–Ko–Rado conjecture for tilings. For sufficiently large $n$, every intersecting family $\mathcal F\subseteq\mathcal T_n^\Lambda$ satisfies

Stanley's conjecture. The polynomial $Q_k(x)$ has distinct roots. This conjecture concerns the simplicity of the poles of the dimer-covering generating functions. It is refuted: th…

Let $k$ be a positive integer and write $S(k)=\{1,2,\dots,k\}$. Let $G$ be a finite abelian group. Say that $S(k)$ splits $G$ if there is a subset $T\subseteq G$ such that the tran…

Domokos–Horváth–Goriely–Regős conjecture. Every polyhedral tiling can be completely softened.

Let $\mathcal T$ be a locally finite tiling with a finite number of polygonal prototiles. Suppose … is a partition into translations of finitely many, not necessarily distinct, fin…

Higher-dimensional parallelogram decomposition conjecture. Every surface in

Let $A,B\subseteq\mathbb{F}_p^3$ be subsets with $0\in A$ and $0\in B$. The notation $A+B$ denotes the sumset $\{a+b:a\in A,\ b\in B\}$, and $\langle A\rangle$ denotes the linear s…

Commensurable-sides conjecture. The tiling must have commensurable sides.

Let $ABCD$ be an ideal trapezoid, and let $(a,b,c)$ be a triple of pairwise coprime integers satisfying … Write $x$ and $y$ for the side parameters of the ideal trapezoid as in the…

Let $(a,b,c)$ be the side lengths of the triangular tile, with angles determined by $a,b,c$, and let $T$ be a triangle with angles $(2\alpha,2\beta,\alpha+\beta)$. Tile-count divis…

Let $(a,b,c)$ be the side lengths of the triangular tile, and say that a positive real number $X$ is equiconstructible by $(a,b,c)$ when an equilateral triangle with side length…

Let $M$ be either of the census manifolds texttt{m125} or texttt{m129}, and let $D$ be a Dirichlet domain for $M$. A finite vertex of valence four is a finite vertex of $D$ inciden…

Robins' conjecture. If

Periodic coloring conjecture. For any $V$, $N$, $\Sigma$, and $\Omega$ as above, the solution set $\operatorname{Coloring}(V,N,\Sigma,\Omega)$ is not aperiodic.