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 $G=(G,+)$ be a finitely generated discrete abelian group. For subsets $A,F\subseteq G$, write $A\oplus F=G$ when every element of $G$ has a unique representation $a+f$ with…

Keller's conjecture. For all integers $d\geq 1$, there does not exist a faceshare-free tiling of $\mathbb{R}^d$.

Let $\Omega\subset\mathbb R^d$ be a set of finite, positive measure. A translational tiling of $\mathbb R^d$ is a collection of translates of $\Omega$ covering almost every point e…

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…

Higher-dimensional parallelogram decomposition conjecture. Every surface in

Let $P$ be a $d$-dimensional parallelohedron, meaning a convex polytope that tiles $\mathbb R^d$ by translations. A Voronoi polytope of a $d$-dimensional lattice $\Lambda$ is the s…

Let $\mathcal A_q^n$ be the ambient Lee-metric space, and let a perfect $(n,M,d)_q$-Lee code be a Lee code with $M$ codewords whose packing and covering radii coincide. Golomb–Welc…

Periodic orbit-closure conjecture. The orbit closure

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

Łaba–Wang conjecture. If $\mu_{\rho,\mathcal{D},{\bf p}}$ is a spectral measure, then ${\bf p}$ is equal-weight, so $p_j=1/m$; $\rho=1/N$ for some integer $N$; and there exists…

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…

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…

Periodic tiling conjecture. The tiling equation $F\oplus X=G$ is not aperiodic; equivalently, whenever $F$ tiles $G$ by translations, it admits at least one periodic tiling.

Let $\mathcal{M}$ be a normal, convex mosaic in $\mathbb R^3$ whose cells have unit volume, and let $\underline{\rho}_1(\mathcal{M})$ denote its lower edge density. Lower edge-dens…

Let $\|\cdot\|$ be a norm on $\mathbb{R}^n$ whose unit ball tiles $\mathbb{R}^n$ by translation. Let $m_1(\mathbb{R}^n,\|\cdot\|)$ denote the supremum of the upper densities of mea…

A tile is a finite non-empty subset of $\mathbb{Z}$, and a tile $T$ tiles $\mathbb{Z}^d$ if $\mathbb{Z}^d$ can be partitioned into translates, rotations, and reflections of $T$. Gr…

Let $V$ be a finite tile in $\mathbb{Z}^n$. A tiling by translations is fully periodic if its translation set has a finite-index period subgroup. Lagarias--Wang conjecture. If $V$…

Lonc's conjecture. If $n$ is sufficiently large and $|P|$ divides $2^n-2$, then

Coincidence Rank Conjecture. The coincidence rank of the tiling must divide the algebraic norm of $\lambda$.

SRS weak-tiling conjecture. The collection

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…

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…