Let $T\neq \emptyset$ be a finite subset of $\mathbb{Z}\times\mathbb{Z}$. We say that $\mathbb{Z}^2 = \mathbb{Z}\times\mathbb{Z}$ is *parquettable* by $T$ if there is a partition $\frak P$ of $\mathbb{Z}^2$ such that for every $P\in {\frak P}$ there is a bijective linear map $\ell_P: \mathbb{Z}^2 \to \mathbb{Z}^2$ and an element $c_P\in \mathbb{Z}^2$ such that for the affine map $f_P:\mathbb{Z}^2\to\mathbb{Z}^2$ defined by $x\mapsto \ell_P(x) + c_P$ we have $$f_P(T) = P.$$

Given $T\subseteq \mathbb{Z}^2$, is it decidable whether $\mathbb{Z}^2$ is parquettable by $T$?