Let $D_1,\dots,D_n \subset \mathbb{Z}^2$ be two-point sets, i.e. 'dominoes' (unlike common dominoes, these are not necessarily connected, but I couldn't come up with a better name).

Does there always exist

finite$B \subset \mathbb{Z}^2$ that can be partitioned into disjoint translates of $D_i$ for all $i \in [n]$?

For instance, if $D_1= {\tiny \square\square} , D_2 ={\tiny \square}\hspace{-6.5pt}{\ ^{^{ _\square}}},$ one can simply take $B=\boxplus.$ However, if we add just one domino $D_3 = {\tiny \square}\hspace{-1.4mm}{\ ^{^{ _\square}}},$ or two dominoes $D_3 = {\tiny \square}\hspace{-1.4mm}{\ ^{^{ _\square}}}, D_4 = \hspace{-1.4mm}{\ ^{^{ _\square}}}\hspace{-1mm}{\tiny \square},$ then the suitable shapes become much more complex: \begin{align} & \large \square \square \phantom{\square \square \square \square \square \square \square} \large \large\square \square\\[-10pt] \large \square & \large \square \square \square \phantom{\square \square \square \square \square} \square \square \square \square \\[-10pt] \large \square & \large \square \square \square \phantom{\square \square \square \square \square} \square \square \square \square \\[-10pt] \large\square \square & \large \square \square \phantom{\square \square \square \square \square} \square \square \square \square \square \square \\[-10pt] \large\square \square & \large \phantom{\square \square \square \square \square \square \square} \square \square \square \square \square \square \\[-10pt] &\large \phantom{\square \square \square \square \square \square \square \square}\square \square \square \square \\[-10pt] &\large \phantom{\square \square \square \square \square \square \square \square} \square \square \square \square \\[-10pt] &\large \phantom{\square \square \square \square \square \square \square \square \square} \square \square \\[-10pt] \end{align}

I expect the answer to be negative in general, though I do not see a way to prove non-existence of the desired $B$ for some specific sets of dominoes. Note that if we don't require $B$ to be finite, the question becomes trivial: one can always take $B = \mathbb{Z}^2$.

Assuming the negative answer, one can further ask how to decide, for a given set of dominoes, if such $B$ exists. Some related polyomino tiling problems are known to be NP-complete, and some are even undecidable. So, this question could also appear to be not so 'elementary' as it might look.

Finally, I am also interested in the natural generalization to $\mathbb{Z}^m$. Observe that the case $m=1$ is trivial. Indeed, if $D_i=\{0,d_i\}$ for all $i \in [n]$, then one can take $B=[2\cdot\text{lcm}(d_1,\dots,d_n)]$, where $\text{lcm}$ stands for the least common multiple.

Perhaps, the questions of this flavor have been already studied, but I was unable to find something related. Though this is not my field of research, so I probably missed something. Apologize in advance if this topic is well known.

with coefficients only $0$ and $1$which is divisible by all the $p_{m_i,n_i}$ and by all the $q_{r_j,s_j}$. This formalizes the problem, but I have no idea whether it helps... $\endgroup$equivalentto tiling by "xxxxxxxx", etc. Perhaps you were assuming that there would be no gaps? $\endgroup$24more comments