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We construct an example of a group $G = \mathbb{Z}^2 \times G_0$ for a finiteabelian group $G_0$, a subset $E$ of $G_0$, and two finite subsets $F_1,F_2$ of$G$, such that it is undecidable in ZFC whether $\mathbb{Z}^2\times E$ can betiled by translations of $F_1,F_2$. In particular, this implies that thistiling problem is aperiodic, in the sense that (in the standard universe ofZFC) there exist translational tilings of $E$ by the tiles $F_1,F_2$, but noperiodic tilings. Previously, such aperiodic or undecidable translationaltilings were only constructed for sets of eleven or more tiles (mostly in$\mathbb{Z}^2$). A similar construction also applies for $G = \mathbb{Z}^d$ forsufficiently large $d$. If one allows the group $G_0$ to be non-abelian, avariant of the construction produces an undecidable translational tiling withonly one tile $F$.  The argument proceeds by first observing that a single tiling equation isable to encode an arbitrary system of tiling equations, which in turn canencode an arbitrary system of certain functional equations once one has two ormore tiles. In particular, one can use two tiles to encode tiling problems foran arbitrary number of tiles.

Undecidable translational tilings with only two tiles, or one nonabelian tile