For example, if $ABC = (0,1), (2, 3), (4, 7)$ and $PQR = (-1, -1), (1, 1), (3, 5)$, then the only way $ABC$ can be superimposed onto $PQR$ is by a translation $1$ unit left and $2$ units down. This would map $A$ onto $P$, $B$ onto $Q$ and $C$ onto $R$.
We could also rename the points around and have a different mapping. For example, $A$ onto $Q$, $B$ onto $P$, $C$ onto $R$. But is it possible to have two different viable translation-mappings for the same triangle?
More precisely, my question is this: Given a triangle $ABC$ and another triangle $PQR$ which are translations of one another, what is the maximum number of translation-mappings? (Not a theoretical maximum, which would have to do with counting permutations, but a real achievable maximum).