Suppose we are given two sets, $S$ and $A$ in the plane, such that $S$ is finite, with a special point, $s_0$, while neither $A$ nor its complement is a null-set, i.e., the outer Lebesgue measure of $A$ and $A^c=\mathbb R^2\setminus A$ are both non-zero. Can we find two congruent copies of $S$, $f_1(S)$ and $f_2(S)$, such that $f_1^{-1}(f_1(S)\cap A)\Delta f_2^{-1}(f_2(S)\cap A)=\{s_0\}$, i.e., $s_0$ is the only element of $S$ that goes in to/out of $A$ when we go from $S_1$ to $S_2$?
My motivation is to extend a Falconer-Croft proof that works for measurable sets, see the details here. It is easy to see that this can be done when the elements of $S$ are the three vertices of an equilateral triangle; in fact, the same trick works whenever $S\setminus\{s_0\}$ falls on a line, which doesn't contain $s_0$.
The problem seems quite similar to the finite Steinhaus problem/Jackson sets, as Laczkovich has pointed out.
