The ringmaster of a flea circus puts three fleas $A$, $B$, $C$ on three different numbers on the real number line, so that flea $B$ sits exactly in the middle between $A$ and $C$.
- Whenever the ringmaster shouts "Hop!", one of the three fleas jumps over one of the other fleas to the mirror point on the other side. (In other words, a flea sitting in point $x-y$ may jump over a flea sitting in point $x$ to the new point $x+y$.)
- While the fleas are jumping around, sometimes two of them may be sitting simultaneously on the same real number. (This is fine, as these fleas are infinitesimally small.)
After some time, the ringmaster notices that the three fleas again occupy the three starting points, but they are now sitting in a different order.
Question: Is it possible that flea $A$ is now sitting on the starting point of flea $B$?