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 $B$?
Answer
No.
I can describe their co-ordinates at the start as -1, 0, 1 on some scale where 0 is defined to be where B is sitting and the unit 1 is half the distance between A and C. There is a simple transformation from this scale to the real numbers where they were placed.
Switching their positions means that A has to reach 0 and B has to reach -1. But A will be jumping in multiples of 2. For example if A is one unit from a flea (as it is at the start) it will jump 2 units and will still be one unit from that flea. It starts on an odd number and will therefore stay on an odd number (on my transformed scale.) Similarly B starts on an even number and will stay on an even number.
By jumping past each other they can move arbitrarily up and down the numberscale but they cannot change the odd/even state of their position using my transformed scale, nor move to a point that is not an integer on my scale.
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