Found this on the net today and lost.
On a table you have a square made of 4 coins at the corner at distance 1. So, the square is of size 1×1. In a valid move, you can choose any two coin let’s call them mirror and jumper. Now, you move the jumper in a new position which is its mirror image with respect to mirror. That is, imagine that mirror is a centre of a circle and the jumper is on the periphery. You move the jumper to a diagonally opposite point on that circle. With any number of valid moves, can you form a square of size 2×2? If yes, how? If no, why not?
I dont think its possible since if it is, then the reverse must be true i.e. say 1/2 X 1/2 scale wise, that leaves the coins in the initial position forever on the vertices.But this is well, the anti proof, how to prove mathematically that 2X2 is not a possibility.