Four frogs are located at the corners of a square on the plane. At each step a frog jumps over another frog and lands in the symmetrically opposite location.
Question: Can the frogs find a way to jump over each other (in any order, even the same frog several times) such that their final places will form a square of larger side (not necessarily parallel to the original square)?
Well, I think this is not possible, I'm not sure yet how to prove it, but just jumping over each other it's simple to show that they can form same size square as they started (just in different place), other than that, I have no clue how to show that it's not possible.