We start with four points on the plain forming a square. We are allowed to move the points around by repeatedly choosing two of the points and reflecting one with respect to the other. Is it possible to form a bigger square?
The answer is no. The process of reflecting one point with respect to another is reversible, and thus it is possible to make a bigger square if and only if it is possible to make a smaller one. But one clearly cannot make a smaller square, and so we are done.
As a bonus, much harder, question: given two configurations of $n$ points on the plane, when can one get from one to the other?
