There are four stones, positioned on the ground at the vertices of a square. At any time, you may pick up a stone and "hop" it over another one so that it lands an equal distance beyond the hopped stone. Can you find a series of hops which will make these stones form the vertices of a larger square? If so, how, if not, why?
To clarify what a "hop" is: if there is a stone a point $p$, you are allowed to move it to a point $p'$ provided there is another stone at the midpoint of $p$ and $p'$.