four checkerboard pieces are arranged in a square of sidelength one

Question

This puzzle is from Terrence Tao's book Solving Mathematical Problems:

Suppose four checkerboard pieces are arranged in a square of sidelength one. Now suppose that you are allowed to make an unlimited amount of moves, where in each move one takes one of the checkerboard pieces and jumps over it, so that the new location of that piece is the same distance from the piece jumped over as the original location (but in the opposite direction, of course). There is no limit as how far two cheerboard pieces ca nbe in order for one to jump over the other. Is it possible to move these pieces so that they are now arranged in a square of sidelength two?

It comes with a hint: There is a particularly elegant solution to this problem, if you just think about it the right way.

I tried to find an invariant of the jumps, but to no avail. Anyone pls throw some lights?

Answer 1

HINT: Color the checkerboard in the usual way, with alternating colors. When you move a piece, it always lands on a square of the same color as the one on which it started.