Where Have All the Grasshoppers Gone?

Let P be an N-element point set in the plane. Consider N (pointlike) grasshoppers sitting at different points of P. In a "legal" move, any one of them can jump over another, and land on its other side at exactly the same distance. After a finite number of legal moves, can the grasshoppers end up at a point set, similar to, but larger than P? We present a linear algebraic approach to answer this question. In particular, we solve a problem of Brunck by showing that the answer is yes if P is the vertex set of a regular N-gon and N not equal 3,4,6. Some generalizations are also considered.