  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 $N\neq 3, 4, 6$. Some generalizations are also considered.
