  Consider two regions in the plane, bounded by an $n$-gon and an $m$-gon,
respectively. At most how many connected components can there be in their
intersection? This question was asked by Croft. We answer this asymptotically,
proving the bounds $$\left\lfloor \frac{m}{2}\right\rfloor \cdot \left\lfloor
\frac{n}{2}\right\rfloor\le f(n,m)\le \left\lfloor \frac{m}{2}\right\rfloor
\cdot \frac{n}{2} + \frac{m}{2} $$ where $f(n,m)$ denotes the maximal number of
components and $m\le n$. Furthermore, we give an exact answer to the related
question of finding the maximal number of components if the $m$-gon is required
to be convex: $\left \lfloor \frac{m+n-2}{2}\right\rfloor$ if $n\ge m+2$ and
$n-2$ otherwise.
