A friend of mine asked me the following question.
Whats the minimum number of rectangular planks of unit width (and infinite length) needed to cover a circular hole with diameter $n$?
Obviously, $n$ is an upper bound because the hole can be covered by putting $n$ planks in parallel next to each other. Also it is easy to see that for small $n$, in particular $n=2$ and $n=3$ one cannot make do with fewer than $n$.
Is there a proof that in the general case $n$ planks are needed?
On
Extend the circulare hole to a sphere and extend each plank vertically (both up and down). With each of the planks associate the surface area of that sphere it covers. Since the surface area of a spherical cap is proportional to its height, we conclude that each plank covers at most $\frac1n$ of the sphere surface. Since they jointly shall cover the whole sphere, we need at least $n$ planks.
This is a theorem of Bang from 1950-51, solving Tarski's Plank Problem. See Wikipedia for description and links.