The Tarski Plank Problem:

> *Unit disc can be covered by $n$-rectangles $1\times n^{-1}$. Prove that it cannot be covered by a smaller number of such rectangles.*

*Solutions.* Take the unit sphere of the same center as the disc. Let $\pi$ be the orthogonal projection on the plane containing the disc. Taking $\pi^{-1}$ of a strip (assuming not too much of it is "outside") we get a "ring" on the sphere. The point is that the area of that ring depends only on its width which is $n^{-1}$ and not on its  position. This is the [Archimedes Hat-Box Theorem][1]. Since we cover sphere by sets of equal areas, they cannot overlap and the problem follows. $\Box$

Of course, this solution is too short, only an idea.

  [1]: https://mathworld.wolfram.com/ArchimedesHat-BoxTheorem.html