I just read this question link and asked myself, if there is any easy way to decide how many charts you actually need to cover a given compact manifold in $\mathbb{R}^3$, maybe at least in this special situation there is an easier answer to the question. I mean, the answer accepted in the cited question seems to use advanced techniques.

What I would be interested in is to understand why some manifolds like the torus or the real projective space cannot have $2$ charts. I mean, $1$ chart is of course wrong by compactness and the same argument tells us that if we have two charts, then the intersection of the open sets on the manifold cannot be empty, but what is the problem with two charts on these manifolds. What is it that distinguishes them from the sphere let's say? Maybe it is even possible to extend this argument further for general manifolds in $\mathbb{R}^3$?