I know this might sound like standard qualifying exam exercise in algebraic topology, so I apologize from start if this post might be inappropriate. The thing is, I've been doing some elementary combinatorial topology around Sperner's Lemma/Brouwer's fixed point theorem and came across the following result, which I don't really know how to prove in its full generality, since I'm kind of new to algebraic topology arguments... so I would really appreciate a full proof which I could understand by following Hatcher or something.

It goes like this:

Prove that it is impossible to cover the $n$-dimensional torus with $n$ simply-connected open sets. However, show one can do it with $n+1$.

The reason I'm asking this is related to Sperner's Lemma. The above has a cute consequence that no matter how you cover the $n$-dimensional cube with open sets, there will always be a path from one face to its opposite lying entirely in one of the open sets. As mentioned, I only managed to do this with Sperner, so I'm taking it as a non-trivial result...

PS. I know how to do first part for the $n=2$ case using van Kampen's theorem. I don't see how to do it for higher $n$'s...