A very famous problem is the following:
The Three Utilities Problem. There are three houses and three utilities: You must draw a line from each house to each utility, without the lines ever crossing. Can you connect the houses to the utilities?
The answer is no, and this can be proved in several ways. The most popular ones perhaps are using Jordan Curve Theorem or Graph Theory.
I found this problem in a list of relatively difficult exercises of Complex Analysis, and I was wondering if anyone can help or provide a reference. I suspect that, if such a solution exists, it should be using index numbers.