"Random" facts that math can prove (eg, there are at least 2 people on the planet with the exact same number of body hairs)

Question

I just learned that you can prove mathematically that in the entire planet, there are at least 2 people with the exact same amount of body hairs.

I was fascinated by this and I searched for more seemingly "random" facts that mathematics can prove, and I found that you can prove that there are always 2 places on the planet with no wind blowing.

I continued searching but I couldn't find any more of this facts that math can prove.

Does anyone know any "random" or cool facts that mathematics can prove?

Answer 1

If you have an accurate map of a place, no matter how much you scrunch it up while in that place, there will always be a point on that map directly above the point it represents.

Answer 2

There's also the 4-colour theorem, which states that any map with contiguous countries can be coloured with 4 colours or fewer so that no 2 countries that share a border have the same colour (although two countries with the same colour can meet at a corner).