# Brouwer and the Mountain

A few years ago, one of my friends told me the following riddle:

A mountain climber starts up a mountain at 8am. They get to the top that day, and camp there. In the morning, they start hiking down the mountain at 8am on the same trail.

Is there a time of day at which they're at the same spot on the trail the second day as they were on the first?

I thought about this a while before finally asking for the answer (which I won't repeat here). I will say that you don't have to make any assumptions about hiking speed, rest breaks, or even that the hiker always heads in the same direction.

When I learned about Brouwer's fixed point theorem, I immediately thought back to this riddle. The answer to the riddle is a straightforward application of Brouwer's theorem.

It turns out that Brouwer's theorem is used in all sorts of places. It was one of the foundations that John Nash used to prove the existence of Nash equilibria in normal form games (for which he won the Nobel).

The moral of the story is: the more riddles you solve, the more likely you are to get a Nobel prize.