What is your favorite application of the Pigeonhole Principle? The pigeonhole principle states that if $n$ items are put into $m$ "pigeonholes" with $n > m$, then at least one pigeonhole must contain more than one item.
I'd like to see your favorite application of the pigeonhole principle, to prove some surprising theorem, or some interesting/amusing result that one can show students in an undergraduate class. Graduate level applications would be fine as well, but I am mostly interested in examples that I can use in my undergrad classes.
There are some examples in the Wikipedia page for the pigeonhole principle. The hair-counting example is one that I like... let's see some other good ones!
Thanks!

In a finite semigroup $S$, every element has an idempotent power, i.e. for every $s \in S$ there exists some $k$ such that $s^k$ is idempotent, i.e. $(s^{k})^2 = s^k$. For the proof consider the sequence $s, s^2, s^3, \ldots$ which has to repeat somewhere, let $s^n = s^{n+p}$, then inductively $s^{(n + u) + vp} = s^{n + u}$ for all $u,v \in \mathbb N_0$, so in particular for $k = np$ we have $s^{2k} = s^{np + np} = s^{np} = s^k$.
This result is used in the algebraic theory of automata in computer science.
