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!

Let $p=6k-1$ be a prime.
Theorem: Modulo $p$, if $a^3=b^3$ then $a=b$.
Proof. Suppose $x\ne 0$. Then, by Fermat's little theorem, $x^{p-1}=1$, so $x=(x^{p-1})^2x=x^{2p-1}=x^{12k-3}=(x^{4k-1})^3$ is a cubic residue mod $p$. 0 is a cubic residue, too, so every residue is a cubic residue. By the pigeonhole principle, every residue is a cubic residue exactly once. Thus $a^3=b^3$ implies $a=b$.
Here the $x^3$ mod $p$ are the pigeons and the $x$ mod $p$ are the holes.
