The following puzzle can be solved by the same technique. A mountain range is a piecewise linear function $f$ defined on a closed interval $[a,b]$ which satisfies $f(a)=f(b)=0$, and $f(c) > 0$ for all $c \in (a,b)$. There is hiker $A$ at the point $(a,0)$ and a hiker $B$ at the point $(b,0)$. The two hikers begin moving along the mountain range, the only restriction being that they must always be at the same elevation. Prove that $A$ and $B$ can always meet at some point of the mountain range.
On a slightly more serious note, I vaguely remember that the existence of Nash equilibria can be proved by a parity argument (via Sperner's lemma).