TITLE: Applications of Brouwer's fixed point theorem
QUESTION [30 upvotes]: I'm presenting Brouwer's fixed point theorem to an audience that knows some point-set topology. Does anyone have any zippy / enlightening / cool applications or consequences of it? So far, I have:

Physical realizations: stuff involving maps of my city, crumpled pieces of graph paper, a stationary gin molecule in a cocktail shaker, etc.
That every n*n real matrix with all-positive entries has a positive eigenvalue.

Thanks! :)

REPLY [3 votes]: The proof of the fundamental theorem of algebra using the Brouwer fixed point theorem is given here:
 B.H. Arnold, "A topological proof of the fundamental theorem of algebra" Amer. Math. Monthly , 56 (1949) pp. 465–466