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! :)