TITLE: Proof of Brouwer's Fixed Point Theorem.
QUESTION [5 upvotes]: What is the simplest way to prove Brouwer's Fixed Point Theorem in three dimensions?

REPLY [6 votes]: The simplest proof (not only for $n=3$, but for every $n$) is the Analysis proof of John Milnor (i.e., no use of Algebraic Topology), and in particular, its simplified version by C.A. Rogers:

Milnor, John, Analytic proofs of the "hairy ball theorem'' and the Brouwer fixed-point theorem, Amer. Math. Monthly 85 (1978), no. 7, 521–524.
Rogers, C. A., A less strange version of Milnor's proof of Brouwer's fixed-point theorem, Amer. Math. Monthly 87 (1980), no. 7, 525–527. 

A simplified version of Rogers' proof can be found here.