40 years or so ago, a kid named Jeremy showed me a proof of the two-dimensional Brouwer fixed-point theorem, which used what I have since come to know is called "Sperner's lemma." The two-dimensional proof he showed me was actually stronger than Sperner's lemma. It used a Stokes' Theorem-type argument to show that the complete triangles can be assigned a value +1 or -1 depending on their orientation (1-2-3 vs 3-2-1, viewed from some consistent orientation), and that the total of these values must be 1. Sperner's lemma follows by taking this mod 2.
I would love to have a published reference for this. I'm sure it holds in higher dimensions if you're willing to go to the painful trouble of orienting your simplices.
And Jeremy, if you're out there, I'd like to thank you.