Euler's characteristic formula

This is one of my favorite proofs of all time, for a few reasons.

  • The culmination of relevant facts into a proof comes as quite a surprise (or at least it did for me when I first heard it).  Such surprise constitutes the heart of any "slick proof".  
  • It touches on the notion of duality, which is an incredibly far-reaching concept.  Duality is undeniably beautiful in the symmetries it produces, and generally unknown outside the higher echelons of mathematics education.  
  • It provides a nice different perspective on a very classic theorem.