I just ran across this, offered by Morton C. Schwartz in an old issue of Pi Mu Episilon Journal:
Take any number of zeros and any number of ones and place them in a circle, in any order. Reproduce the circle a second time, concentrically with the first. Rotate either circle, and any number of places. The number of zeros opposite ones will always be even.
(Morton C. Schwartz, “An Amazing Parity Theorem,” Pi Mu Episilon Journal 5:7 [Fall 1972], 338.)