A problem from Peter Winkler’s excellent collection Mathematical Puzzles, 2021:
Four bugs live on the four vertices of a regular tetrahedron. One day each bug decides to go for a little walk on the tetrahedron’s surface. After the walk, two of the bugs have returned to their homes, but the other two find that they have switched vertices. Prove that there was some moment when all four bugs lay on the same plane.
Call the bugs A, B, C, and D. In the initial position, when D looks at the other three bugs, she sees triangle ABC labeled clockwise, but in the final position she sees it labeled counterclockwise (or vice versa). Since the bugs move continuously, some position must arise during the walk in which either D is on the plane ABC or ABC do not determine a plane. In the latter case they’re collinear, which means they share a plane with any other point, including D.
