TITLE: How many triangulations of a regular octahedron are there, without introducing new vertices?
QUESTION [6 upvotes]: It is easy to find three triangulations, each consisting of four tetrahedra. Are there more?

REPLY [15 votes]: No, these are all. The edge graph of the octahedron has no $K_4$ subgraph, so you have to add a new edge to make a triangulation. The only possible places for a new edge are connecting opposite vertices. You can only add one such edge, as any two meet in their interior. So every triangulation of the octahedron (without new vertices) adds exactly one of the three edges $e$ between opposite vertices. The only $K_4$'s in that graph are the four tetrahedra arranged around $e$, so any traingulation using $e$ must use a subset of those tetrahedra. We need all of them to fill in the octahedron.