The amplituhedron is a geometric object introduced by physicists to compute scattering amplitudes, certain probabilities that describe what happens when particles with given momenta collide. I'll give a gentle introduction to the amplituhedron and to the magic number conjecture, which says that the cardinality of a tiling of the amplituhedron is the number of plane partitions which fit inside a particular box. (This is a generalization of the fact that triangulations of even-dimensional cyclic polytopes have the same size.) I will also introduce some interesting lattice polytopes called Parke-Taylor polytopes, as well as cyclic partial orders and circular extensions, which are cyclic analogues of the well-known notion of partial order and linear extension. Finally I'll explain how these polytopes and cyclic partial orders are related to the magic number conjecture.