Abstract: Consider a finite set of points in the Euclidean plane. How many ways are there to equipartition it by a straight line? What is the maximum number of such partitions for any point set of a given size?

The goal of this talk is to convey some of the flavor of "discrete" or "combinatorial" geometry by discussing this (still unsolved) question, the algorithmic problems that motivate it, and connections to other fundamental notions and questions in the area.