The thread "Hard" is about a problem which, I think, demands a probabilistic solution but for which I can only find a sort of geometric one. This problem is the reverse: it's stated in geometric terms, and the only nice solution I can think of uses probability theory.
Consider an n-dimensional cube of volume 1. The intersection of this hypsercube and an (n-1)-dimensional hyperplane is an (n-1)-dimensional "polygon". What is the minimum possible volume of this polygon, and for what hyperplane is it attained?
(For example, consider the area of the polygon formed by the intersection of a plane and a cube, and generalise to more dimensions.)