proof

In 1777, Georges-Louis Leclerc, Comte de Buffon posed the following question:

Suppose we have a floor made of parallel wooden boards, each the same width, and we drop a needle onto the floor. What is the probability that the needle will lie across a line between two boards?

He solved it with some fancy calculus, which is a fine way of doing it, but not particularly satisfying to me. However in 1860 a man named Joseph-Émile Barbier came up with this super slick proof, which is without a doubt the coolest proof I've ever seen.