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My beautiful binaries! (With apologies to Eric Rosen)

Yesterday a friend came to me with a problem. While managing some backups, he accidentally deleted /bin, /boot, /dev, /tmp, /srv, /usr, and /opt from the filesystem itself instead of the backups.

This left an interesting problem: without the binaries from /bin and /usr, it was impossible to do almost everything you normally do. Picture the situation. You do not have a head or a tail, let alone an entire cat! No chmod, certainly nothing fancy like curl or sshd. If you listen closely though, you might hear an echo...

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.