Buffon's Needle

Georges-Louis Leclerc, Comte de Buffon, 1777

A floor is ruled with parallel lines a distance d apart. You drop a needle of length ℓ at random. What is the probability it lands touching a line?

Turn that around and you have a way to measure π by throwing things on the floor:

This is the first Monte Carlo method ever invented, roughly 170 years before the name existed. No circle is drawn, measured or mentioned. π appears purely because the needle's angle is uniformly random over a half-turn — and once you integrate over all angles, you have integrated around a circle whether you meant to or not.

Where does the π actually come from?

A needle at angle θ covers a horizontal span of ℓ·|sin θ|. Averaged over every angle from 0 to π, the mean span is 2ℓ/π — and that average is exactly the probability of straddling a line spaced d apart. The π is smuggled in by the average of a sine, which is where circles hide.

Things to try

• Set needle length equal to line spacing. The hit rate should sit at 2/π ≈ 63.66%.
• Make the needle longer than the gap. The simple formula stops working — a long needle can cross two lines at once. The page tells you when you've left the safe zone.
• Throw 10,000 and watch how slowly the third decimal place settles down. Getting π right to six digits this way would take roughly a trillion needles. Buffon's method is beautiful and completely useless for actually computing π.

In 1901 an Italian mathematician, Mario Lazzarini, claimed 3,408 tosses gave π = 3.1415929 — accurate to seven digits. Statisticians have been politely skeptical ever since; 355/113 is suspiciously exactly the answer he got.