Drop a Needle, Find Pi

Take a needle. Any needle. Drop it on a floor with parallel lines spaced apart by the needle's length.

Do it again. And again. A few hundred times.

Count how many needles cross a line. Divide the total drops by the crossings.

You get pi.

Not approximately pi. Not "close to" pi. The ratio converges on pi as you keep dropping. 2/p, specifically. Which means if you drop 1,000 needles and 637 cross a line, you get 1000/637 = 1.57. Multiply by 2. That's 3.14.

This is Buffon's Needle. Georges-Louis Leclerc, Comte de Buffon, 1777. Almost 250 years ago. Before electricity. Before statistics was even a proper field.

He proved it with pure geometry. The needle can land at any angle. It can land at any position between the lines. Work out the probability that those two random variables produce a crossing, and pi falls out of the integral. No circles involved. No diameters. Just a needle and some lines.

Pi is supposed to be about circles. It's in the definition. Ratio of circumference to diameter. But here it shows up in a problem about straight lines and random drops.

The trick is the angle. The needle can land pointing any direction from 0 to 180 degrees. That range, in radians, is π. And whether the needle crosses a line depends on the sine of that angle. Integrate sine over that range, normalize, and you get 2/π as the crossing probability. Pi was hiding in the rotation the whole time. A straight needle on a flat floor, but it can spin. And spin is circular.

Buffon didn't have Monte Carlo simulations. He didn't have a computer to drop ten million virtual needles. He had the math. The experiment came later, when people wanted to see if the universe actually agreed with the proof.

It does.

Mario Lazzarini claimed he did it in 1901. 3,408 throws. Got 3.1415929. Six decimal places. Too good, honestly. Statisticians have side-eyed that result for over a century. But the method itself holds up.

You can try it today. Toothpicks on lined paper. Drop enough and you'll watch the number settle into something familiar.

Pi hiding in the gaps between parallel lines. Found by a French nobleman who just wanted to know what happens when you drop a needle.

Happy Pi Day.