From Buffon's Needle to Buffon's Noodle
Drop a needle of length $L$ onto a hardwood floor with floorboards of width $W$. On average, the needle crosses $2L / \pi W$ lines between floorboards, a classic result of Buffon. But that $\pi$ in the formula means there’s a circle hiding somewhere. The trick to finding it? Bend the needle into a noodle. A single noodle, before being dropped Floor with ruled lines, with one noodle highlighted Drops K 0 Avg crossings (observed) — 2L / πW — Total turn Θ° 75° Segments N 4 Length L/W 2.0 Drops K 200 Reset Needle (Θ = 0°) π-circle From needle to noodle The usual approach to Buffon’s problem involves a double integral. Respectable, but this hides the circle at the heart of the solution, and frankly, I don’t love doing integrals. Instead, we’ll derive the result1 by going from a straight needle, to a curvy noodle, to a circle. All we need is some basic geometric reasoning and probability. ...