From Buffon's Needle to Buffon's Noodle

3 days ago

Buffon's needle problem shows that a needle of length L dropped onto a floor with boards of width W crosses lines an average of 2L / πW times, revealing a hidden circle.
By linearity of expectation, the expected number of crossings, f(L), is linear: f(L) = cL for some constant c, deduced from welding needles end-to-end without requiring independence.
Bending the needle into a noodle (any curve or polygonal line) extends the result: the average intersections depend only on the curve's length, not its shape.
Using a circle of radius W/2 with circumference πW, which always crosses a line twice (except tangency with probability zero), solves for c: c = 2 / (πW).

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