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Buffon's constant

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Buffon's constant is . Its decimal expansion is 0.63661977236758... (see A060294) and its continued fraction is

(see A053300).

Like many other numbers involving , Buffon's constant can be expressed as an infinite sum or product:

(where is the th prime number)[1];

It is also the limit , where is the expected number of real zeros of a random polynomial of degree with real coefficients chosen from a standard Gaussian distribution.[2]

Theorem. The probability that a needle of length will randomly land on a line, given a floor with equally spaced parallel lines at a distance apart, assuming that the angle and the position of the fallen needle are independently and uniformly random, is .

Proof. If the needle always fell perpendicular (angle radians) to the parallel lines, we would have . So we have

as specified by the theorem. □

  1. David Blatner, The Joy of Pi. New York: Walker & Company (1997): 119, circle by upper right corner.
  2. S. R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, p. 141