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# Buffon's constant

Buffon's constant is ${\displaystyle \scriptstyle {\frac {2}{\pi }}\,}$. Its decimal expansion is 0.63661977236758... (see A060294) and its continued fraction is

${\displaystyle {\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{3+{\cfrac {1}{31+{\cfrac {1}{1+\ddots }}}}}}}}}}}}\,}$

(see A053300).

Like many other numbers involving ${\displaystyle \pi }$, Buffon's constant can be expressed as an infinite sum or product:

${\displaystyle {\frac {2}{\pi }}=\prod _{n=2}^{\infty }{\frac {p_{n}+2-(p_{n}\mod 4)}{p_{n}}}={\frac {2}{3}}\times {\frac {6}{5}}\times {\frac {6}{7}}\times {\frac {10}{11}}\times \ldots }$

(where ${\displaystyle p_{n}}$ is the ${\displaystyle n}$th prime number)[1];

${\displaystyle {\frac {2}{\pi }}=1+\sum _{n=1}^{\infty }(-1)^{n}(4n+1)\left({\frac {\prod _{i=1}^{n}2i-1}{\prod _{j=1}^{n}2j}}\right)^{3}=1-5\left({\frac {1}{2}}\right)^{3}+9\left({\frac {3}{8}}\right)^{3}-13\left({\frac {5}{16}}\right)^{3}+\ldots }$

It is also the limit ${\displaystyle \lim _{n\to \infty }{\frac {z(n)}{\log n}}}$, where ${\displaystyle z(n)}$ is the expected number of real zeros of a random polynomial of degree ${\displaystyle n}$ with real coefficients chosen from a standard Gaussian distribution.[2]

Theorem. The probability ${\displaystyle \scriptstyle P(l,\,d)\,}$ that a needle of length ${\displaystyle \scriptstyle l\,}$ will randomly land on a line, given a floor with equally spaced parallel lines at a distance ${\displaystyle \scriptstyle d\,\geq \,l\,}$ apart, assuming that the angle and the position of the fallen needle are independently and uniformly random, is ${\displaystyle \scriptstyle P(l,\,d)\,=\,{\frac {2}{\pi }}\cdot {\frac {l}{d}}\,}$.

Proof. If the needle always fell perpendicular (angle ${\displaystyle \scriptstyle \theta \,=\,{\frac {\pi }{2}}\,}$ radians) to the parallel lines, we would have ${\displaystyle \scriptstyle P_{\perp }(l,\,d)\,=\,{\frac {l}{d}}\,}$. So we have

${\displaystyle P(l,d)\,=\,\left(\int _{0}^{\pi }\sin \theta \cdot {\frac {d\theta }{\pi }}\right)\cdot P_{\perp }(l,d)=\left({\frac {-[\cos \theta ]_{0}^{\pi }}{\pi }}\right)\cdot {\frac {l}{d}}={\frac {2}{\pi }}\cdot {\frac {l}{d}}\,}$

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