Template:Sequence of the Day for June 22

Intended for: June 22, 2013

Timetable

• First draft entered by Alonso del Arte on March 28, 2012 based on a comment by Benoit Cloitre ✓
• Draft reviewed by Daniel Forgues on May 30, 2012 ✓
• Draft approved by May 22, 2013

Yesterday's SOTD * Tomorrow's SOTD

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A060294: Decimal expansion of Buffon’s constant

2 π

.

0.63661977236758...

Buffon’s needle problem:

Theorem (Buffon’s needle problem, problem first posed in 1733, solved in 1777). (Georges-Louis Leclerc, Comte de Buffon)

The probability

P (l, d )

that a needle of length

l

will randomly land on a line, given a floor with equally spaced parallel lines at a distance

d   ≥   1

apart, is

P (l, d ) = 2 π  ⋅   l d

.

Proof. (assuming that the angle and the position of the fallen needle are independently and uniformly random) If the needle always fell perpendicular (angle

θ = π 2

radians) to the parallel lines, we would have

P⊥(l, d ) = l d

. So we have

P (l, d ) = ∫ π 0 sin θ   ⋅   d θ π   ⋅  P⊥(l, d ) = − [cos θ  ] π0 π  ⋅   l d = 2 π  ⋅   l d  .

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