

A089491


Decimal expansion of Buffon's constant 3/Pi.


27



9, 5, 4, 9, 2, 9, 6, 5, 8, 5, 5, 1, 3, 7, 2, 0, 1, 4, 6, 1, 3, 3, 0, 2, 5, 8, 0, 2, 3, 5, 0, 8, 6, 1, 7, 2, 2, 0, 6, 7, 5, 7, 8, 7, 4, 4, 4, 2, 7, 3, 8, 6, 9, 2, 4, 8, 6, 0, 0, 4, 0, 6, 4, 3, 5, 3, 3, 8, 0, 7, 8, 5, 8, 0, 5, 3, 5, 9, 2, 1, 0, 5, 4, 0, 6, 8, 2, 8, 1, 6, 5, 9, 7, 5, 1, 8, 5, 1, 5, 7, 3, 6, 4, 3, 7
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OFFSET

0,1


COMMENTS

Whereas 2/Pi (A060294) is the probability that a needle will land on one of many parallel lines, this is the probability that a needle will land on one of many lines making up a grid.
The probability that the boundary of an equilateral triangle will intersect one of the parallel lines if the triangle edge length l (almost) equals the distance d between each pair of lines. This follows directly from the Weisstein/MathWorld Buffon's Needle Problem link's statement P=p/(Pi*d), where P is the probability of intersection with any convex polygon's boundary if the generalized diameter of that polygon is less than d and p is the perimeter of the polygon. (Take d=l, then p=3d.)  Rick L. Shepherd, Jan 11 2006
Related grid problems are discussed in the Weisstein/MathWorld BuffonLaplace Needle Problem link.  Rick L. Shepherd, Jan 11 2006
The area of a regular dodecagon circumscribed in a unitarea circle.  Amiram Eldar, Nov 05 2020
For any nonobtuse triangle ABC (see Mitrinović and Oppenheim links):
(a/A + b/B + c/C)/(a+b+c) >= 3/Pi,
(a^2/A + b^2/B + c^2/C)/(a^2+b^2+c^2) <= 3/Pi,
where (A,B,C) are the angles (measured in radians) and (a,b,c) the side lengths of this triangle.
Equality stands iff triangle ABC is equilateral. (End)


REFERENCES

Joe Portney, Portney's Ponderables, Litton Systems, Inc., Appendix 2, 'Buffon's Needle' by Lawrence R. Weill, 200, pp. 135138.


LINKS

A. Oppenheim, Problem E 2649, American Mathematical Monthly, 84 (1977), p. 294.


FORMULA

Equals Product{k>=1} cos(Pi/(6*2^k)).
Equals Product{k>=1} (1  1/(6*k)^2). (End)


EXAMPLE

3/Pi = 0.95492965855137201461330258023508617220675787444273869248600...


MATHEMATICA

RealDigits[ N[ 3/Pi, 111]][[1]]


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KEYWORD



AUTHOR



STATUS

approved



