A075838 -id:A075838

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A352453

Decimal expansion of the area of intersection of 4 unit-radius circles that have the vertices of a unit-side square as centers.

+10
4

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

0,1

COMMENTS

The solution to a problem in Jones (1932): "At each corner of a garden, surrounded by a wall n yards square, a goat is tied with a rope n yards long. Find the area of the part of the garden common to the four goats." (When the square is taken to be of unit size, the common area is this constant.)

The perimeter of the shape formed by the intersection is 2*Pi/3 (A019693).

The solution to the three-dimensional version of this problem is A352454.

LINKS

Table of n, a(n) for n=0..104.

Donald L. Chambers, Problem 3684, School Science and Mathematics, Vol. 77, No. 5 (1977), p. 443; Solution by J. Philip Smith, ibid., Vol. 78, No. 4 (1978), pp. 354-355.

Amiram Eldar, Illustration.

Samuel Isaac Jones, Mathematical Nuts: For Lovers of Mathematics, 1932, Problems 9 and 10, pp. 86, 301-302.

Missouri State University, Problem #8, Finding the Area (resp. Volume) of Overlapping Circles (resp. Spheres), Advanced Problem Archive,; Solution to Problem #8, by Raymond Roan.

Bruce Shawyer, Problem 6, APICS 1999 Mathematics Competition, The Academy Corner, Crux Mathematicorum, Vol. 25, No. 8, 1999, p. 453; Solutions by Richard Tod and Catherine Shevlin, Vol. 26, No. 4, 2000, pp. 193-194.

Charles W. Trigg, Problem 686, Crux Mathematicorum, Vol. 7, No. 9, 1981, p. 275; Solution by Jordan Dou, Vol. 8, No. 9, 1982, p. 294.

FORMULA

Equals 1 + Pi/3 - sqrt(3) = 1 + A019670 - A002194.

EXAMPLE

0.31514674362772045262676811958729526112291787931465...

MATHEMATICA

RealDigits[1 + Pi/3 - Sqrt[3], 10, 100][[1]]

CROSSREFS

Cf. A002194, A019670, A019693.

Cf. A075838, A133731, A192930, A352454.

KEYWORD

nonn,cons

AUTHOR

Amiram Eldar, Mar 16 2022

STATUS

approved

A173571

Decimal expansion of constant related to Goat Problem, Donkey Problem, Tenenbaum and A173201.

+10
1

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

0,1

LINKS

Table of n, a(n) for n=0..104.

M. Fraser, A tale of two goats, Math. Mag., 55 (1982), 221-227. Has extensive bibliography. [From N. J. A. Sloane, Jul 12 2011]

Gerd Lamprecht, Iterationsrechner Beispiel 3

FORMULA

x = sqrt(1-A072112^2) = sqrt(1-(1-A133731^2/2)^2); A075838 =(x+PI/2-asin(x))/(A133731^2); with A072112=1-A133731^2/2; A133731=cos(A173201/2)*2;

EXAMPLE

0.9528478646... = A075838 =(0.9444433782055...+PI/2-asin(0.9444433782055...))/(A133731^2);

KEYWORD

cons,nonn

AUTHOR

Gerd Lamprecht (gerdlamprecht(AT)googlemail.com), Feb 22 2010

STATUS

approved

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