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%I #16 Mar 17 2022 05:49:56
%S 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,
%T 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,
%U 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
%N Decimal expansion of the area of intersection of 4 unit-radius circles that have the vertices of a unit-side square as centers.
%C 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.)
%C The perimeter of the shape formed by the intersection is 2*Pi/3 (A019693).
%C The solution to the three-dimensional version of this problem is A352454.
%H Donald L. Chambers, <a href="https://doi.org/10.1111/j.1949-8594.1977.tb09283.x">Problem 3684</a>, School Science and Mathematics, Vol. 77, No. 5 (1977), p. 443; <a href="https://doi.org/10.1111/j.1949-8594.1978.tb09373.x">Solution</a> by J. Philip Smith, ibid., Vol. 78, No. 4 (1978), pp. 354-355.
%H Amiram Eldar, <a href="/A352453/a352453.jpg">Illustration</a>.
%H Samuel Isaac Jones, <a href="https://archive.org/details/MathematicalNuts/page/n155/mode/2up">Mathematical Nuts: For Lovers of Mathematics</a>, 1932, Problems 9 and 10, pp. 86, 301-302.
%H Missouri State University, <a href="http://people.missouristate.edu/lesreid/Adv08.html">Problem #8, Finding the Area (resp. Volume) of Overlapping Circles (resp. Spheres)</a>, Advanced Problem Archive,; <a href="http://people.missouristate.edu/lesreid/AdvSol08.html">Solution to Problem #8</a>, by Raymond Roan.
%H Bruce Shawyer, <a href="https://cms.math.ca/publications/crux/issue?volume=25&issue=8">Problem 6</a>, APICS 1999 Mathematics Competition, The Academy Corner, Crux Mathematicorum, Vol. 25, No. 8, 1999, p. 453; <a href="https://cms.math.ca/publications/crux/issue?volume=26&issue=4">Solutions</a> by Richard Tod and Catherine Shevlin, Vol. 26, No. 4, 2000, pp. 193-194.
%H Charles W. Trigg, <a href="https://cms.math.ca/publications/crux/issue?volume=7&issue=9">Problem 686</a>, Crux Mathematicorum, Vol. 7, No. 9, 1981, p. 275; <a href="https://cms.math.ca/publications/crux/issue?volume=8&issue=9">Solution</a> by Jordan Dou, Vol. 8, No. 9, 1982, p. 294.
%F Equals 1 + Pi/3 - sqrt(3) = 1 + A019670 - A002194.
%e 0.31514674362772045262676811958729526112291787931465...
%t RealDigits[1 + Pi/3 - Sqrt[3], 10, 100][[1]]
%Y Cf. A002194, A019670, A019693.
%Y Cf. A075838, A133731, A192930, A352454.
%K nonn,cons
%O 0,1
%A _Amiram Eldar_, Mar 16 2022