OFFSET
0,2
COMMENTS
The problem is two-dimensional. Three identical disks are placed in a plane with positions and orientations that are independently and uniformly selected at random. Given that the second disk overlaps the first and the third disk overlaps the second, this constant is the probability that the center of mass of the second and third disks falls within the area of the first disk.
LINKS
Patricia R. Allaire and Antonella Cupillari, Artemas Martin: An Amateur Mathematician of the Nineteenth Century and His Contribution to Mathematics, The College Mathematics Journal, Vol. 31, No. 1 (2000), pp. 22-34. See p. 31.
Benjamin F. Finkel, A Mathematical Solution Book, Kibler, Cokely & Co., Kidder, Mo., 1888, pp. 344-345.
Benjamin F. Finkel, A History of American Mathematical Journals, National Mathematics Magazine, Vol. 16, No. 6 (1942), pp. 284-289. See p. 286.
Artemas Martin, Solution of a Problem in Probabilities, The Mathematical Visitor, Vol. 1, No. 1 (1878), pp. 9-10.
Artemas Martin, Problem 120, The Mathematical Magazine, Vol. 2, No. 6 (1892), pp. 100-101.
FORMULA
Equals 1/16 - (3/(16*Pi))*((3/16)*sqrt(15) - 2*arccosec(4)).
EXAMPLE
0.0493205462476131363353194330521558309611298502588085...
MATHEMATICA
RealDigits[1/16 - (3/(16*Pi))*((3/16)*Sqrt[15] - 2*ArcCsc[4]), 10, 120, -1][[1]]
PROG
(PARI) 1/16 - (3/(16*Pi))*((3/16)*sqrt(15) - 2*asin(1/4))
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Amiram Eldar, May 21 2026
STATUS
approved