Let S be a spherical shell radius 1. Find the average straight line distance between two points of S. [In other words S is the set of points (x, y, z) with x2 + y2 + z2 = 1).
It is sufficient to fix one point and to find the average distance of the other points from it. Take the point as P and the centre as O. Now consider a general point Q on the surface. Let angle POQ = θ. The distance PQ is 2 sin θ/2 and this is the same for all points in a band angular width dθ at the angle θ. The band has radius sin θ. Hence the average distance is 1/4π ∫0π (2π sin θ) (2 sin θ/2) dθ = ∫ 4 sin2θ/2 d(sin θ/2) = 4/3.
18th Putnam 1958
© John Scholes
25 Feb 2002