### 18th Putnam 1958

**Problem B4**

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 x^{2} + y^{2} + z^{2} = 1).

**Solution**

Answer: 4/3.

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 ∫_{0}^{π} (2π sin θ) (2 sin θ/2) dθ = ∫ 4 sin^{2}θ/2 d(sin θ/2) = 4/3.

18th Putnam 1958

© John Scholes

jscholes@kalva.demon.co.uk

25 Feb 2002