A telephone wire stretched tight between two poles placed ten meters apart is a favorite resting spot for a flock of crows.

Suppose one morning two crows land on the wire, each at a random spot (the probability is uniformly distributed). With a bucket of paint and a brush you mark the stretch of wire between them. A certain length of wire will have been painted.

On average, what length of wire would you expect to have painted? Assume that each bird is a single point along the line, and so has no width.

Suppose instead that a dozen crows landed on the wire, each at an independent, random location, and you painted the stretch of wire between each bird and its nearest neighbor. On average, what total length of wire would you expect to have painted now?

And if a thousand crows landed?

A computer-generated solution could be found, but bonus points will be awarded for a formal proof!

(In reply to re(5): Full solution by Charlie)

It seems we have different interpretation.  I interpreted that the wire in-between each pair of birds are painted.  You intepreted that only the nearest of the two neighbors of each bird are painted.

Of course our interpretations will be the same for n=2 case.  For n=3, your interpretation is uncomprehensable to calculate.

Posted by Bon on 2004-08-05 16:12:26