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!

When you state the part about painting the section between each bird and its nearest neighbor, do you mean we start at the left and fill in alternating sections, or do you mean literally what the question asks?

In the case of birds A, B, C, and D (sequentialy on the power line) with B and C being closer to each other than they are to A nd D respectively, then in case 1, sections AB and CD get painted, but section BC is not, but in case 2, all three section get painted.

Am I missing something?

Posted by Erik O. on 2004-06-07 12:55:59