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 Bird on a Wire (Posted on 2004-06-07)
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!

 No Solution Yet Submitted by Sam Rating: 3.7000 (10 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
 re: Some thoughts on the 2 bird problem. (partial solution) | Comment 5 of 42 |
(In reply to Some thoughts on the 2 bird problem. (partial solution) by Erik O.)

It definitely does approach 3.33333 for two birds. I also don't know why completely, but I can foresee the end solution being continually divided by some fraction for each bird added to the wire.
 Posted by Eric on 2004-06-07 14:04:32

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