The Farmer Problem

The following "Farmer Problem" appeared in Shine [2006]:
A million farmers are randomly scattered on a square, flat plain, 1000 kilometers on a side. One farmer wins the lottery. He tells his nearest neighbor, who in turn tells his nearest neighbor, and so on.

(a) Not counting the lottery winner, what is the most probable number of farmers who get the message?

*(b) What is the expected number of farmers who eventually hear the news?

Of course, on sociological grounds based on other means of communication, the answer to part (a) is "all of them." However, we stick to the mathematical considerations.

Underlying assumptions are that

• The chain of nearest neighbors ends when a farmer's nearest neighbor is the farmer from whom (s)he just heard the news. The two farmer neighbors are termed isolated nearest neighbors, a nearest-neighbor pair, reflexive nearest neighbors, or an isolated pair. We term both such farmers isolated.

• No farmer faces a choice between equidistant neighbors.

• The farmers' locations follow a uniform probability distribution. For a large number of farmers, the binomial process of uniform distributions on each axis produces a Poisson process as a limiting distribution.