Farmer Brown lives in a land where farms stretch far and wide. One day he wins the lottery and wants to tell his neighbors. He tells his nearest neighbor of his good fortune. What follows is an odd sort of transmission of the news: each person who hears it (as well as farmer Brown himself) tells only his nearest neighbor, no one else.
Whenever that nearest neighbor is the person who told him the news, that branch of the transmission is closed, as there's no point in telling the person who told you, and there's no substitution of the second nearest neighbor.
-
What's the probability that the only person to get the news is the one farmer Brown called himself, due to farmer Brown being his nearest neighbor's nearest neighbor?
-
What's the expected number of people, besides farmer Brown, who will get the news before the transmission dies out altogether?
Consider the land where this happens an infinite plane, with each farmer a randomly placed point with uniform probability density.
(In reply to
Simulation by Larry)
Dear Larry,
I'm sure you're right.
The simplifying assumption I made was that the chains would essentially be linear, i.e. f-f-f-f. For the overwhelming part, they are.
But now I have actually drawn some graphs by hand, one of my examples threw up this arrangement:
f-f-f-f
|
f-f
and another example:
f-f-f
|
f
so although the problem does resolve to a grid arrangement, the grid is not necessarily linear, as it can contain these 'intermediate nodes'.
Certainly the number of dimensions is low, because in principle each square kilometre contains around 5 farmers, so there is a limit in principle to how many farmers could be closer to a given node than each other. On an infinite plain with farmers distributed as directed, at most 6 can be closer to an intermediate node than each other, even assuming they could all be equally distant, contrary to the problem. So a complete graph in the case of large n might require 3 dimensions, but not 4.
To explore further would require computing resources well beyond my limitations of web Excel and hand-drawn graphs.
Overall I am very happy with the solution I gave and how well it has stood up to more refined analysis!
|
|
Posted by broll
on 2025-03-26 00:27:42 |
re: Simulation
