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.
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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?
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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.
I made a simulation using a rectangular grid of N points with coordinates distributed between (0,0) and (N,N).
For Question 1, I expected to get 2/3, but instead consistently found numbers around 0.62
For Question 2, the numbers were around 1.5
For one run with N=10,000
the number of each length branch (1st entry is zero):
[0, 6226, 2773, 796, 172, 30, 2, 1]
and Expected Value: 1.5017
I speculate that the answer to question 1 might be φ-1, 0.618.
If the probability of farmer Brown being his nearest neighbor's nearest neighbor were exactly 2/3, no matter how many neighbors, then adding the sums of:
(2/3) + 2(1/3)(2/3) + 3(1/9)(2/3) + 4(1/27)(2/3) + ... = 1.5
Just as expected. This equation goes on forever even if the telephone chain reaches someone who has already been called.
In the sim, I kept track of who had been called, so my phone chains terminated more quickly than the theoretic equation was assuming.
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Posted by Larry
on 2025-03-25 23:08:54 |
Simulation
