The graph I generate for #3 is quite different from the one above. For example, P(r) never reaches 1. Here's my semi-analytic, semi-numeric solution made while thinking of the answer to the probability challenge:Building off of an approach similar to @mfb in post #91 Let N(r) be the number of inhabitants in generation r. There are N(r) inhabitants who know the rumor and cannot pass it on to themselves, and n + 1 - N(r) inhabitants who do not currently know the rumor (let's call these inhabitants naive).For case 2 (naive inhabitants), they have N(r) opportunities to hear the rumor in the next round. The probability that one rumor spreader passes the rumor to one particular individual is 2/n. Therefore, the probability that a naive individual does not hear the rumor is ##(1-2/n)^N##. The expected number of naive individuals to hear the rumor in the next generation is ##(n+1-N) (1 - p^N)##, where p = 1-2/nFor case 1 (rumor spreaders), they have N - 1 opportunities to hear the rumor as they cannot spread the rumor to themselves. Therefore, the probability that they hear the rumor during the next round is ##1 - p^{N-1}##, and the expected number of rumor spreaders who become part of generation r+1 is ##N(1-p^{N-1})##.In total ##N(r+1) = (n+1-N) (1-p^N)+N (1-p^{N-1})## which simplifies to ##N(r+1) = (n+1) - (n+1 + \frac{2N}{n-2})p^N##We can express, the fraction of inhabitants part of generation r, F(r), through the following recurrence relation: $$F(r+1) = 1-\left(1+\frac{2F(r)}{n-2}\right)\left(\frac{n-2}{n}\right)^{(n+1)F(r)}$$Using the recurrence relation and F(1) = 2, I can plot F(r) for various values of n:Solving the recurrence relation analytically would give a solution to the probability challenge, but I'm not sure how to do that.