This surely isn't what you had in mind either, but just one word: Collatz. (OK, more than one word: Assume that the 3n+1|n/2 process is completely random. Odd->even, even->1/2even+1/2odd. Now it is obvious the random process should converge, since the Markov chain equilibrium is 2/3:1/3 and 3^(1/3)<2^(2/3). Not much of an insight, I admit...)