Gareth:

*Two* things are special in the 10/even/odd case: we always reach the same cycle, and that cycle has length 1. I think the best generalization is "describe other situations where there is a unique such cycle and it has length 1".

--------------------------------------------------------------- Hello, I think I have one: "Sixteen, seven, nine" There are indeed 16 letters all together in the above line with 7 even-rank-in-the-alphabet letters (..xt..n, ..v.n, n.n.) 9 odd-rank-in-the-alphabet letters (si..ee., se.e., .i.e) Don't know if this is the unique such fixed point... Best, É.