On Friday 15 October 2010 15:49:35 Victor Miller wrote:
So how about the following generalization?
We're working in base B, we have some fixed partition of the digit set {0,...,B-1} into r pieces, numbered from 1 to r.
For any base B number we make a new number the same way:
first write down the base B representation of the number of digits, concatenating the base B counts of the digits for the r pieces given above. It's clear by the same reasoning that eventually we'll reach a cycle. So the challenge is to describe other situations where the cycle is length 1.
*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". -- g