The first of five articles concentrating on whole number dynamics, ideas of general dynamical systems are introduced and seen in concrete cases.
This article extends the discussions in "Whole number dynamics I". Continuing the proof that, for all starting points, the Happy Number sequence goes into a loop or homes in on a fixed point.
In this third of five articles we prove that whatever whole number we start with for the Happy Number sequence we will always end up with some set of numbers being repeated over and over again.
You may like to denote the red beads by 0 and the blue beads by
1. Then the new bead is given by the sum modulo 2 of the two
adjacent beads because two the same colour are given by 0+0=0 and
1+1=0 and two different colours are given by 1+0=1 and 0+1=1.