### Whole Number Dynamics I

The first of five articles concentrating on whole number dynamics, ideas of general dynamical systems are introduced and seen in concrete cases.

### Whole Number Dynamics II

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.

### Whole Number Dynamics III

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.

This is a dynamical system. The important idea here is that there are a finite number of states so the system must eventually reach a state it has been in before and subsequently the states must change in a periodic cycle. It may reach a steady (fixed) state or it may cycle through the same p states (where $p> 1$) in the same order over and over again, a cycle of length $p$ (a p-cycle). If the number of beads is a power of 2 then the system always reduces to a steady state with all red beads whatever the initial state but the proof is rather subtle.