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.

Number Chains

Stage: 5 Challenge Level:

The sequence can start anywhere and it is only practical to test a few starting values. The problem is to prove results that show that all the number chains reduce to a few that can be listed.

This problem is a simple example of a dynamical system, demonstrating fixed points and cyclic or periodic behaviour. It is instructive to work with such 'whole number dynamics' as an introduction to dynamical systems leading to useful applications as the Logistic Equation for population dynamics and to fractals and the well known Mandelbrot set.