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.
Encourage questions like "what could come before . . . . . and
what before that" ?
Working backwards from the goal is a mathematical thinking
Especially encourage explanation - accounting for pattern.
There is lots of interest in this simple mechanical process -
look out for the triangle numbers!
Above all, if possible, let each insight prompt fresh questions
to consider, each new question will illuminate more of the