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.
Three beads are threaded on a circular wire and they
are coloured either red or blue.
You repeat the following actions over and over again. Between
any two of the same colour put a red and between any two of
different colours put a blue, then remove the original beads.
Discuss all the possible outcomes.
What happens when you do the same thing with 4 beads, 5 beads or