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.
This could be a useful extension activity helping students to
break away from too readily expecting odd
or even to be the important characteristic. Odd or even-ness can be
seen more generally as the remainder after a division by two, and
this problem depends on remainders using a different