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.
$2$ - $8$ - $17$ goes to $4$ - $7$ - $16$
and from there to $3$ - $9$ - $15$
Consider the numbers in each circle in modulo $3$.
Modulo $3$ means the remainder amount when you divide a number
In the first arrangement ($6$ - $9$ - $12$) the modulo $3$ value
of each pile is $0$.
On each move you take $1$ from $2$ of the piles and add $2$ to
the third so the numbers which were all $0$ in modulo $3$ now all
become $2$ in modulo $3$, and after that $1$ in modulo $3$, then
finally $0$ again.
After that the cycle just repeats over and over again.
For four of the arrangements the initial numbers are all equal
in modulo $3$ and whatever you choose as the next move they will
stay equal in modulo $3$.
But $4$ - $9$ -$14$ is $1$ - $0$ - $2$ in modulo $3$ and so
cannot turn into any of the other four arrangements or be reached