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.
So, in the illustration, if a stone is removed from the $4$ and
the $10$ circles and added to the $13$ circle, the new distribution
would be $3$ - $9$ - $15$
Here are five of the ways that $27$ stones could be distributed
between the three circles :
$6$ - $9$ - $12$
$3$ - $9$ - $15$
$4$ - $10$ - $13$
$4$ - $9$ - $14$
$2$ - $8$ - $17$
There is always some sequence of "moves" that will turn each
distribution into any of the others - apart from one.
Identify the distribution that does not belong with the other
Can you be certain that this is actually impossible rather than
just hard and so far unsuccessful?