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.
Turn $2$ - $8$ - $17$ into $4$ - $7$ - $16$
then $4$ - $7$ - $16$ into $3$ - $9$ - $15$
This example isn't the same thing but might give you a clue
about the kind of thinking to try.
In a $4$ circle problem and using $26$ stones the distribution
$1$ - $4$ - $7$ - $14$ cannot be turned into $3$ - $5$ - $7$ -
To understand why notice that in the first there are two odd and
two even numbers while in the second the numbers are all odd.
On a "move" one value goes up by $3$ and the others go down be
What will happen to odd and to even numbers?
The Odd Stones problem isn't about odd or even numbers but a
similar kind of thinking could be useful.