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.
One possible distribution of the sticks is 4 - 1 - 5, but there are lots of other arrangements possible.
Next make one new heap using a stick from each of the heaps you have already.
Our example now becomes 3 - 3 - 4 (notice how the heap with just one stick vanishes).
Then keep repeating that process : one from each heap to make the new heap.
Continue repeating this until you see the distribution settle in some way.
You can of course begin with more, or less, than three heaps.
Could the arrangement 7 - 1 - 1 - 1 ever turn up, except by starting with it?
That's the main question, but you may like to pose yourself other questions about this situation.
Let us know what you try and what you find out.
Mathematicians like to notice patterns and then try to explain them, can you explain any of the things you noticed?
Eleven sticks, twelve, any number, explaining as much as you can about what you notice.