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.
The line before 7 - 1 - 1 - 1 either had 1 or 7 heaps.
If 1 heap (of ten) then the next line would have to be 1 - 9
which isn't what we want.
But 7 heaps will work :
Three of those seven have to be twos so they'd go to ones at the
That leaves four of the ten sticks for the other four spaces -
so it has to be one in each place.
Hamish from New Zealand had similar
reasoning to Fiona (well done Hamish).
He then wondered about a
If 7 - 1 - 1 - 1 can have some thing before it, will any number
of heaps ( arranged as n and the rest ones ) always have something
Exploring this conjecture, combinations such as 8 - 4 - 1 and 6
- 1 - 1 - 1 - 1 are possible by starting with
2 - 2 - 1 - 1 - 1 - 1 - 1 - 1 and 2 - 2 - 2 - 2 - 1 - 1 - 1 - 1