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.
Published July 1998,February 2011.
The whole number $6712$ has digits $6$, $7$, $1$ and $2$. Given any whole number take the sum of the digits, and the product of the digits, and multiply these together to get a new whole number; for example, starting with $6712$, the sum of the digits is $6+7+1+2=16$, and the product of the digits is $6\times 7\times 1\times 2=84$. The answer in this case is then $84\times 16=1344$. If we do
this again starting from $1344$ we get
$$1344\rightarrow (1+3+4+4)\times 1\times 3\times 4\times 4=576$$
and yet again
$$576\rightarrow (5+7+6)\times 5\times 7\times 6=3780$$
At this stage we know what the next answer will be (without working it out) because, as one digit is $0$, the product of the digits will be zero, and hence the answer will also be zero. Whenever a digit is zero, the next answer will be zero!
How often do we end up at $0$? Here is a case in which it doesn't happen. Start with $332$, then we get
$$332\rightarrow (3+3+2)\times 3\times 3\times 2=144$$ and $$144\rightarrow (1+4+4)\times 1\times 4\times 4=144$$
Thus if we reach $144$ we stay there however many times we apply this rule. We say that $144$ is fixed by this rule. Now try $233$; what does this go to? What do you notice about $332$ and $233$? What happens if we start with $98$? Can you find some other numbers that go to $144$?
There is another number that is fixed by this rule; it is $1$ (because the sum of the digits of $1$ is $1$, and the product of the digits is $1$ so, starting with $1$, the answer is $1\times 1=1$).
Now here is something interesting. We only know of one other number (apart from $0$) that is fixed by this rule, and $1$, $144$ and this other number are the only numbers that are fixed by this rule; such numbers are sometimes called SP numbers. What is this other number? You can find it for yourself, but to help you I will tell you that it lies between
$110$ and $140$.
There are a few numbers that have the property that when we apply the rule repeatedly, we end up at $1$, $144$ or this other number (which by now you should have found). It seems that most numbers will eventually end up at $0$ when we apply the rule repeatedly, but again, no-one has yet proved this. Try some numbers for yourself and see if they end up at $0$. If you find something surprising,
show your teacher because you may have discovered something that has never been noticed before!
SP Numbers Continued, published in October 1998, is a follow-up (much harder) article on this topic.