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Sums and Products of Digits and SP Numbers

Stage: 5
Article by Alan Beardon

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 we believe that $1$, $144$ and this other number are the only numbers that are fixed by this rule; such numbers are sometimes called SP numbers. No-one has yet been able to prove that these are the only three 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.