PROFESSOR COPELAND: So, the number 197.
And what I'm going to do is I'm going to
add the three digits.
So 1 plus 9 plus 7 is 17.
Now what I'm going to do is, I'm going to form a number out
of the last two digits of the original
number and the new number.
9 plus 7 plus 17, and that is 33.
And now I'm going to do the same again.
So, it's 7 plus 17 plus 33, and that gives me 57.
You're beginning to get the idea.
17 plus 33 plus 57, and that gives me 107.
And now--
ready, you've been very patient--
the Keith number is about to emerge.
33 plus 57 plus 107, and look what I get when I do that.
I get 197.
BRADY HARAN: That's quite interesting.
I can't decide if these would be really, really rare or
really, really common.
Intuitively, it strikes me that this would be rare.
PROFESSOR COPELAND: They're very rare.
This is a number with three digits.
There are only 80 of order 80 numbers, 80 Keith Numbers,
with less than--
BRADY HARAN: Of order 80.
Hang on.
I don't like this of order.
PROFESSOR COPELAND: There are in fact 84.
84 Keith numbers with less than 26 digits.
BRADY HARAN: Once you get more than three digits, how do you
start this process?
PROFESSOR COPELAND: So you start with 2785, for example.
Then you would begin your process with 2 plus 7 plus 8
plus 5, which is equal to--
10, 17--
22.
And the next stage along the line is you take the preceding
three digits and add the fourth from the new number.
So it would be 7 plus 8 plus 5 plus 22.
There are, for example, two-digit numbers.
OK, so if I take d equals 2, and 14 is a Keith number,
you'd go 1 plus 4 is 5.
4 plus 5 is nine.
5 plus 9 is 14.
Mike Keith, who's a mathematician and discovered
them in 1987--
for me, this is recreational mathematics.
It's how I got into mathematics.
My hero wasn't a physicist early on, it was actually a
guy called Martin Gardner.
I'd never heard of [INAUDIBLE]
earlier on when I was at school in Gauss.
But Martin Gardner was this guy, brilliant mathematician,
that worked--
I knew him as the guy working for "Scientific American." And
I subscribed to "Scientific American," and I never
understood as a single article I read.
I kept trying to, and so I'd reread it, and it'd be a blur.
But you had these mathematics puzzles at the back, and I
generally couldn't do them either, but I was
fascinated by them.
They were little numerology things.
And then I went and bought some of his books and just got
into it, just playing around with numbers, and you just get
to love them.
BRADY HARAN: So the people who watch Numberphile and get a
bit sniffy about our recreational maths videos,
what do you say to them?
PROFESSOR COPELAND: Oh, you shouldn't.
You should just enjoy it.
Embrace it.
And just have fun.
Just have fun playing around with them.
There's plenty of occasions when you doodling thinking,
what shall I do.
And just play around with numbers, and you'll be amazed
some of the coincidences that can come up.
And out of some of these things come major
breakthroughs.
We're not quite finished.
There's just a couple more things which are around there.
There's one called the reverse Keith, and this is where--
so again, it's best just to show you an example of the
reverse Keith number.
It's where the reverse number appears in the sequence.
So if I begin say with 12.
OK.
So we just follow the prescription that we've had.
So we go 1 plus 2, 2 plus 3, 3 plus 5, 5 plus 8, 8 plus 13,
and that's equal to 21.
And that's your reverse Keith number.
You've reversed the order of the integers in the original.
There's even fewer of what you could call Keith clusters.
Mike Keith has got a conjecture about these Keith
numbers, his numbers.
He's allowed to have a conjecture about his own
numbers, of course.
And they're to do with the clustering of these numbers.
And what he's done is he's said, how many ways can you
cluster the Keith numbers such that in any cluster, which
could have--
of course, a cluster has to have more than one number-- so
it could have two numbers, three numbers, four numbers.
Each number in that cluster is an integer multiple of the
lowest number.
His conjecture is the following.
There are three that are known, just three, and he says
he believes that's all there are.
So I think it's quite an interesting challenge for some
of the Numberphile people to go and maybe find some more.
So--
BRADY HARAN: So what's a Keith cluster?
PROFESSOR COPELAND: So here's an example
of the Keith cluster.
So the number 14 and 28 are both Keith numbers, and of
course, 28 is twice 14.
And another Keith cluster is the number 1,004--
1,104 and then if I double that 2,208.
You've also got to have the same number of
digits in the sequence.
There is, remarkably, one cluster of three Keith numbers
which is 31331, of course, 62662, 93993.
I think that's amazing.
Each of those are Keith numbers.
They all satisfy the properties that we've been
describing, and they form this cluster.
So the conjecture is that's it.
I used to do these puzzles on the bus going to school.
I had to get two buses, and I'd just try and do, well,
whatever I could.
Usually, I got stuck, especially with the
geometrical ones.
He has geometrical puzzles as well as sort of basic
algebraic ones.
And then he has his own puzzles and paradoxes.
How can you get a infinite number of people in a hotel
with an infinite number of rooms and then double the
number of people and things?
It was great.