What's the next term in the following sequence? It goes 1,
4, 8,
48, 88,
488 - what's the next term? Which actually is very easy, and why? That's not so obvious.
Well, it's a little strange. 4, 8, 48, 88,
488, and obviously the next one is going to be 888, and then
4888.
The right way to look at this is to stare at these numbers. Look at this number. What do you see?
Well you see 8, but more important you see 2 holes. When you look at 48 you see 3 holes.
88 has four holes in it. 4 has one hole, if you draw your 4s that way, and
1 has no holes in it. So the
definition of this sequence is it's the smallest positive number that has n holes in it. - (Brady: People always get really funny about)
(sequences and numbers that are sort of base 10 specific, this is, this is how you draw your numbers specific! Like, if you draw your 4)
(non enclosed-)
It would be- it would change the sequence. In that case, if your 4 is non enclosed, then you would make it 1, 6, 8,
68, 88, 688 and so on.
And if you do your 2s in a very fancy way with a hole in them,
then the 6 could be replaced by a 2. This is topology,
what- um, what we're doing is partitioning the piece of paper into parts, with- using numbers.
So if you write down a 4,
you've partitioned the plane into two parts: this part, and the outside.
So it's topology.
This one is another base 10 sequence. So...
but it's lovely. 61, 21,
82,
43,
3.
And the question is, what comes next? Don't get distracted into looking for something that's too complicated.
Watch carefully. There's nothing up my sleeve,
I'm just gonna move the comma.
Then I'm gonna move that comma, and that comma,
and that comma.
And there's an invisible zero here, and so the next one,
get it? - (Brady: 6, 12, 18, 24, 30, so 36)
(So that one after that would be-) - 42!
They're multiples of 6. Nothing could be simpler.
But if you were misled
into looking for something complicated you wouldn't get it. - (Brady: That is sneaky that one.) - Yeah, yeah.
Yeah, I like it. It's misleading, its deceptive, sleight of hand.
One - we get 0, two - we get 0, three - we get 0, four we get 0. Hmm,
this doesn't look too promising.
All right, is it ever gonna get more than 0? Yea- 5 we get 4.
6
we get 9.
7 we get 5.
8 we get 1.
9 we get 1.
10 - I'll do a couple more to make it eas-10 we get 0.
11,
you'll never guess what we get for
11- 55. What's the rule?
Let me show you the answer. If I put these squares on top of the numbers and
you look through the window - you have to think like a Roman.
V- IV,
4.
IX, 9. V, 5. 8, 1 - there's a single 1. 9, 1.
10 there's nothing. 11 is 55,
and so on. - (Brady: I could- that would have taken me a year.)
So, of course, it might not be a legal Roman number. You might have the Is and the V's and the Ms in the wrong
order. In which case the way the sequence would be defined is you say that's zero, for that term.
Two closely related sequences, in the sense that they both have names, they both have
slightly misleading names,
titles. And the first one are the eban numbers.
Well, you know what the even numbers are:
2, 4, 6, 8. But I would like you to figure out what the eban numbers are. 2, now
I have to be very careful, 4,
6,
7 - 7 is not eban. 8, 9, 10, 11, 12, 20, 19, 20-
30! Is the next eban number. And then 32, 34, 36,
40, 42, 44,
50,
52, 54,
56,
are you getting the idea?
60 - let me continue - 62, 64, 66, 2000.
Those are the eban numbers.
(Brady: 2000? That was a jump!)
Why are those the eban numbers?
What do you notice about these numbers? For one thing they're all even.
The second thing is that if you say, if you write out the number in English,
there are no E's in them. One is missing because one has an E in it.
Three has an E in it, eight has an e in it. In fact, there's a theorem.
It's an old theorem of mine. Every odd number
contains an e. And you can easily prove that by looking at all the odd numbers, and
check. - (Brady: Why are they called eban numbers?) - E is banned.
Any other questions? Let me show you another one with a name.
These are the emirps. And again, the name is a hint. And in this case it's actually a legitimate hint.
The emirps, as you might guess - well, let me show you what they look like first of all and then you can guess. 13,
17, then
it's not 19 - you might have thought I was, I was going to write 19 but it's actually, umm,
31,
37,
71 - and that's probably enough of a hint.
Well, obviously they're primes. And
if you look at the name, you'll see it's prime backwards.
So these are the primes which, when you read them backwards, are
still primes - but different primes. So we don't put down 11 because it's palindromic.
These are primes which, when you read it backwards, gives you a different prime.
Brady: I know what you're all thinking everyone,
what is the largest known emirp? Well as far as I can tell it's this, or
its decimal expansion is this. Now if you like these fun little puzzles
I've actually got one left over, one that got cut from this video, but it's still pretty good fun.
I've put it over on the Numberphile2 channel, there's links on the screen and in the description, and we'll have Neil Sloane back
really soon talking about more amazing sequences from his online encyclopedia. Stay tuned for that.