And it seemed to me th- the stupidest
problem you could conceivably imagine
that led to the most complicated answer
you could conceivably imagine.
I was in Cambridge; I finished a lecture course
two lectures early, and then I sort of
hinted, you know, I said the University
regulations or something demand that I
turn up on Friday but you don't have to
turn up on Friday. And I thought they'd
taken the hint, because when I get to the
classroom there was usually- you know, a classroom
with 150 people in it you can usually
hear some chatter and so on, but it was
all quiet. And then I opened the door and
they said 'surprise!' And they'd equipped it
all for a party. Some of the girls wore
nice long party dresses, none of the boys
wore long party dresses or if they did
they were indistinguishable from girls.
And- but anyway, one of them came up
to me with this problem,
he said 1, 1 1, 2 1, 1 2 1 1; and I gathered
that I was supposed to guess what came
next. And I didn't succeed;
111221. And he looked at me, and I still didn't guess.
312211. And I knew
from the way he was saying it that
somehow I was supposed to be able to
guess, I still didn't; 13112221.
Anyway, it was quite interesting at this
party but I didn't guess - and I suppose I
should cut the story short a little bit -
in the end he had to tell me. So there's
no need to be embarrassed if you didn't
get the rule. What's this sequence consist of? 1- no,
how many 1s? One 1, oh yes. What's this
next sequence consist of? Oh two 1s,
oh yes that's right. This next one is one 2
followed by one 1, oh yes. One 1, one 2, two 1 - you
get the rule.
And that's quite an interesting sequence.
And I was just going off at the end of
term and on the plane to, Boston I think
I was going to, I fiddled with this and
came up with a really rather remarkable
conclusion. On average each of these
sequences is about 1. - I hope
I'm getting it right - 1.357
times the one before it.
I mean, not exactly, but it gets closer to
being this fixed number times the length
of the one in front of it. So it gets
bigger and bigger and bigger by powers
of 1.357 all the
time. And I proved that that number - I
hope I've got it right by the way - was an
algebraic number, don't worry if you
don't know what that means, of degree 71.
And it seemed to me this- the stupidest
problem you could conceivably imagine
that led to the most complicated answer
you could conceivably imagine. Anyway,
that's been called by some people a
look-and-say sequence. I callled it- I wrote a
little paper about it in the
undergraduate magazine of Cambridge. And
I called it the weird and wonderful
chemistry of audio active decay. Audio
active because you say this, and then in
some sense the structure decays although
the thing's getting longer all the time.
And it's like radioactive. And I showed
that there are some sequences; in fact if
I go one step further, or maybe two steps
further, 1113213211, yes
this is it. If I take a red
pen so that you can see it more easily;
I'm putting a red spot there. It turns
out that if you develop this sequence
here it always ends in a 2. Because, you
know, you've got one two at the end of
it next time. Here you've got maybe
it's two 2s, I don't know. But anyway
it'll be 2s. So the the left-hand
sequence always ends in a 2. And the
right end sequence, well look: it goes 1
1 1 3 1 2 - something like that -
and then 3 1 1 3 1 1; and then 1 3 2 1 1 3
etcetera; and then 1 1 1 3 1 2. And here
we are - this sequence and this sequence
begin in the same way. And you can see
that- so this sequence will begin 3 1 1 3
and so on and this is 1 3 2; what
I'm trying to say is that this will
always begin with a 1 or a 3. And I
remarked before that this always ends in the
2. So what's interesting about that is,
if you follow this sequence as if it's
all there is, and you follow this sequence as
if it's all there is, and then you stick the
results together it's correct. And so I
say this sequence splits, it's a compound
of two atoms or elements. And it turns
out that there are precisely 92 common
elements in this thing. And you know there are
92- historically there were 92 common
elements in the world. And that just took
my fancy, there's no connection
whatsoever, and but I call these elements:
hydrogen, helium, lithium, beryllium, boron,
carbon,
nitrogen, oxygen and so on. - (Brady: All the way)
(through to uranium?) - Yes. This sequence of
length 10 is not an element, it's
the compound of this sequence 1 1 1 3 2
and the sequence 1 3 2 1 1 in that if
you follow the- let's call those L and R. If
you follow L and follow R
independently you still get the right answer.
You only need to consider elements that
aren't compounds or sequences that aren't
compounds and we call those elements or
atoms. And it turns out there are 92 of
them. I mean you can do various other
things; for instance I gave a lecture on
this in Acadia University in Nova Scotia.
There was this Nobel Prize winning
physicist, David Park, at the lecture and
he said, what- have you thought about
Roman numerals? One, one one. Oh this is
two ones, this is three ones, oh this is
four ones - now the Roman numeral for four
is IV. And then this is 1 1 1 5 1 1. This
guy gives rise to another problem of the
same kind and it turns out that this
depends on the root of a twentieth degree equation.