Brace yourself, this is gonna be big.
2, 2, 3, 4, 7 -
looks easy right - 14,
42, 213,
2837,
175450,
139759600,
6837625106787
You want one more? They get big!
266437144916648607844: and the next one gets a lot bigger.
And they keep getting bigger and bigger
and bigger.
What are these numbers? Well you don't
have a clue right?
I wouldn't have a clue myself. Where did
they come from?
Yeah, okay, well the where they came from
is actually a puzzle,
a genuine puzzle sequence. And
it's a triangle, so I show you the
beginning of the triangle
and I ask you to explain how I'm getting
from one row to the next. The zeroth row
just has a single number in it - 2. The
first row
is 1, 1. Second row is
1, 2; then 1, 1, 2 yep?
Then 1, 1, 2, 3, and then
1, 1, 1, 2, 2, 3, 4.
They're getting bigger and bigger, the
next one:
1,1,1,1,2,2,2,3,3,4,4,5,6,7;
and so on. This is Lionel Levine's
triangle; and how is it defined?
Can you figure it out?
Give up? Okay, let's see how we would go
from 1,1,2: you read it from the
back,
and this tells you how many 1s there
are, so there are two 1s in the next row.
This says how many 2s there are,
there's one 2,
and this says how many 3s. So the
next row is
1,1,2,3 okay. How do we go from
1,1,2,3
to this row? We read it from the back, the 3
says we begin with three 1s; the 2
says the next we have two 2s;
that 1 says we have one 3 and this
says-
that 1 says we have one 4. And we
keep going in that way;
so here again we have four 1s, three
2s, two 3s, two 4s,
one 5, one 6, and one 7 - that's the
next row and you keep going.
That's the triangle and now you add up
the numbers in each row.
So here we get a 2, 1 plus 1 is 2;
1 plus 2 is 3,
then we get 4; so these numbers are the
sums
of the numbers in the rows. And it's an
innocent looking triangle but when you
follow it for a few rows
you get a triangle which is bigger than
any computer memory could possibly hold.
(Brady: That is fiendish, I don't think)
(anyone would get that in a job interview.)
No unless they could cheat.
Just in case you're wondering what would
come next in this sequence, because
obviously there are more than nine
digits now, what we do is we have
seven 1s, six 2s, five
3s, four 4s, four 5s,
three 6s, three 7s, a couple of
8s, a couple of 9s and a couple of
10s, and then 11
12, 13, and 14. So we're using full
natural numbers
not digits. So for the sum of this row
we add the natural numbers not the
digits, so for example 11 is treated as
11 not two 1s
and the total of the whole row is 213;
and so it continues. You can find out
more about this
sequence and many other sequences on the
Online Encyclopedia
of Integer Sequences; and this is entry
number
A011784. And we've got plenty more videos
with Neil Sloane who's the founder and
keeper
of the OEIS. There are links on the
screen and in the video description.
"We have to always go to a new square"
And by the way if you're into podcasts we
have a Numberphile podcast,
have you listened to that yet? And I have
another podcast called the Unmade
Podcast with a friend of mine in
Australia called Tim.
I'll have links on the screen and down in
the video description.