Pascal's triangle. It's one of my

favorite things in math - particularly

because whenever I talk to other

mathematicians they always have some

interesting patterns they like talking

about. It's almost never the same one.

So Pascal's triangle is sort of like an

addition triangle. So we start with

the number one and every time I draw one

assume that everything around it is a

bunch of zeros so what we do is we have

one and then we add whatever is to the left

of it - so it's neighbor here is 0 and we just

drop that down in between - then I do

the same thing on the right.

Alright cool now I have two ones in this row

and I'll do the same thing so

there's a 0 we can imagine here so we

drop that down.

There's one in this neighbor now so

we can have a two and then there's one

and the zero to its right we make another one.

And we just keep doing this for a while

and you can see that the triangle just

keeps getting bigger this is nothing

that's gonna stop. So we can just keep

doing this for as long as we want and

for me as long as I want is when the

first double-digit number shows up

because that's when adding gets hard.

So Pascal was also a combinatorist so

in particular like problems where "If I have

N things and I want to choose K of

them, how many different ways can I do

that?" And it turns out that Pascal's

triangle is really like "childlike adding

game" as seems to be actually encodes N

choose K. So if I have N things -

in particular let's say i have four things

and I want to choose let's say one of

them then I go to the fourth row I move

over one and that's how many ways I can

do it four ways. I if I have four things

let's say I have like four circles and

I want to only choose one of them - there are

four ways i can do that right. I can

choose this one, I can choose that one

I can choose that one, I can choose that one

And it turns out Pascal's triangle says if I go down to the

fourth row and - where it like this is the

0th row - and I choose one thing so I move

one over then that's exactly how many

ways I can do it.

(Brady Haran) So if i told you to choose two circles-

Then there are six ways to do it.

You notice that the sides are always ones

Like, this isn't going to change because

we just keep doing 1+0 and then oh, look

1 + 0, 1 plus 0. And this is just gonna keep

happening so we can define this entire

diagonal as "+0" because every time I

go down 1 i'm not changing anything

But then we get to this row right and we have

one plus one plus one plus one because

I have 0 1 2 3 4 5 so i can define this row

I felt like it as "+1" then this

pattern looks less fun right. I have One, a Three

But the difference between three and one

is 2. The difference between six and

three is three the difference between

10 and 6 is 4 so it turns out this pattern is

+ 1 + 2 + 3 keeps going. So around here

One may be tempted to look for more

patterns like right here is like plus

what three plus six but that's when you

start to notice what we've been doing

all along which was just adding the

number to the top-left like Pascal

wanted us to anyway. Let's just look at

something else one that might be tempted to

do when they see a bunch of numbers and

that's add them in a different way.

So this is 1. One plus 1 is 2. One plus 2 plus 1 is 4.

One plus 3 is 4 plus 3 is 7 plus 1 is 8.

You'll notice that some point I just stopped adding

because we can actually rewrite everything over here as

the powers of 2. So the sum of the rows of

Pascal's triangle is two - like if I'm on

the Nth row and I call the first one row zero then

the sum of that row is 2 to the N.

Let's just read these as numbers let's not add

them let's not treat them as anything else

So the first row is 1 the next row is 11

The next row is a hundred and twenty one. The next row is thirteen thrity-one. Then we get

Then we get fourteen six-four-one

And then we get to this

*laughs*

So ,right now these are actually the powers of 11 too

So this next row looks pretty frightening

because if I wrote it out just

as it is, I get a pretty big number right

like once I put in these commas,

this is around 15 million

(brady) That's not 11 to the 5

It's not. But it turns out Pascal

you know his triangle doesn't quit that early

so what's happening here is that

when I was writing these as digits

before, I was actually considering that

place value right - like the ones place

the tens place the hundreds place

So I'm just going to do the same thing where I rewrite these numbers

as ones and tens and hundreds place

So the first one will be one times 10 to the 0

because this is the 0th row

Then we have 1 times 10 to the 1 plus 1

times 10 to the 0 (1x10^1+1x10^0)

so this one's 1 this one's 11.

Uh, let's do 1 times 10 squared plus 1 times

te- oh 2 times 10 to the one my bad - plus 1

times 10 to 0. That"s a hundred twenty-one

So let's let's skip on down to our friend here

We do 1 times 10 to the 5 Plus 5 10

to the 4 plus 10 10 to the 3 plus 10(10) squared? - yeah

5 to the 10 to the 1 and 1 times 10 to the 0

and this is 11 to the 5

(Brady) So it is 11^5

It is 11^5, it's just hiding

So now I have this really cool thing

that I found extremely recently called

um, the hockey sticks method i think

some people call it. It's just a really

cool trick that you can do on Pascal's

triangle another result of the fact that

it's the coefficients of like a binomial

expansion which is all of - it will make

more sense once I draw hockey stick

What you do is you pick a one somewhere on the

edge and you just go down diagonally so

i'm going to pick this one and then i'm

going to go down diagonally and then I'm just

gonna keep going and then at some point

I'm just gonna veer off in this case I have to

veer off because I didn't draw any more

rows. So the number that I veer off to

is the sum of all the numbers I traced down

1 plus 2 Plus 3 plus 4 equals 10

So this row - or this diagonal rather - equals 10

So I'm gonna start with this one

and it's gonna be awful real quickly

actually I'm just gonna all right to

work on small ones. Uh, let's use this one

Because... why not. We can do the other direction, right

and we get 1 plus two plus three equals 6

I'm gonna do Pascal's triangle mod 2

So, brief aside - mod 2 in this case is just

going to mean that every time a number

is odd I write it as a one and every

time a number is even I write it as a zero

(Brady) Cool

cool right

(Brady) That's still a Pascal Triangle, is that?

Yeah, just mod two. It doesn't like

preserve the actual value of the numbers

as much, but it totally tells you whether

or not something is even or odd and that's

all we care about right now.

So then I'm going to outline all of the

diagonals that are all ones and all the

rows are all ones

So i'm going to do this - which isn't the world's prettiest

thing but it's pretty nice

and this shape should maybe start to

look a little familiar if you played - I dunno -

legend of zelda before. It's th-also the Triforce

but more importantly it's sierpinski's triangle

so for those who don't recognize sierpinski's triangle

I start with an equilateral triangle

I inscribe another one upside down and

then i can just keep doing this on all

of the upward facing triangles as many

times as I want

so yeah that's can just go on for a long time

(Brady) Cool, yeah?

And it turns out that the

way that the evens and odds show up

in Pascal's triangle when I convert them

to mod 2, I can make Pascal triangle as

big as I want and I'm just going to keep

getting iterations of sierpinski's triangle

so that's really cool

(Brady) That's awesome!

That's really cool by itself but it gets cooler using the mod 2 construction

There's something else we can do in mod two though

Um, in particular we can write down the same

things but let's write them now as - we've got ones and zeroes, right?

And for most people ones and zeros kinda remind them of binary

So, let's just see what happens when we transcribe these from binary

You know one binary - 1 and then 11 in binary is 3

101 is 5

111 is 15 and then the next one, 10001 is 17

Now these may not immediately appear pretty cool

but if I kept going the next one is 51

The next one is 85

After that is 255 and here I'm gonna stop and I'm

going to look at what these numbers are

three is prime

Five is prime

15 is 5 times 3

17 is prime

51 is 17 times 3

85 is 17 x 5

And 255 is 17 x 15

So you'll notice that every time I get a prime i go back and

just multiply by all the things that came before it

and I can generate the next you know n minus 1 entries

So the primes that we have here actually at least the ones

that I'm - that we're certain of - are actually Fermat (pronounced fer-mat) primes

Or Fermat (pronounced fer-mot) primes rather

Which means that they can be written of the form 2 to the 2 to the n plus 1

Somehow every time I talk to a mathematician-

even 10 minutes ago when i asked a different mathematician for an idea

he told me something I've never heard before

So even if maybe somewhere out in the world all of the things you could

do with this triangle are done

I don't know if there's any one person

who knows them all and maybe there's

only one person who knows one of them

and we just have to keep exploring the

triangle and seeing if we can

collaboratively explain everything

inside of it but it's an infinite

triangle right so maybe there are things

on the hundredth line down that we

haven't explored yet

I feel like it wouldn't be a numberphile video if somehow

the Fibonacci sequence didn't show up

So i'm going to show you where the Fibonacci sequence is in the Pascal's triangle because...

this is a numberphile video

yeah so what we're going to do is we're

gonna add up the shallow diagonals

So what that means here is the triangle kind of

inscribed diagonals already right like I

could look at this one and just kind of

follow this but a shallow diagonal means that

i'm going to kind of go off at a

slightly higher angle that I would have

if I was just following the way I drew

the triangle so for example the first