Brown's Criterion - Numberphile

JAMES GRIME: We're going to try out a little mathematical
trick with you today, Brady.
It's nothing special.
I mean, it's nothing new.
But we'll try it out.
I'm going to ask you to pick a number between 1 and 30.
Don't say it out loud, but pick a number
between 1 and 30.
Have you done that?
OK.
BRADY HARAN: So we'll put it on the screen, right?
JAMES GRIME: Yes.
BRADY HARAN: There it is.
JAMES GRIME: Yeah, great.
So we put it on the screen, and the guys at
home can play along.
OK, so we're going to use Brady's number.
I don't know what his number is at the moment, but I'm
going to hold up these cards.
And they have numbers on them, of course.
If you see your number, let me know.
Say yes.
And if you don't see your number, say no.
Say pass, something like that.
OK, so let me know.
BRADY HARAN: Yes
JAMES GRIME: Yes.
Do you see your number?
BRADY HARAN: No.
JAMES GRIME: No.
Do you see your number?
BRADY HARAN: No.
JAMES GRIME: No?
Do you see your number?
BRADY HARAN: No.
JAMES GRIME: No.
Do you see your number?
BRADY HARAN: Yes.
JAMES GRIME: Yes.
Do you see your number?
BRADY HARAN: Yes.
JAMES GRIME: Yes.
Are we back at the beginning?
BRADY HARAN: Yes.
JAMES GRIME: Yes.
OK, so in that case, we're back at the beginning.
I can stop.
So I know that your number was 17, is that right?
BRADY HARAN: Yes.
JAMES GRIME: Yes, right.
So now, how did we do that?
It is an old trick, but if you haven't seen it before, what
you do is I know which was Brady's number because I was
adding up the numbers in the top left corner.
And now look at the top left corners.
I actually do it this way.
If I spread out the cards, they all have different
numbers on them.
But look at the top left corner.
Now if you notice, the top left corner are
all powers of 2.
Starting from 1, they're powers of 2.
So 1, 2, 4, 8, and 16.
Though 1 is a power of 2, it's 2 to the power of 0.
I knew what Brady's number was going to be because I added on
these left numbers for each one he said yes for.
So Brady said yes to this one.
The red one.
And he said yes to this one, the blue one.
And he said no to the others.
And I knew that I could add those two numbers together,
the 1 and the 16.
And they added Brady's secret number.
Now the point is, every number can be written as a sum of
powers of two.
That's the point.
So whatever number you pick, these have been designed so
that if you add up the powers of two, it will give you your
secret number.
Let me show you what I mean.
If I write down the powers of two, so I just
start with a 1, though.
I am going to write it right to left.
So 2, 4, 8, 16.
OK.
And we'll pick a number like, I don't know, 20.
Now, which of these two do I need to use to make 20?
Which do I need to add on?
I'm going to use the 16 and the 4.
I've written a 1 here to indicate I'm using
the 16 and the 4.
Don't use the rest.
I'm going to write a 0 to show that I don't use the rest.
Now, this is binary.
This is the famous binary, which computers use.
It's the count.
I'm going to give you a twist to this.
I'm going to show you a new way to do this trick.
A less well known way to do this trick.
Let's try it out.
I've got a new set of cards here.
This is a less well known version of the trick.
Same trick.
Shall we do it in the same way?
BRADY HARAN: Yeayh.
JAMES GRIME: All right, so here's my cards again.
You're going to pick a number, of
course, keep it to yourself.
But we can put it on the screen, everyone can play.
Have we done that?
BRADY HARAN: Yeah.
JAMES GRIME: OK.
Right.
So hopefully we'll do this again.
OK, tell me yes and no when you have your number.
OK, yes?
BRADY HARAN: No.
JAMES GRIME: Sorry, no for that one.
BRADY HARAN: No.
JAMES GRIME: No for that one.
BRADY HARAN: No.
JAMES GRIME: No for that one.
BRADY HARAN: No.
JAMES GRIME: No for that one.
BRADY HARAN: Yes.
JAMES GRIME: Yes for that one.
BRADY HARAN: No.
JAMES GRIME: No for that one.
BRADY HARAN: No.
JAMES GRIME: No for that one.
Did you pick 8?
BRADY HARAN: Yes.
JAMES GRIME: He picked 8.
He picked this one on his own.
OK.
Now, did you notice what the top left
numbers were this time?
They weren't the powers of 2 this time.
Did you notice what they were?
They were Fibonacci numbers.
I can tell you that we can do the same trick
using Fibonacci numbers.
What I mean is, we can write every number as a sum of
Fibonacci numbers as well, like we did with binary.
Let me try it out.
So let's write out the Fibonacci numbers.
There's 1.
There's 2.
There is 3.
And there's 8.
No, there's 5.
Get that right, James.
And then there's 8.
Then there's 13.
And 21?
Shall I go up to 21?
There we go.
Right.
Now, let's pick a number.
What shall we pick?
BRADY HARAN: 19?
JAMES GRIME: 19.
So I'm going to pick the number 19.
What do I need to make 19?
Let's use the 13, because that's quite a big number.
And wow, 6 now.
So a 5 and a 1.
So I've written 1's to indicate what to use.
I'm going to write 0's everywhere else.
And again, it looks like a binary sum.
It's looks like my binary is a binary sum.
It uses 1's and 0's, but this time the 1's indicate which
Fibonacci number to add together.
Using this idea, you can make any number you want using
Fibonacci numbers.
And if you don't use consecutive Fibonacci numbers,
so here I used a 5.
Now I could have used, instead of 5, I could
have done 2 and 3.
Well, if I don't use consecutive ones, it's unique.
So there's only one way to do this.
So you can do this trick with Fibonacci, but now I'm going
to add an extra twist to this.
Because although I don't have the cards for you, I'm afraid,
you can do this trick again with prime numbers.
Every number can be written as a sum of
prime numbers, as well.

So I'll show you how I made those cards, then.
And then everyone can make them.
Because you can make them at home.
So I'll use the powers of 2 card.
I'll show you how to make them.
And then if you learn this, you can make your Fibonacci
ones as well.
I started with the powers of 2 in the top left corner.
Now I said, OK, I've got 1.
I've got 2.
3.
Well, to make three, it has to be on this card and on this
card, because those numbers add up to 3.
So you write down your 3 at that.
4 is here.
Great.
5.
Well, how do I make 5?
1 and 4?
So I have to write the 5 on the 1 and the 4 card.
Then the 6, well, it's 4 and 2.
So I have to write the 6 here and here.
So let's go back to Brady's 17.
As we know that 17 is, well, think about the powers of 2.
16 plus 1 would do it.
And that's why I wrote 17 here on the 16 card, and I wrote 17
here on 1 card.
That's how I made them.
Now, that fact that you can do it with Fibonacci numbers is
not obvious.
It's called Zeckendorf's theorem.
That's quite a cool name.
But it's actually misnamed.
The first guy who came up with this was called Lekekeke, and
that was his name.
That's fun that everyone can enjoy.
Lekekeke.
The prime version of it is also another surprise.
And altogether, if you want to know how to do it in general,
it's called Brown's Criterion.
These aren't the only sequences for
which this works for.
But they're the pretty cool ones.