All right this is pretty exciting - we're
going to look at order out of chaos.
We're not just going to pick numbers out
of the air, I've actually organised
something here for us - I've got some Uno
playing cards. Okay, I've got nine and
they're all different which is important.
So first of all what I'm going to do is
I'm going to give it a very light shuffle.
Let's put it out here:
so we've got 9, 6, 4, 5, 1,
7, 2, 3 and 8. Okay so let's
look at these, so this is these are nine
cards in a particular order, particular
permutation, let's see if we can find
four cards that are in ascending order.
So ascending order; so we can see we've
got a 4 here, we've got a 5, we've
got a 7, and we've got an 8 okay?
So they didn't need to be right next to
one another but we found 4, 5, 7, 8.
Okay how about four cards in descending order?
All right so 9 is a good start
because that's as high as you can get so
we've got 9, 6, 5 and 1. Or we
could have had 2 or we could of had 3.
So look at this here. Now Brady let's
look at this: we've got nine cards so
that means that we have nine factorial
different ways to organise them, which is
362880 permutations. But there is 1764
ways that you can organise these nine
cards such that there are no four
numbers in ascending order or descending
order. And I know you want what you're
thinking: why is that a square number?
Yeah? - (Brady: I wasn't, but I am now.)
Because it is, it's 42 squared;
and the reason why is because it
actually follows this marvellous little
formula here: 9 factorial on 1 to the
1 by 2 to the 2 by 3 to the 3 by 4 to
the 2 by 5 to the 1 all squared. But the
point is, this is some serious maths. So
1764 divided by 362880 is
approximately equal to 0.5%. So at
chance if you just take it at random
there's a 0.5% chance that
you can actually put this in an order so
there's no four increasing or decreasing.
Okay, so Brady would you like to make
some money? I'm willing to put down 20
pounds, okay? 20 pounds against a
probability of 0.5% - at random but
I think you're pretty smart. So if you're
able to organise this in a way that
there's no four increasing or decreasing,
that's yours. - (Gosh, well how long have I got?)
You've got as long as you like!
Okay my 20 pounds is looking safe.
Do you want to have another go? It'll be a
smaller denomination. All right here we go,
here we go. A little bit of- that's the
only time I'm interested in gambling,
actually just testing you. Okay so this
is it, this is the way we do it. What you
do is you split it up into groups of
three, and you go like this: you go duh duh,
duh duh, duh duh, okay? So looking at it from left to
right what I've done is that you can see
it goes 3, 2, 1; 6, 5, 4;
9, 8, 7 - so the maximum of descending
is 3, 3, 3 and then if you want to go for
ascending all right what you do is you
basically you can only take one from
each of these groups. So it doesn't
matter which one, so you could go like
2, 5, 8 for example. Or you could pick any
one so you could go 1, 4, 7 right?
But if we add one more card, okay,
the beautiful zero, all bets are off
literally. And I'll tell you why; it's
because if you have ten cards what
happens is if I put my zero here
wherever I put it I'm always going to
create four ascending or four descending
right? So if I put my zero here you can see
it goes 0, 1, 4, 7 in that
particular permutation. If I put it here
it doesn't change, if I put it here it
doesn't change, okay? If I put it here
alright I no longer can do that here but
now look what I've got: 3, 2, 1, 0.
And that doesn't change if I put it
here, here, here, here, or here okay? Now,
that is not only just for that
particular permutation, it so happens
that for ten factorial permutations,
which is 3628800
there are now zero ways to find no
four numbers ascending or descending
order.
No, all right? Which is a strong result,
which means we've got order out of chaos,
why should that be the case?
There's a second video in which Simon explains all
the mathematics behind what's going on
in this video. If you'd like to see it
there'll be a link in the description
and on the screen; it's about eight or
nine minutes long so I thought I'd do it
separately and keep this video a bit
shorter. If you'd like to see my really
embarrassing attempt at arranging the
cards which cost me twenty pounds I will
make that available but there's going to be
a bit of a catch, have a look in the
video description and I'll explain how
you can see it.