..7. And I've got some more numbers,
let's do those. 19 and let's do 13... So
what I've got here is a honeycomb, right,
of hexagons. But, have a look, it's a magic
hexagon; so if we add up the rows,
they all add up to the same number - 38. So
let's do that if; I say 9 plus 11 is 20
plus 18 is 38. Or if I add up a longer
one, let's do a row of four like this one here,
I could have 12 plus 2 is 14, plus 7 is 21,
plus 17 is 38. Or I do a big one here,
it's got five right down the middle; 18
plus one is 19, plus five is 24, plus 4
again is 28, plus 10 again is 38. So
it's called a magic hexagon, every row
adds up to 38, it's like a magic square
where the rows and the columns and the
diagonals all add up to the same number.
Which is, you know, nice that's kind of nice.
Now, the thing is, this is the magic
hexagon. This is the only magic hexagon.
Okay, so magic squares, there are many
types of magic squares but this is the
only magic hexagon. You can rotate it and
reflect it but apart from that it is
unique. So we could do a bigger size; this
has size three. So so we've got three
layers; we've got one in the middle, then
we've got a layer around it and then a
third layer. We could have a magic
hexagon- or we could try and get, a magic
hexagon of size 4, size 5 or something
else - they don't exist. This is the only
magic hexagon, it is unique, it is special,
and I hope to show you why. So what I
want to do first is I want to have a
look at magic squares just for a second,
Brady, if you want to sort of edit and cut
to point- So, you could have a
magic square like that, it's 3x3. So every
row and every column should add up to
the same number - 15. Let's do this one
here, 7 plus 8 yeah, so this is equal
to 15. If I did this column, 15. If I did this
column - 15. And so should the rows and the
diagonals too.
You can do different sizes, so let's do a
quick look at on a different magic
square. This time all the rows and
columns and diagonals add up to 34,
all of them...yeah,
equal to 34. Let's do a 5x5; and
again I think the magic number is 65 for
this one. So all the columns and rows
should add up to 65, all right. How do
you work out the magic number? What we're
going to do is add up all the numbers in
the square, all right? So it's gonna be
1 plus 2 plus 3 plus 4, right. So we
get 1 plus 2 plus 3 plus- and so
on. And that will go up to- well the
biggest num- if it was nxn it
goes up to n squared. So we're going to add up all
these numbers together; so they go all
the way up to n squared, so n squared
would be 9 here. It would be 16 here and
it's 25 here. So we're going to add them all
together; there's a nice little trick
to do that which is very simple to
explain. If you add the first number and
the last number together what do I get? I
get n squared plus 1. If I add the second
number and the second to last you get
the same thing, n squared plus 1. You can
actually add them up in pairs, you always
get the same value. How many pairs are
there? It's n squared divided by 2 pairs.
So that's how much we get. But if we look
at this in a different way, we're going
to come at this from a different angle,
what about this? We know each row is
equal to the magic number, right, we'll
call it M. So this is equal to M. The
second row is equal to M, the third row
is equal to M; every row is equal to the
magic number. How many rows are there?
Well there's n of them. So if you add all
this together, another way of thinking
about that is it's n lots of the magic
number. And that means we can work out
what the magic number is; if I do that
using this that must mean the magic
number is - we just divide look - it's going
to be n divided by 2 times n squared
plus 1. So we learn how to do the magic
number, look let's do it! And let's have
this one, n equals 3,
the magic number is: 9, 3 squared equals 9,
plus 1 is 10 and times by 3 is 30, divided
by 2 is 15 - that's how you work out the
magic number, yeah. So a kind of nice little
thing about maths there is looking at the
same problem in two different ways to
come up with an answer, that's kind of nice.
Now the reason I wanted to show that is
that's what we're going to do with the magic
hexagon, it's going to help us a lot, right.
So with the magic hexagon we're going to
add up all the numbers. So it's going to
be 1 plus 2 plus 3 plus...now
what's the last one here? It's not as
simple as n squared, it's actually three
n, n minus 1, plus 1. That's kind
of come out of nowhere; I can actually
show you why that's true, there's a nice
way to show you why that's true, which is
why I got the prop here. So I'm going to
use the colours on this to show you why
that's true. I'm going to destroy this
now, let's do this. So if I rearrange-
So I've rearranged the colours and I've made
the honeycomb again, but notice we've got
like three rectangles here and each
rectangle is going to be n long and n
minus 1 on the short side. So long side
is n, the short side is n minus 1. So each
rectangle is n times n minus 1; there's
three of them so it's three lots of that
and then there's one in the middle,
that's what you get when you make a honeycomb.
So here that's why the final number,
that's how many hexagons there are in a
honeycomb: 3 lots of n(n minus 1)
plus 1 in the centre; alright. So we can
do- we can do this, we can add this
together, there's a good trick for
that. It's the same thing, you do it pair
by pair. So if we add 1 to the last one
it's 3n(n minus 1) plus 2
because I've added those together. And
how many pairs do I have? It's this thing
plus 1 divided by 2. Which
is not so nice, it's not as nice as the
one I did with the magic squares,
that's why I wanted to show you the
magic squares first. But it's the same
idea, completely the same thing. On the
other hand, with the magic hexagon we
know that each row is equal to the magic
number. So how many rows do we have? We
have 2n minus 1 rows and they're all
equal to the magic number, M. All right, so
we solved that, we can work out what the
magic number is. I'll skip to the end and
I'll tell you what it is. 32 lots times
the magic number is equal to something
not particularly nice but don't worry
about it too much, it's not a very nice
formula. 72 n cubed, minus 108 n squared,
plus 90 n, minus 27. Here's the important
bit: plus at the end, a little fraction,
it's 5 divided by 2 n minus 1. Right.
(Brady: That's a beast!) -It's a nasty looking
thing; actually most of it you don't
have to worry about. This is the
important thing to notice: the left-hand
side is a whole number. All this nasty
stuff here is going to be a whole number
as well. So the important observation is
this thing here at the very end has to
be a whole number, otherwise it won't
work, so this has to be a whole number.
When is that last thing there a whole
number? It has to be n equals 1, or n
equals 3. when n equals 1 you've got 5
divided by 1, which is a whole number. And
when n equals 3 you've got 5 divided
by 6 minus 1 so 5 divided by 5, is equal
to 1. And that's the only size magic
hexagon you can have. So n equals 1; I
didn't mention that, I kind of was a bit
sneaky about that. So if you want to know
the magic hexagon of size 1, there it is.
Let's do a- let's just check. So if we add up all the
rows we get 1; if we add up the columns
it's 1; and just check the diagonals
it's 1, yeah. So that
works, so that's our magic hexagon.
Ignoring that one we've got a magic
hexagon of size 3. What is the magic
number, by the way? If you put it into
this formula, if you had n equals 3,
you got the magic number I told you at
the start, it was 38. A magic hexagon, if
it exists, has to be of size 3. So
it's three layers: you've got the one in the centre,
you've got your second layer, and your
third layer. We could fill it- maybe
we could fill it in different ways, so
that we could get different magic
hexagons? And it turns out you can't.
There is only one way to fill it; and let
me show you how to work out. So first of
all, if we add up all the numbers
together - so that's adding up 1 to
19 - it's 190. Or, and I'm going to use
the colours here, I could add all the
pinks together; all the pink hexagons. So
it's going to be this one here on the
corners, all these pinks together, we're
going to call that P for pink. By the way,
that centre one we'll not call that pink,
I think that's lilac, so that's not a
pink one. Let's add up all the blues
together as well; so we add all the blues
together we'll call that B for blue.
We'll add up all the yellows together,
there's all the yellows. We'll call that
Y for yellows. And then we add in the
centre one; what d'you want to call it?
L for lilac or? - (C for centre?)
C for centre, yeah, I'm
happy with that. C for centre. We add them
all together, that's 190. I hope you agree.
Right, here's another way of looking at
it; this time I'm going to add all the
rows of three hexagons. So like this row
together, and I add this row together, and
this row together; there's actually six
of those. And if I add them together
a row a three is equal to the magic
number 38, I know that. So let's do that.
But notice, by the way, if I add this
together and I add this row, this is
actually getting counted twice. So let's
do that. If I add all the rows of three,
let's write that down,
rows of three. All the blues get added
together, but do notice that pinks get
added twice. So we've got two lots of the pinks
plus the blues. There's six of them so
it's six lots of the magic number, 38. If
we do the same thing for the rows of
four and the rows of five, and then we're
nearly done. Let's do the rows of
four; okay this is a row of four. So we are
adding the blues together, we've got some
yellows here; and that's going to be
equal to 38. There's actually six of those
as well, rows of four, there's six of those.
And I think for this, yeah, each blue is
going to get counted twice and each
yellow gets counted twice. So we've got two
lots of the blues and two lots of the
yellows- and there's six of these rows, so six lots
the magic number 38. And for the rows of
five, they go straight through the centre
like that, 1- actually three of them: 1,
2 and 3. So only three of those. The
pinks get added once each time; the
yellows get added once; the centre one
actually gets added three times, because
you go through the centre three times
over. So you're going to get pinks plus
the yellows plus 3 times the centre; and
that's three lots of the magic number 38.
We've got a few equations there. This
is where I say 'and skip to the end'. I
won't go through all the working but
you've got enough information there now
to work out that the numbers that you
have to put in the magic hexagon have to
be in this order. There is still some
work to do, it's not completely obvious.
You do have to try a few options and
you'll notice that there's only a few
options, and then in the end you realise
there can be only one option. And then
that's the magic hexagon. That is The
magic hexagon.
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