MATT PARKER: See, most people
forgot about the Rubik's Cube
back in the '80s, so it had
its heyday and then
disappeared.
But mathematicians remain
obsessed with it.
KATIE STECKLES: I mean, people
probably are aware that there
is some math involved
in Rubik's Cube.
JAMES GRIME: The reason I not
interested in the Rubik's Cube
was I wanted to use as
a teaching tool.
In my day job I study something
called group theory.
KATIE STECKLES: If you go to
any kind of Wikipedia page
about anything to do with group
theory, there will be,
in the top corner, a picture
of a Rubik's Cube.
And the reason for that is that
it has got quite a nice
example of a group associated
with it.
And the group is the number of
different combinations that
there are on the Rubik's Cube.
It's all the different
ways that you can
scramble up a cube.
JAMES GRIME: So for example,
if you had a square, and I
turn it a quarter turn, you
still get a square, don't you?
And even if I take a square, and
I flip it over, you still
get square, right?
So it has that symmetry.
Now, this is what I study.
So I started to learn how the
Rubik's Cube worked, because
the mathematics is the same.
So if I turn this here, the
right hand side a quarter
turn, I can do that.
I do it twice, you get a half
turn. you do it get three
times, that's equal to
a quarter turn in
the opposite direction.
If you do it four times, you get
back to where you started.
LUCAS GARRON: Hey,
Numberphile.
My name is Lucas, and
I'm a speed cuber.
That was about 9.6 seconds,
although I got
a little bit lucky.
MATT PARKER: Other things
do have permutations and
symmetry, but there's just
something nice and tactile,
and you can grab it,
and you can see
everything as it happens.
So there are a lot of other
things with symmetric, but
nothing quite as tangible
as a Rubik's Cube.
JAMES GRIME: The mathematics
of the Rubik's Cube is rock
hard, but we'll try and do
a few things about it.
We'll talk about the
combinations, first of all.
How many ways can you mix
up the Rubik's Cube?
Let's do that.
ANKUR SADHOO: This is my Rubik's
Cube [INAUDIBLE] for
Numberphile.
Here goes.
MATT PARKER: I'm adequate at the
Rubik's cube, so I solve
it on average, about a
minute and a half.
On a good day, if you give me a
mixed a Rubik's Cube, I can
do it in a minute.
I will almost always get
it done in two minutes.
Personally, my goal, I like to
be able to solve it talking to
other people.
So it's my goal to solve the
Rubik's Cube while having a
conversation, and I can do that
between two and three
minutes, give or take.
BRADY HARAN: You've set
yourself up, you know?
You know what I'm going to do.
Give it here.
I'm going to mix it up.
JAMES GRIME: So imagine I can
take these cubes off, these
little sub cubes here.
These, what are they called?
Cubelettes?
Cubits?
We can take those off, and just
arrange them all on the
table, and we'll put it
back together again.
And we'll do the corners first,
let's say, for the
first corner I'll eight choices
of corner cubes.
I'll have eight corner
cubes to put there.
For the second corner, I'll have
seven corner cubes left.
For the next one, I'll have
six corner cubes left.
Then five, then four, then
three, then two, then one.
Now, you multiply
that together.
times 2 times 1.
That's called 8 factorial,
which is an 8 with an
exclamation mark after it.
BRADY HARAN: So I've mixed
up this cube for Matt.
He told me that he can solve it
while talking, so I'm going
to give it to him while I talk
to him about the Rubik's Cube.
There you go.
MATT PARKER: Right.
So I mean, me and
my big mouth.
I did admit that I can
solve it while
talking to someone else.
So I believe, Brady, you're
going to attempt to engage me
in conversation.
JAMES GRIME: We can do the
edges in the same way.
There are 12 edges.
We're going to put the edges in,
and there are 12 factorial
ways to arrange the edges.
So we've got 8 factorial
and 12 factorial.
Now for each corner, there are
three ways to orientate it.
So there are three colors there
on each corner, and you
can, you know, if you have
free choice, you
can move them around.
There are three ways to
do that for each one.
Three choices for that, three
choices for that, three
choices for that, three choices
for that, three
choices for that, three choices
for that, three
choices for that, three choices
for that, and you
times them.
What I've done is that's
Times 3 times 3 times
MATT PARKER: When you are mixing
this up, there's a
number of different ways you
could have mixed this up
before you give it to me when
I start solving it.
So there's only one arrangement
of all these
pieces which counts as a
solved Rubik's Cube.
JAMES GRIME: And now finally,
let's look at the edges.
And each one has two
ways to arrange it.
And so there's two ways,
and two ways, and two
ways, and two ways.
And you times them together.
And because there's 12 edges
altogether, there are 2
to the power 12.
ADAM HADASH: 3, 2, 1.
MATT PARKER: These two corners
are where I want them, but
these two are the wrong
way around.
And so I know if I do that,
that, that, that, that, that,
and then double twist,
that will have
swapped those two pieces.
This is the total number of
ways if I could take the
pieces off and then put
them back on again.
Now, how many ways is
that all together?
It's a very large number,
so we work it out.
Well, 5.2 times 10 to the 20.
It has 20 zeroes after it.
But this is not the
right answer yet.
BRADY HARAN: Matt, are you doing
maths here, or are you
just kind of, like a monkey
having learned
something by rote?
MATT PARKER: A combination
of the two.
I've memorized, oh, yes, it's
solved, by the way.
I've memorized a lot of
combinations of twists.
That's pure rote.
And then what I'm doing is I've
just got to spot what I
need to do, and then do
it at the same time.
And so my method is very lazy.
I'm spotting one bit I want to
do at a time and doing it.
The more advanced people will
spot two different things they
want to do the same time and
then do those together.
ED BANKS: I love Numberphile,
it's great.
And I saw the post about you're
making a video on
Rubik's Cubes, so I'm just
going to mess this up
and then solve it.
JAMES GRIME: Not all
permutations are possible.
I can't reach every combination
by these twists.
In fact, if I take one of those
edges off and turn it
around, and then start twisting
it, I get a whole new
set of combinations that you
won't usually get to on a
Rubik's Cube.
It's called an orbit.
A whole new orbit.
A whole new universe
of combinations.
So if I do that with one
of the edges, you get
this whole new set.
If I do that with the corner,
there are three ways to turn
the corner.
And again, for each one of
those, you get a whole new
universe of combinations.
And finally, if I take
two pieces, like two
edges or two corners.
If I took them off and swapped
them over, you again get a
whole new universe that you
can't really reach on a
standard Rubik's Cube.
Now, because of these
universes,
there's 12 of these universes.
So I'm going to divide
that number by 12.
Some people call that 43
quintillion, but I don't like
things like that.
It's a massive number, though.
It's a massive number
of combinations.
MATT PARKER: If you want the
exact number of different ways
you can mix up a cube, it's
The number of possible ways.
And to get a sense of how big
that number is, there's not
enough raw material in
the UK to make 43.25
billion billion cubes.
So you couldn't make enough
to put one in every single
possible arrangement.
In fact, there's not enough of
the UK to make enough Rubik's
Cubes to have one in every
single arrangement.
If you started just breaking
off anything around you, so
just grab anything and break
it off into a 57 millimeter
cube, the exact size
of a Rubik's Cube.
And then if you panted those,
all the different possible
combinations, by the time you
had made all 43.25 billion
billion, you would have
a UK-shaped hole
over 20 miles deep.
It's just, it's a mind numbing
number of possible
arrangements.
LEONARD KOCH: And there we go.
Brady, this is your solved
Rubik's Cube.
JAMES GRIME: So this was a
big crazy in the 1980s.
Big craze.
In the 1880s, exactly 100 years
earlier, there was a
similar puzzle that was
also a big craze.
It was this.
This is called the 15 puzzle.
It's got the numbers 1-15,
and it's like the
Rubik's Cube, though.
You just move the
pieces around.
The Rubik's Cube is
essentially a
Now, there was an American
puzzle expert called Sam Loyd.
He would sell this puzzle.
But he would set
it very meanly.
He would set it with the 14
and the 15 swapped over.
This was how he would sell it.
And he would say, can
you solve it?
And he offered a price of $1,000
to anyone who could
solve this puzzle.
What he knew, though, was that
it was like the Rubik's Cube.
If you take one of those pieces
off and turn it around,
you get a whole new universe.
You can't get back to
the solved Rubik's
Cube if you do that.
And that's what Sam
Loyd was doing.
But that is essentially a
simpler version of this craze
BRADY HARAN: Can I just say, for
the record, Matt actually
wrote the number on
the board there.
But you do that without--
MATT PARKER: I did it without,
yeah, yeah, yeah.
BRADY HARAN: Let the
record show, he did
that without looking.
MATT PARKER: Just knowing
it was on the board
was enough for me.
KATIE STECKLES: I guess
it comes in the
official Rubik's packaging.
And you know, this is the
original design that they had
when they invented it in--