---
title: 'Entirely Ridiculously Big Numbers - Numberphile'
source: 'https://youtube.com/watch?v=SAtFZPtbMpI'
video_id: 'SAtFZPtbMpI'
date: 2026-06-28
duration_sec: 1166
---

# Entirely Ridiculously Big Numbers - Numberphile

> Source: [Entirely Ridiculously Big Numbers - Numberphile](https://youtube.com/watch?v=SAtFZPtbMpI)

## Summary

This video explores the mind-boggling number of possible states (scrambles) for large Rubik's Cubes, starting with a 2x2 cube and progressing to a 10x10 cube. The presenters, Richard and Brady, calculate the exact number of scrambles for each cube size, showing how the numbers become astronomically large, far exceeding the number of atoms in the universe. The video concludes with a general formula for calculating the number of states for any even-sized cube.

### Key Points

- **Standard 3x3 Cube's Scrambles** [0:44] — The standard 3x3 Rubik's Cube has 43 quintillion (4.3 × 10^19) possible scrambles.
- **Calculating 2x2 Cube States** [2:08] — For a 2x2 cube (8 corner cubelets), the number of states is calculated as 8! × 3^7 / 24, which is about 3.6 million.
- **4x4 Cube States Breakdown** [5:57] — The 4x4 cube has three types of pieces: corners (same calculation as 2x2), edges (24 edge pieces with 24! arrangements), and centers (24 center pieces with 24! / (4!^6) arrangements). Corners: C = 8! × 3^7 ≈ 88 million.
- **4x4 Cube Total States** [10:36] — Total states for a 4x4 cube = C × E × K / 24 ≈ 7 × 10^45, which is far larger than the 3x3's 4.3 × 10^19.
- **6x6 Cube Complexity** [11:11] — The 6x6 cube introduces two types of edges (central edges and wing edges) and four types of centers. Total states = C × E^2 × K^4 / 24 ≈ 1.6 × 10^116.
- **General Formula for Even Cubes (2n x 2n)** [14:44] — For a 2n x 2n cube: states = C × E^(n-1) × K^((n-1)^2) / 24. This shows that the number of states grows exponentially in n squared.
- **10x10 Cube's Astronomical Number** [17:54] — For a 10x10 cube (n=5), number of states ≈ 10^349. This is ridiculously large – even counting every atom in the universe four times over still gives a smaller number.

### Conclusion

The number of possible scrambles for large Rubik's Cubes grows incredibly fast, exceeding all intuition and universe-scale comparisons for a 10x10 cube. The general formula also applies to odd-sized cubes, which are covered in an extended video on Numberphile 2.

## Transcript

So, this is an entirely ridiculously
ridiculously big number. You've spoken
to me about bigger numbers. I have
spoken to you about bigger numbers, but
I've never held one in my hand.
I thought we might talk about Rubik's
Cubes today, but not about this Rubik's
Cube. Instead, how about
that Rubik's Cube? Oh, that's got that's
got more cubes in it. Yeah, that's What
do you call the the elements of a
So, the opinions are divided. I think
what most people nowadays call it is a
cubie, but I'm a bit old school and I
call it a cubelet.
Yeah. And that's got more cubelets. It's
got more cubelets. You you can see it's
six cubelets wide, while your your your
classic is three cubelets wide, right?
So, you might call this a six cube. So,
the standard
three cube has got an awful lot of
scrambles, lots of different
possibilities. Do you remember how many
different scrambles, Brady? I believe 10
seconds ago, before we started the
filming, you told me it was 43
quintillion. 43 quintillion on your the
standard Rubik's Cube.
>> That's Surely it's not a rounded all
zeros. Is there like a more exact number
than that?
>> more exact, yeah. What's the exact
number? Uh I don't remember, I'm afraid.
I'll put it on the screen. There it is.
And what does that number refer to? So,
it's the number of different possible
scrambles or different possible states
it could be, different ways it can be
mixed up, including of course the
unscrambled solved state. Is this one of
those situations where there are like,
you know, reflections and mirrors and
the same ones exist multiple times and
you have to do that or Yes,
that's right. So, we want to make sure
when we're counting different states,
we're not overcounting by
counting um
ones which look identical
um to each other. In fact, this is going
to be more important than with this,
because you can see it's got all these
center pieces.
And if they get jumbled up between
themselves, it doesn't matter, right?
It's it's still going to be the same
state. So, we won't have to worry about
that yeah.
>> So, if you swapped two of those whites
with each other, I wouldn't know.
Exactly. So, we can't we should count
them as that as being the same the same
state. But I thought it might be fun to
try and work out how many different
states there are for a big cube like
this. Okay, so I'm actually having said
that, I think it's best to start with a
slightly smaller cube. So, I'm actually
going to start with this cube here, two
cube. And it's going to be actually
easier to do the calculation for cubes
with even
width. The question is, how many
different states are there for this? So,
this is a 2 by 2 cube. Let's scramble it
up a little bit.
You can see there are eight
cubelets right?
And there are eight positions that a
cubelet could be in,
right? So,
you can put any of the
cubelets in any of the positions. So,
let's just pick a slot,
maybe this one here, that position,
then there's eight choices of which
cubelet to put it in. So, we can write
that down, eight. And then once we've
got that one,
for the next slot, there's seven
choices, right? So, it's times seven.
And then for the next one, there's six
and so on. So, you can see how it's
going to go. The number of choices is 8
* 7 * 6 * and so on down to 1, which is
usually written as 8 factorial with an
exclamation mark. That's what that
exclamation mark means, okay? So, that's
the start, but that's not the whole
answer, because once we've decided maybe
we put this white, green, and red
uh cube cubelets into this slot,
it can actually be one of three ways
round. So, if you just look at
concentrate on that cubelet,
if I do that, it's in the same position
again, but it's in a different
it's been rotated, right? And we can do
it again, okay? So, actually there's
um
for each
each of these cubelets in each of these
positions, it can be one of three ways
round, right? We need to multiply by
three for each of these. So, you think
it's going to be times three to the
eight, right? Because there's eight of
them. But actually it's not times three
to the eight, because it turns out that
once seven of them have been fixed,
the the the final one is automatically
fixed. You can't do anything more with
it. So, it's actually times three to the
seven. So, that's the the number of
states for the two cube. Well, maybe not
quite, because this point we have to
rewind and think about
every cube's favorite thing, the one
cube. And the question is, how many
states does the one cube have, right?
And I think the
what feels like the natural answer is
it's got one state, yeah? But of course,
I mean you could argue that it's got 24
states, because you think, okay, which
which face will I put on the top?
I'll put the the white face on the top.
Um and then which face should I put on
the front? So, it could be the orange
one or the green one and so
There's four choices there.
So, you could argue that the one cube
has got 24 states, 6 * 4, yeah? But we
probably don't want that. We probably
want to count them all as being the
same, which means we need to Likewise
for this, a lot of these states are
going to be the same, but just rotations
without actually doing any Right, so let
me call this number we've got so far C,
okay? I'll explain why it's called C in
a moment. That's around about 88
million. But then to get the number of
states for the two cube, we need to
divide that by 24, okay? So, the two
cube is C divided by 24. The answer is
that's about 3.6 million, just for this
little cube, okay? Wow. When we think
about uh the bigger cubes, so I mean the
one I'll do next is the four cube,
because it's easier to work with um even
sides. With a two cube, all the cubelets
are basically equivalent, right? Any can
go in the position of any other. That's
not the case with here. Um in fact,
there's three different types of
cubelets here. You've got the corners,
and you've got the edges here,
and then you've got the centers. You've
got three different types. And to work
out the number of states for the four
cube, what we need to do is work out the
number of
possible arrangements of the corners and
the number of possible arrangements of
the edges and the number of possible
arrangements of the centers and then
multiply them together. That's that's
the plan. And I want to start with the
two cube, because
the corners of the four cube in or
indeed the corners of any cube
are
basically exactly the same argument as
for the two cube, because of this is
these are corners, right? The first
thing for the four cube, we need C
again, that same number, right? So, the
next thing to think about is the
edges. So, for this cube, the sort of
overall shape the the the cubic shape
has got 12 edges and along each edge
there's two cubelets, right? So, we've
got 24 edge pieces. So, we've got 24 in
total.
And you can put one in any of the slots.
So, we've got 24 slots. So, for the
first slot,
we've got 24 choices of cubelets. So,
24.
And then for the next slot, we've got 23
and so on. You can see how it's going to
go. It's going to be 24 factorial.
Actually, that's it for the edges,
because you might think, well, we also
need to consider the fact that one edge
can be either way up. But actually it
can't
in the sense that
um once you fit a fix a slot to put this
uh this edge piece in, it can only go
one way up in it. The mechanics of the
puzzle mean that. So, I'm just going to
put a dot on
this edge.
And now I can flip those two
edges there. So, don't worry about the
other pieces, just look at those two
edge pieces.
So, I flipped them over, yeah? But you
can see in flipping them over, I've also
moved the one that was here to the one
that was there. The one with the dot was
there before, now it's there.
Um and it's always going to be that way.
So, once you've put the cube the cubelet
in the slot, that's it. So, that is the
that is the arrangement for the edges,
okay? So, the number of edge
arrangements is just that, it's it's
just 24 uh
which actually is a pretty big number.
This is around uh six
times 10 to the 23.
So, that's 620 sextillion. The next
thing is the centers. Again, it sort of
starts off as the same as the edges,
because we've got six faces of the cube.
Inside each face, we've got four center
pieces. There's 24 center pieces all
together. So, I'm going to call this I'm
going to call this number K, which is
because C was already taken. The number
of we can just pick a a slot here and
then the number of choices of which
center piece to put in it is 24.
And then pick another one and the next
one is 23 choices. So, it's going to be
24 factorial again. But this time we do
have to consider something else, which
is that these four
cubelets are identical to each other.
And if they mix around between
themselves, we don't really care, right?
We don't consider it as a different
scramble if you know, if I was to swap
those two or something. It's all it's
all the same. So, that means
every possible scramble
is going to have basically
um different versions of it where the
orange centers are swapped around. So,
that's an overcount at the moment. So,
we have to correct that.
How many how many um times is each one
counted? Well,
how many arrangements are of these
four centers are there? Well, there's
four choices
for which one to put there times 3 times
2
times 1, so 4 factorial, which is 24,
okay? Yeah.
So, we need to divide by So, this to get
the number of total arrangements of all
the centers in the cube, we need to
divide by 24. Well,
that's for the orange centers, okay? And
then we have to do the same thing for
the green centers and for the yellow
centers. So, we have to divide by 24 six
times. So, we divide by by
to the six. And that gives us the number
of arrangements of the centers. This
comes out to be around 3 * 10 to the 15,
which is 3 quadrillion if you like those
sorts of words, okay? Okay, so now we
can all to work out the number total
number of possible states of the 4 cube,
we just have to multiply those together,
okay? So, we've got the number of
possible arrangements of the corners, C,
times the number of possible
arrangements of the edges, E, times the
number of possible arrangements of the
centers K.
And then we have to divide by 24 for the
same reason as previously, right? Okay,
so that's the answer and that gives us a
total answer here of uh roundabout 7 *
10 to the 45. So, that's the number of
possible states of a 4 cube. How does
that differ from our 3 cube, that that
number that I was supposed to remember?
Septillion.
43 quintillion. Quintillion, that's it.
43 quintillion. How does it differ? Is
it What sort of magnitude is that? For
the standard cube, it's 4.3 * 10 to the
19. And now we've jumped up to 7 * 10 to
the 45. So, it's a very substantial
increase, okay? Really significant
increase. Yes.
Shall we have another substantial
increase? Always. Okay, this is now a 6
cube. We can get started because the
corners
are exactly the same as previously. So,
we start with the same number C. So, now
we think about the edges. So, there's
two fundamentally different kinds of
edges on this cube, right? You've got
the ones here which are uh
blue and yellow and you've got the ones
which are yellow and green. They're like
They're in two sort of different types
of corridors, aren't they? They're in
two corridors and you can never switch
between the corridors. So, once you're
in one corridor, you're always in that
corridor. The central edges are
different from sometimes they're called
wing edges, right?
If you're in one of the wing edges, you
can get to anywhere else in any of the
other wing edges.
And if you're in one of the central
edges, you can get to anywhere else in
any of the central edges, okay? What
that Can the one on the left Can can you
If you're in a central edge, like those
two, can you change between left and
right? You can. Yes, you can.
>> So, you're not stuck in your corridor,
you're stuck in your brand of corridor.
>> You're You're stuck in your brand of
corridor, yeah, exactly. That's right.
That's right. Um And what What that
means is that if you just look at the
central edges, the um the argument is
exactly the same as the edges for the 4
for the 4 cube, right? Because the the
the edges here are just the central
edges, right? So, the the number of
possibilities for the central edge
is that number E as we had before,
right? So, times E. But then the number
of possibilities for the wing edges is
the same again, so it's times E again.
So, we times by E squared. Okay, so now
we've got to look about the centers.
They call it centers. It means
everything which isn't a corner or an
edge. So, actually quite a lot quite a
lot of cubes cubelets in here. There's
um there's sort of 16. What are the
What's the sort of corridor structure?
How can you get Which can you get
between and which can't you? Yeah, how
are they constrained, yeah? How How are
they constrained? Yeah. If you look at
this white
white cubelets, right?
The question is where can it get to?
So, it can
on this face,
there's four places it can sort of
obviously get to, right?
Those four. So, it can be just above the
bottom left corner, just to the right of
that corner,
just underneath that corner, right? So,
there's four four places it can get to.
Are there any others? And actually the
answer is no. So, you might think you
would be able to get it get the white
cube into that position there, but
actually you can't. And likewise, you
can't get it down there. That's sort of
maybe more believable.
And you can't get it there either. Just
use a different face. So, actually the
sort of key thing here
is
if you look at this blue square of four
cubelets, those are the four
fundamentally different kinds of
centers. So, there's four kinds of
centers.
And then for each kind of center,
the argument goes exactly as the centers
for the 4 cube, okay? So, the number of
ways of arranging
those kinds of centers is that number K
we had. And then the number of ways of
arranging those kinds of centers is also
K, and those kinds of centers are also
K, and those kinds of centers are also
K. So, overall we get K to the 4. Then
of course, we do divide by 24,
which matters less and less as these
numbers get bigger and bigger. It's a
bit of a formality at this point. But
when you work that out, we're up to 1.6
* 10 to the 116. We're like beyond atoms
in the universe. We're beyond atoms in
the universe. Yeah, there's there's more
possible scrambles of this puzzle than
there are atoms in the universe.
This is my biggest cube.
Um it's a recent and rash purchase. Um
this is a 10 cube.
And I haven't actually yet dared
properly scramble it cuz I dread to
think how long it would take me to
solve. Um at some point I will. Um So,
but I think it might be fun to work out
how many I mean, has it gotten to a
point now where for even numbered cubes,
we can
create an algorithm here or Yeah, we
can. We can. So, the way to think about
it is let's call it a 2n cube, right?
So, um I'll do this as an example. So, a
2 * 5 cube, so that's my 10 cube, right?
We're going to let n be sort of half the
size. You could let n equal 10, but then
the algebra comes out harder. You've got
C, that's the number of corners, that
never changes. Okay, now we think about
the edges. How many different edges have
we got? Now, let's put some different
colors on.
Okay, but the point is how many
different types are there? Well, there's
the central ones,
then the next ones out, then the next
ones out, and then the wings, okay? So,
we've got four,
right?
1 2 3 4 fundamentally fundamentally
different types of edges.
Um and in general,
so here in this cube, n equals 5. So, in
general, you're going to have n minus 1.
Yeah? And the reason it's n minus 1 is
because n is half the width of the cube,
and we just don't want the the outermost
ones, so so n minus 1, yeah? Okay? So,
that means the number of possible
arrangements of the edges is E to the n
minus 1. And now we've got to think
about the centers. Yeah, just give me a
second.
Sorry, it's just a bit slow, this thing.
So, now if we think about the centers,
this white square of 16 cubelets, that's
representative of the different
corridors. So, each of those is
different to each of the others. You can
never get from any of one of those to
any of the others. And once we've chosen
one of those types, so there's 16 types
here,
um or in general, that's n minus 1
squared, half the width, so that's n,
then we take away the edge, so that's
that width there is n minus
1, then we square it, okay? So, there's
n minus 1
kinds of centers. That kind, by the way,
is my justification for using K.
Um pretty weak, I think. And then the
argument for each one is exactly as the
previous day, that's the number of
possible combinations. So, we need to
then multiply here by K
to the n minus 1 squared, right?
And K, remember, is 3 * 10 to the 15.
And that's why you can sort of see here
why these numbers are getting so big so
quickly cuz it's exponential in n, but
it's not actually just exponential in n,
it's exponential in n squared. And
that's why why it gets so big. Of
course, we have to go through the
formality of dividing by 24. Of course.
Of course, of course. Um Okay.
>> that does knock one off the power,
doesn't it? It It knocks one off the
power, yeah. It knocks one off the
power. Um so, for 10, if you plug in um
n equals 5 into this, well, you get C
times E to the 4
times K to the 4 squared, so 16, divided
by 24. Work it out, you get to 10 to the
349 or thereabouts, okay? Which is a
which is really ridiculously
ridiculously big number. I mean, that's
the sort of place where,
you know, if you took every atom in the
universe and replaced it with a whole
universe,
and then repeated that
four times,
and then counted the number of atoms in
that whole thing,
this has still got more scrambles than
that, right? Okay, so it's it's an
entirely ridiculously ridiculously big
number. You've spoken to me about bigger
numbers. I have spoken to you about
bigger numbers, but I've never held one
in my hand.
You may have noticed we've only dealt
with cubes with an even number of pieces
along the edges. If you want to find out
how it gets a bit more technical and
mathematical with an odd number,
like that,
go and have a look on Numberphile 2.
We've got an extension of the video
there. We go into way more detail, but
before you do that,
go and order one of these. Order a copy
of Huge Numbers. It's Richard's new
book. It's available now to order or
pre-order, depending on where you are.
The cover might look different,
depending on where you are, but it's a
fantastic read. It deals with really big
numbers, like you've been seeing just
now.
I'll also put some links below to other
things that may interest you, previous
videos we've done about Rubik's Cubes,
previous videos we've done about big
numbers, and of course, the link to that
extra video,
and the order link for Richard's book.
Go and check it out, people. Thanks for
your time, and thanks for watching.
puzzle can be solved in 20 or 15