Today I'm going to tell you about something that's really close to my heart.
It's a super simple trick that allows you to find your own solutions to pretty
much any Rubik's Cube on that shelf.
That sounds too good to be true but it's actually true, it's really quite amazing.
So, what I'm going to do is I'm going to explain the trick using the normal
3x3x3. My audience is going to be this guy here my 11-year old, Karl. Say hello
Karl. (Karl) Hi. (M) He is there in the background and what he can do is he can solve the
first layer of the Rubik's Cube and the main message of this whole video is going
to be: "If you can solve the first layer of any of those cubes,
you can solve them. You don't have to look up any sort of recipe." With the
trick that I'm going to tell you about you will be able to design your own
solutions.
Anyway, he can do the first layer,
a lot of people can do the first layer, probably millions have gotten to the
stage where they've solved the first layer. So first layer, again.
You look at a Rubik's Cube you basically see one of those things,
completely solved.
Why is it so hard to get beyond that stage.
Well what's really, really difficult there is that as soon as you do
something now, pretty much anything, you're going to cut into what you've
just solved and you're gonna destroy it. That gets very frustrating very quickly
and there's pretty much three different outcomes now for most people
who got there.
The first one is you give up, that probably happens in the majority of
cases, second one is you go to internet and look up somebody else's recipe to go
beyond that and the third outcome is you actually persist and
somehow get your own solution.
So what I want to do today is to enable as many as possible
among you who are watching this video to actually get into this third
category where you find your own solutions of the Rubik's cube.
Ok, let's get going.
Anything you do now is going to destroy things. So what we're looking for
at this stage is some magic moves,
really sequence of moves,
what the magic moves are going to do is they are going to leave pretty much all the
Cube intact and only touch and manipulate small parts of it.
For example, one thing you might want to do is you want to look for magic move
that just flips that edge here.
Just flips that edge here, how hard is that?
Hmm, very hard, just think about it. There is just no way I'm going to do be
able to do this. To find a sequence of moves that just flips one
edge is actually not too hard if you just have to worry about the first layer
so just flip that edge and leave the rest of the top layer unchanged.
In fact, if you are a master of the first layer you can permute that first
layer any way you want. You can actually do this. You may not be aware of this
but you can as long as you don't worry about the bottom. So let's just do that.
So, go home do it, right?
I'll show you an example of how to do this. So what we can do is
we can maybe do something like this
turn things out of the way so the sides come up again and we turn this guy over,
we do that, we do that,
sides back and, well, we've just found a move that just flips this edge here and
leaves the rest of the top layer unchanged.
Ok everybody here, Karli, still understand it? (Karl) Yes.
Very good, alright, 'cause this is for you, Karli,
you're supposed to understand this, alright?
Now let's unleash the whole thing on a solved cube and see what happens.
Your move,
let's unleash it on a solved cube and see what happens.
Well it's getting messy but basically the top layer is fine, right,
the top layer is fine except for this one flipped bit. Of course the bottom is messed up.
And now comes the really really, really important question: "How can we restore
the cube really, really easily? Karli? (Karl) You do it in reverse!
(M) Exactly and we didn't even rehearse this.
Ok, so we do it in reverse.
Obviously if we do the whole thing in reverse it's going to solve the thing,
right?
So let's do this in reverse, here we go, doing it in reverse, and, doesn't come as
a surprise,
the whole thing is back to normal. Let's just put in the move
again.
Put in the move again. Now what does the reverse move actually do?
So in the top layer what does it do?
Well it leaves everything in peace except for this one edge which gets flipped. And
what does it do to the bottom part Karli?
Does it fix it? (Karl) No (Karl's sister in the background) It fixes it Karl!
Ok, so what does it do to the bottom part, Karl? (Karl) It fixes it.
(M) Exactly, it fixes the bottom part.
Very good, very good, we're getting there. It fixes the bottom part. Now we're not
going to run it in reverse straight away. Here comes to trick and it's so
simple
you won't believe it, really. What we do is, we just give the top layer a twist
and now we're running it in reverse. And now we're going to try and predict
what's going to happen and then we have it happen. So what is going to
happen is, well, if you run your move in reverse now
well it should just flip that edge piece on the top and it should restore the
bottom.
Let's just see what happens. So run it in reverse, here we go.
Alright
that already looks very very promising. We can make it look even more promising
if we undo that twist of the top, so let's just do that and turn it around.
And now in total
we've come up with one of those magic moves, a magic move that only affects two
edge pieces, flips those two edge pieces.
Pretty amazing, right? So, for example, if we've got a messed-up cube like
this and our aim is to just flip those two edges, we can just use this
combined move now to achieve this.
So, we're doing your move, then we're doing the top twist, then we're doing your
move in reverse and then we're doing just the top in reverse. And then at the
end that whole thing would look exactly the same, except that these two guys are
flipped. Good! Important thing is that the only thing that really requires your input
here is to design this move that just affects the top layer.
You don't really have to worry about everything else, okay. Let me give you
another example.
So, maybe you want to twist some corners in isolation, right? So maybe you want to
twist that corner here. Now if you just worry about the top layer and not about
anything else
you know it's pretty easy maybe something like this, right?
If you look at the top now really only that corner has been twisted. If we
unleash that move on a solved cube what do we get? This thing, obviously the bottom is
is messed up.
Now how do we turn this whole thing into a magic move. Well we turn the top
a little bit.
Now we the run things in reverse and so what's happening here?
Well let's have a look.
Only those two corners are affected,
those two corners get twisted, nothing else is affected. Again, you can
design something like this very easily.
Just one more example. So far we've just been kind of stepping on
the spot, twisting things, flipping things but obviously you really
want to also find some magic moves that move stuff around.
So let's just do that,
let's move some things around. Let's have another close look at the top layer.
Well if, for example, we move that edge piece here somewhere.
Well, it has to go somewhere so whatever is there has to go somewhere else and
well the simplest thing we can really do is to of swap those two
pieces. And again if you just worry about the top layer you can do this, design
something that does exactly that. Here we go,
just an example, you probably come up with something totally different,
it doesn't matter. The main thing now is, you've recorded your move,
you give the top a twist
and you run your move in reverse.
Okay.
And untwist the top and let's just have a look around.
Everything's fine except three pieces have been moved around and
I'll just show you exactly what happened here.
This guy here moved over there, that guy here moved over there and that
guy here moved over there.
Like that, ok? So this particular magic move what it does is it cycles three
edge pieces
and leaves everything else unchanged. And you can do the same sort of thing
for corners, you can designa magic moved that moves corners. And so in
total what we've done now is we've designed four magic moves, one that flips edges,
one that twists corners, one that moves edges and one that moves corners. And
together with a bit of a common sense this is a complete recipe for
solving the Cube. Might not be completely obvious but obviously, you know, if it's
supposed to be YOUR solution so
now it's really time to sit downand fill in the gaps.
I'm going to do a footnotes video where I give some more explanations of all
this stuff, but that's basically it.
Alright now this guy's really happy. Karli are you happy? (Karl) Yes. (M) Very good he's happy.
Did you understand everything? (Karl) Yeah. He's probably just pretending but anyway. (Karl) Hey!
(M) :) I am going to test you on this later, okay? So I claim that this kind of works for pretty
much anything here on the shelf and so just to give you an example let's have a
close look at this puzzle here, it's called a Magaminx. Very nice
puzzle. This thing also has a top layer, here it is and again. Very very easy
to restore just this top layer and it's also very very easy to design a
move that flips one of the edges if you don't worry about the bottom.
And then you take this move, you give the top a twist, run your move in
reverse and untwist the top and the effect this has is exactly the same as what
we've seen before with the Rubik's Cube. And just like with the Rubik's Cube, you can
design a total of four magic moves and with those for magic moves you can solve
this thing.
Now these magic moves are examples of what's usually referred to as commutators.
That's actually a mathematical term. So it's expressions of the form A B A inverse or A reverse B reverse
In our case the A corresponds to your move,
the B corresponds to twisting the top, then your move in reverse and then
top in reverse. But that A and B can actually stand for any sequence of moves.
So say A is a sequence of moves and B is a sequence of moves and if we then do
A first and then B and then A reverse and then B reverse that tells us
something about the two moves.
What does it tell us? Well, order matters when it comes to the Rubik's cubes. So
usually it matters whether you first do A and then B or whether you first do B
and then A. Usually you get totally different outcomes.
Unlike when you multiply numbers. So for example if you've got 2x3
well that's the same as 3x2. With Rubik's Cube moves
it's different. For example, if we take this Megaminx here
and we twist the top here and then we twist that side here
the outcome is very different from first twisting the side and then twisting the top.
So what does this compound move tell us about A and B. What it tells us whether
they commute or not.
So take a solved cube, ok, so maybe a solved cube like this guy here and then we have
a sequence of moves A and we've got a sequence of moves B.
Now we do A, then we do B,
then we do A in reverse and then we do B in reverse, unleash it on this thing here
and see what happens.
Well, if nothing happens, so if that Cube basically stays unchanged by this
compound move, what that means is that the two moves A and B commute, that
it doesn't matter whether you first do A and then B or first B and then A.
The commutator is a measure for how close two moves are to commuting.
So the less stuff gets shuffled around here
as a result of ABA inverse B inverse, the closer the two moves are to
commuting. Very very important in mathematics, for example, in group theory
but also in physics and quantum mechanics but, you know, that really goes
beyond this video.
So that's it for today.