## A simple trick to design your own solutions for Rubik’s cubes

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) ðŸ™‚ 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 2×3 well that’s the same as 3×2. 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.