Rubik's Cubes

At some point this will become a thread about Rubik's cubes, when I don't have to go to bed an hour ago. Also at some point I'm just going to start talking about my particular views on solving the Rubik's cube and other Rubik's type puzzles. There's a lot of strategy and planning involved and I'm sure someone could make some metaphors for life or some garbage like that out of it.

Spoiler: My current collection

Standard (3x3x3)
4x4x4
5x5x5 (somewhere, probably in a box)
7x7x7
Void Cube
Helicopter Cube
Rex Cube

I can solve all of those, although the Helicopter gives me some trouble. I've also recently learned how to solve a Gear Cube, and can theoretically solve all of the regular face-twisting ones. I've played with up to an 11x11x11, although that thing was heavy, unwieldy, and took forever.

Current opponent: A Super Square One cube owned by a colleague. Hopefully he hasn't undone the progress I made while procrastinating today.
[EDIT 03/21]: Solved the Super Square One. I think I can consistently and deliberately solve it once it's in a cube shape, but not necessarily get it into that state.

 

So, the Rubik's Cube:



It looks like a cube that has been cut into smaller cubes, 3 to an edge. It allows for rotations along all three major axes and allows each layer on each axis to rotate independently.
Invented by Erno Rubik, Hungarian professor of architecture and design, to demonstrate 3-dimensional engineering, and also because he wanted to figure out if such an object could be made.
It twists, it turns, it took him a month to get it back to the solved position.

In the 80s it became very popular, and is the world's best selling puzzle toy; it might even be the world's single best selling toy overall. Wikipedia has sources for those claims that I haven't looked at.

It has been experiencing something of a revival "recently", where I've been hearing about this revival for at least ten years so I don't know how much of a "recent" or a "revival" it really is anymore.

As with many things, there are of course competitions for solving Rubik's cubes and Rubik's Cube-like puzzles, which I will generally call "twisty puzzles" because of trademark reasons. The current speed record for solving a single scrambled Rubik's cube is 4.73 seconds, set Feliks Zemdegs on 16 December 2016. He also holds the record for middle-three-out-of-five with 6.45 seconds on average on the three solves, and best single solve one-handed at 6.88 seconds.
There are also records for solving it blindfolded, for solving it with one's feet, and solving some of the common variants like the 4x4x4 or the 5x5x5.

I can talk a bit more about speed-solving, but it's not a topic I'm generally all that interested in. What I am interested in is not speed but move-number: how many twists does it take to get from a scrambled state to a solved one. The record for fewest moves taken to solve a single scrambled cube given an hour is 19, achieved by Tim Wong on 11 October 2015. This is really interesting to me because it was proved in 2010 that the smallest number of moves it takes to solve any scrambled cube is 20.
To clarify: it will never take more than 20 quarter-turns or half-turns to get from a scrambled state to a solved state, and there are scrambled states that require 20 quarter-turns or half-turns to get to the solved state. This means that Wong got a state that wasn't maximally scrambled.
The reason that this is interesting is that how to find the the shortest path from scrambled to solve is in general completely unknown. Except for a few cases, we don't even know how to determine how many moves it would take to solve any given scrambled cube, much less which moves those would be. An algorithm that takes a scrambled cube and tells you how to solve it in the fewest moves is referred to as "God's Algorithm", because of course God would use the fewest moves possible. The number 20 is then called "God's Number" in the context of Rubik's cubes, as that is the highest number of moves that God's algorithm will ever require.

In the next post, or whenever, I'll start on the mechanics and terminology of the cube, in preparation for talking about the mathematics of the cube and some strategies for solving it.

 

While most people first starting on the cube think of the stickers as the fundamental component of the Rubik's cube, the actual minimal piece that moves independently is more than just a single sticker. For instance, in the picture above, the piece in the center has a red, a white and a blue sticker on it, and those three stickers will always be attached to that bit of plastic and hence in that position relative to each other.
We have pieces on the corner of the Rubik's cube, called "corner pieces" or sometimes just "corners", we have pieces in the middle of the edges of the cube, called "edge pieces" or just "edges", and a piece in the middle of each face, called a "face piece" (note that when we say "face", we generally mean the entire face of the Rubik's cube, not the piecein the middle).

The 3x3x3 has 26 relevant piece: 8 corner pieces, 12 edge pieces, and 6 face pieces. You might object that 3 * 3 * 3 = 27 and that is a true statement, but that supposes a piece in the center of Rubik's cube. But we can't see the piece in the middle at all or do anything to it, so we don't really care about it.

A corner piece has 3 stickers on it, an edge piece 2 stickers, and a face piece 1 sticker. We call the set of pieces that have a sticker on a particular face a "layer", so that we have a 3x3 block of pieces, i.e. 4 corners, 4 edges and a face piece, that compose a layer.
A piece can be wrong in two ways: in the wrong position, meaning that it sits where another piece should be instead, and the wrong orientation, where the piece as a whole is in the right position, but is twisted. For instance, a corner piece could be in the right position for that piece, but twisted so that its stickers are pointed in the wrong direction.

The basic move is to twist a layer (relative to the rest of the pieces), by 90 degrees, 180 degrees, or 270 degrees. Some people don't consider the 180 degree twist to be a basic move, but I will do so here; notably, God's Number referenced above is in terms of the set of basic moves that include the 180 degree turn as single moves. We call twists by 90 or 270 degrees "quarter turns" and twists by 180 degrees "half turns".

When you twist a layer, the face piece doesn't actually go anywhere, it just rotates in place. On a standard Rubik's cube, this does nothing interesting, because the sticker is a single solid color. The four edge pieces move around, going to positions formerly occupied by one another, and similarly the corner pieces also move to positions formerly occupied by one another; none of the basic moves will move an edge piece to a position that used to have a corner piece in it.
Because the face pieces don't change position, just orientation, a lot of people consider them to be entirely stationary. If you start out with the face piece with the yellow sticker on top, you always leave it there, and if you start with the face piece with the red sticker facing you, you always leave it there.
In contrast to the face pieces, the other pieces all can move to other positions as well as change orientation.

Some numbers:
When you scramble the cube, you can move a given edge piece to any of the twelve positions that can hold edge pieces, but of course once you've decided that a particular edge piece goes into a particular position, you only have 11 positions open for the second edge piece. And so on, so that gives 12*11*....*2*1 = 12! possible position arrangements for the edge pieces.
Similarly, for the corner pieces you can move a given corner to any of the eight positions that can hold corner pieces, and so you end up with 8*7*...*2*1 = 8! possible position arrangements for the corner pieces.
For any given edge piece, it can have two orientations, so that gives 2¹² arrangements for the orientations of all twelve edge pieces, and for any given corner piece, it can have three orientations, so that gives 3⁸ arrangements for the orientations of all eight edge pieces.
So that would give 12!*8!*2¹²*3⁸ possible states of the cube.
But! Not everything is independent.
For instance, we can't actually reach all 2¹² possible orientation arrangements. Once we've decided the orientations of eleven of the edge pieces, the orientation of the last one is actually set for us. Similarly, if we've decided on the orientations of seven of the corner pieces, the orientation of the last one is set. Finally, the positions of the edge pieces and the corner pieces aren't entirely independent either; if you've set the positions of all twelve edges and six of the eight corners, the positions of the last two corners are also set.
So we actually only have 12!*8!*2¹²*3⁸/(2*3*2) possible states of the cube, which is somewhat over 43*10¹⁸.

So that's the problem we're facing: getting each piece into the correct position and orientation, with more than 43*10¹⁸ possible starting positions and only one solved position.

Next time: musing about solving the Rubik's Cube.

 

rubix cubes are fun. im not great at them but i know someone who goes to rubix cube competitions!

 

I have never solved on to this day though I neared it once. The cube mocks me. I tell it to fuck off. It continues to mock.

The numbers of the cubes are interesting though, even if I don't quite get them. Also the things REALLY PRETTY.

 

For those who want a rough sense of the total number of states, there are about 7*10⁹ people alive on Earth at the moment. So the number of states is like if each person on Earth was given their own planet full of people; the total number of people on all of those planets is about 49*10¹⁸, which is slightly more than the number of ways to scramble a Rubik's cube.

 

huh, that's smaller than i was expecting

 

Today a friend and I tried an interesting experiment: we took a scrambled Rubik's cube and took turns doing five moves each. The only communication we were allowed was strategy, of the form "I want to move this piece over there".

It was tough, because all of the algorithms we knew were more than five moves long, so we would keep getting interrupted. Fortunately we knew roughly the same algorithms.

Unfortunately, I haven't really defined what I mean by "algorithm" yet here, because I haven't talked about solving yet. I should do that next post.

 

So at least in my parlance, a strategy or a method is a general ordering of which pieces to move where. So, for instance, a popular method for people first learning to solve the Rubik's cube is called the "Layer method", where first you get one layer of pieces into the right positions and orientations, and then you solve the edge pieces directly next to those (referred to as the second layer), and then you solve the remaining layer. It's pretty straightforward to plan figure out what piece you need to get into place next, and it's easy to see progress happening, and it's pretty easy to see that, if carried out properly, it will solve the cube.
You can further divide these steps of the strategy into substrategies, such as solving the first layer by putting the edge pieces in first and then the corners.

An algorithm, in contrast, is a specific set of twists for the purpose of moving the pieces in certain positions and orientations to certain other positions and orientations while . So for instance, you might have an algorithm that swaps the position of two corners of the third layer while leaving the first and second layers alone. This algorithm might screw up the edges on the third layer in the process, though.

Most solvers use a strategy that puts some pieces in place, then others, then the rest. But as most people who try to solve the Rubik's cube find out, once you've solve part of it, if you don't want to undo that progress then you lose a lot of freedom of movement. So a lot of algorithms that are supposed to not undo progress require temporarily breaking up previous progress to grant some freedom, then undoing the damage done.
If that sounds somewhat inefficient, it is. In contrast, competition-level solvers tend to use long-view strategies that don't put anything into the proper place until the very end, and have memorized hundreds, probably thousands of algorithms.

People who encounter the twisty puzzles that aren't the standard Rubik's cube often ask if knowing how to solve a Rubik's cube helps with solving the nonstandard ones. The answer is "it depends". Some of the strategy transfers, for instance for the 4x4x4 and 5x5x5 and so on, the nxnxn cubes, can all be solved using roughly the same strategy as the Rubik's cube, but you need a few more algorithms.
The Megaminx, which is a Rubik's dodecahedron, can use the same strategies as the Rubik's cube and the same algorithms, except you have to change the algorithms slightly to accommodate the fact that the faces have five edges instead of four. Again this screws up muscle memory because each "half turn" on a regular Rubik's cube could mean either of two possible things on a Megaminx and it's not trivial to figure out which one is the one you want to use.
For things like the Skewb or the Helicopter Cube, where the types of pieces and types of possible moves are completely different, often both the strategy and the algorithms need to be modified heavily or replaced entirely.

 

Ahh this is really cool info. I got a regular one for my partner as a fidget/stim toy but I've always wanted to learn how to solve it myself.

 

Saw someone on the bus today solving one at really high speed. Appropriate!

 

The Rubik's cube is actually a pretty ingenious object from a mechanical standpoint. How do you make it so that the edges and corners can move freely about the cube without everything falling apart?

The original mechanism of the Rubik's cube is shown here:

The white bit in the center has six arms, two along each axis. The face pieces are held to the corresponding arm by screws. The edge pieces, such as the loose pieces on the far left and right, are then held in by the centers by the protrusions, which are called "feet", which stick under the center pieces; you can kind of see that with the piece on the left side of the cube. The corners are then held in by the edge pieces, as shown by the red-and-purple-stickered piece.
The little feet can slide underneath the face pieces and the main bodies of the edge and corner pieces, allowing for relatively free movement.

The larger cubes like the 4x4x4 and the 5x5x5 work on the same principle: the middle face pieces are held in place by a central piece, the pieces next to those are held in by the middle face pieces, and so on outward in layers. For the cubes with an even number of pieces-per-edge, for instance teh 4x4x4 or the 6x6x6, the middle face pieces have to be held in a little differently than in the standard Rubik's cube or the 5x5x5, because there's no single middle face piece, but beyond that it's the same mechanism.

However, there's an interesting little caveat. If you look at the picture in the introductory post (after my initial blathering) you can see one face of the cube turned about 45 degrees. The corner sticks out quite a bit, doesn't it? Fortunately, it has that little foot sticking out of it to hold it in.
But for larger cubes (by which I mean more pieces-per-edge), the corner is smaller, and thus sticks out more in proportion to its size. For cubes of size 7x7x7 and up, the proportion is actually bigger than a single cube piece.
So how do people make cubes of size 7x7x7 and up? There are two things you can do: make the edge and corner pieces bigger compared to the face pieces, or curve the cube a little bit. Here are examples of each:

Spoiler: 11x11x11 that does both

Some other companies use slightly different mechanisms to hold the pieces together while allowing them to move, which is why the Chinese knock-off 3x3x3s often turn better than official Rubik's cubes.
For instance, V-cube's larger cubes have several shells of pieces holding things in, to provide more stability and spread out the effect of the middle face pieces better:

Spoiler: V-cube 6x6x6 interior

The cubes that aren't face-turning use slightly different mechanisms, but they're all roughly predicated on the idea of some pieces holding other pieces together. Even the 2x2x2 uses the same idea, although with the "middle face pieces" completely hidden.

This is notably why, even though Larry Nichols figured out how to make a 2x2x2 before Rubik's did, Nichols' design, being held together with magnets rather than being purely mechanical, faded out of favor.

 

I have a rubiks cube... I've a system for solving it, sort of, but I don't think it's very efficient, and I haven't seen it in any of the online tutorials I've looked up when I got stuck, so that's a little frustrating.

It mosly relies on getting the corner pieces pointed in the right direction and solving from there. Maybe call it the sandwich method? But I have a rather short attention span so getting the corners both oriented and positioned correctly involves more dumb luck than it probably would if I could stick to a single task for more than three turns.

Given that, the whole thing is more of a color-fidget than anything scientific, for me. In which case I guess it's beneficial that I'm so slow at solving it. Short attention span + pretty colors + nothing else to do = hours of low-stakes hand-occupation.

 

Yeah, I mostly use my 3x3x3 for fidgeting. Well, these days I can also use it for trolling people, because I've got quite a bit more experience with Rubik's cubes under my belt than any of my colleagues. But mostly fidgeting.

 

Fidgeting with them is like the main thing I use them for. They make pleasant sounds when you shift things.

 

If you like fidgety cubes, take a look at the Gear Ball:

The gears force the opposite faces to turn at the same time; in other words, if you spin the yellow face, the face in the back also spins, but in the opposite direction. Also the gears in the middle of the edges spin when you turn the faces that they touch, so that spinning the red face will cause the green-yellow gear to spin. It turns super smoothly and the gears make little clicks when you spin them and it's so much fun to play with.

 

i want this thing even if i will never be able to solve it

 

o/ my SO is really into rubiks cubes so via osmosis i also like them a lot!!! im not nearly as good as he is; it can take me like five minutes to solve it (he taught me all the easy algorithms), and i think his best time is like 8 or 9 seconds ;w;
he went to his first competition in uruguay recently and did pretty ok iirc! i think when he's back stateside he's thinking of going to another one this summer

 

One thing that experienced cubers talk about is the notion of "parity errors".
I mentioned in my numbers post that you can't just flip an edge without changing anything else, or spin a corner, or swap two edges. These are all called "parity errors", in analogy with parity errors in computer science.
They come about due to the mathematics of the cube, in particular an area called "group theory". One thing that group theory tells us is that rearrangements (permutations) of some finite set of objects can be put into two categories, usually called "even" and "odd". Two odd permutations or two even permutations make an even permutation, while an even and an odd permutation together make an odd permutation.
When you spin a face of a Rubik's cube, you move four edges and four corners in what is called a cycle, in that if you keep spinning that same face enough you'll get back to the initial state. It takes four quarter turns to get back to the beginning, so we say that we have a 4-cycle of edges and a 4-cycle of corners. For reasons that I will brush under the rug for the moment, if n is even then an n-cycle is an odd permutation. So a 4-cycle is an odd permutation, but two four cycles make an even permutation. So twisting a face makes an even permutation.
Swapping two pieces makes a 2-cycle, which is an odd permutation. But a combination of even permutations only makes more even permutations. Hence you can't just swap two pieces by twisting the cube.
Similarly, if we just look at edge pieces, each has two stickers, so the on-face stickers form a 4-cycle, and the off-face stickers that are on the edges on the layer also form a 4-cycle, and so we have an even permutation in total. But flipping a single edge will swap the two stickers, which is a 2-cycle, which is odd, and hence impossible.
The corners are a similar argument involving three-ness instead of even-odd, but the argument is similar: rotating a single corner introduces a threeness offset that can't be produced by twisting moves, as those twisting moves don't introduce any threeness offset.
This is one way to tell if the person who "scrambled" the cube actually took it apart or just put the stickers back on randomly: if they actually scrambled it by twisting, then they won't have put in any parity errors, but if they took it apart and put it back together randomly, they only have a 1-in-12 chance of putting it back together without any parity errors.

On larger cubes like the 4x4x4, you can get things that look like parity errors. For instance, the edges now consist of two pieces rather than one, and you can swap those two pieces. Even though a twist is still an even permutation here, the trick is that the four face pieces of a given color are identical, so you can swap them as well: two 2-cycles make an even permutation, but with some of the pieces being identical, only one of the 2-cycles is detectable.
Because of the stickering, if you swap two adjacent edge pieces on an otherwise solved 4x4x4, to most cubers it looks like you flipped an edge, because to most people the edge-pairs look like they go together to form a single edge object, and the four face pieces also look like they go together to form a single face object, as most cubers see a 4x4x4 as just a chubby 3x3x3. If you hand me a 4x4x4, it will probably end up in this state.
You can also swap pairs of edge pieces, because this again is an even permutation since we have edge-pairs now rather than single edge pieces, and so again we can introduce what looks like a parity error. Note that we still can't rotate a single corner, and this is true for all of the nxnxn cubes.

 

The Best State

The best state on a Rubik's cube is the superflip, a state where every piece is in the correct place, but all of the edge pieces are flipped.



The reason that this is the best state is that it's highly symmetrical and also maximally scrambled. Remember a few posts ago when I mentioned that it takes at most 20 moves to get from any state of the cube to any other state if you know what you're doing? It takes 20 moves to get from the superflip to the solved state.

You can sort of see why the superflip is at least locally maximally scrambled, in the sense that any move you do to it can't get you farther from being solved.
Let's talk about symmetry for a moment. Start with the solved state. Suppose the yellow face is on top and the white face is on the bottom. Spinning the yellow face is, functionally, the same as spinning the white face and then flipping the entire cube upside down; in both cases, the layer that's currently on top has been spun and everything else is in the correct place. The only difference is the coloring scheme, but that's not really an important factor; all the colors are equivalent. Spinning the entire cube doesn't affect solvedness at all; everything is considered relative to the center pieces, so if they move, the "correct" positions and orientations move with them.

So now assume that there is a shortest algorithm that will solve a superflip, and it starts by spinning the red face. In other words, spinning the red face gets you one move closer to the solved state.
Well, according to the picture the red face is currently the face that is facing towards you and slightly to the left. But you can rotate the entire cube, which still leaves the cube in the same overall state, so that any of the other faces is pointing towards you and slightly to the left, and then start the solving algorithm. So for any face, there is a version of the solving algorithm that starts by spinning that face. Hence spinning any face gets you one move closer to the solved state!

So that's what we mean by "locally maximally scrambled", in that, at least in only one move, you can't get farther away from solved if you start at the superflip. This doesn't mean that you can't find a longer chain of moves that gets you farther away.

However, as I said before, not only is the superflip locally maximally scrambled, it's globally maximally scrambled, i.e. there is no state that is farther away from solved than the superflip. There are plenty that are as-scrambled, i.e. that also take 20 moves to get to the solved state, but none that take 21 or more. Also, all of those other maximally-scrambled states aren't as nice as the superflip.

Other reasons why the superflip is nice:

* It's an involution, in that if you do a superflip, and then perform the superflip algorithm on that, you get back to the beginning.
* It's central, in the sense that if you have some algorithm, or just some sequence of moves, you can do that sequence of moves, and then perform a superflip on that (i.e. leave all the pieces where they are but flip all of the edges) and that gives the same result as first doing the superflip and then performing that algorithm on the resulting state. Other than the trivial "do nothing" algorithm, the superflip is the only thing you can do to the cube with this property.
So we say that the center of the Rubik's cube group is Z₂, i.e. it has two elements, one being the trivial identity element, the other being an involution.
* It was the first state found that was known to need 20 moves to get to. It was in fact the example that gave us the lower bound on God's number, quite a while before the maximum was known.

Okay, so here's how to do the superflip in 20 moves, where by "move" I mean either a quarter-turn or a half-turn of a single face.
First, some notation:
I'm going to write U for a clockwise spin of the top face by 90 degrees, if you're looking down at it. I'm going to write U² for a half-turn of the top face, and U^-1 for a counterclockwise turn of the top face.
Similarly, D for a clockwise spin of the bottom face by 90 degrees if you're looking up at it and so on.
L for a clockwise spin of the left face by 90 degrees if you're looking straight at it.
R for a clockwise spin of the right face etc
F for a clockwise spin of the front face etc
B for a clockwise spin of the back face etc
All of these are "clockwise if you're looking at that face" and so on. This is called Singmaster notation, and is the standard notation for writing down algorithms for the standard Rubik's cube, although sometimes you'll see, for instance, R2 instead of R² and R' instead of R^-1. I use the exponents because that matches with mathematical notation.
Be careful to keep track of which face is which, because it's easy to screw up if you have to keep turning the cube around to actually look at the faces head on.
Anyway, here it is, 20 moves from solved to superflip:

U R² F B R B² R U² L B² R U^-1 D^-1 R² F L R^-1 B² U² F²

If I come across a Rubik's cube that isn't obviously off-limits, I usually leave it in the superflip state. If you're going to scramble a Rubik's cube for someone, might as well leave it maximally scrambled.