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Solving the Rubik's cube.
blaisepascal
I've been solving Rubik's Cubes for over a quarter century now. It's something I do in my idle time. It wasn't unusual to see cubes around the house in places I frequent, ready to pick up and fiddle with (that has stopped mostly because skitten stopped complaining about it and started throwing cubes away, a behavior I don't think she'd like if I did that with her stuff). My college roommate, knowing I tended to not like to let a scrambled cube stay scrambled, would often take my cubes and scramble them when I wasn't around, just to see how long it would take me to notice and solve. I own or have owned several Rubik's Revenge cubes (4x4x4 instead of the 3x3x3 of a standard Rubik's cube) and a Professor Cube (5x5x5). I can solve them all.

There's a common misconception about the Rubik's Cube which can work to my advantage. The misconception is that it requires great smarts or intelligence to solve the cube. As such, demonstrating that you can solve it to someone who can't works well as a surrogate to demonstrating real brains. The truth is... It really isn't that difficult. The methods people use for solving the cube mostly rely on memorized sequences of moves ("patterns") which accomplish a known subtask in the overall method. You typically work in stages, solving the cube bit by bit, and what patterns you use vary by stage. The main method I use has 6 stages and there is about 8 patterns I use to solve it. I'm not counting how I do the first stage which probably has patterns but I don't think of them that way. Other methods have different stages and patterns. Once you sit down and learn a method, committed to muscle-memory, solving the cube is easy. It just looks impressive.

What's hard, what does take smarts, is figuring out from scratch a method, figuring out which series of stages to use, and what patterns will take you from one stage to another. But that work doesn't show when you pick up a cube and solve it in front of someone. And most people who try and fail to solve a cube never consider that it needs to be done in stages, and that one needs to find patterns that advance what you are doing without destroying work already done. As for me, well, I learned my first method out of a book.

It's a lot easier to figure out patterns once you already know how to solve a cube. Before you know a method, every attempt of working with the cube is fraught with fear of messing it up, of losing what progress you have made. If you make a mistake, you won't be able to get it back. But once you know how to solve a cube reliably, you feel free to experiment, to see what happens when you do this or that. If it doesn't work, you can always recover by your old method. I'm fairly confident that the first method used many cubists involved a screwdriver. It eliminates the fear, but overall it's cheating in my opinion. Once a cubist can solve it without the screwdriver, few go back to it. The screwdriver method helped me figure out the Rubik's Revenge, but a large part of it was utilizing what I knew about the regular cube.



The workhorse method I've used for the past couple of decades follows a sequence of stages. Of course, one starts with a scrambled cube.

photo.jpg

The first stage is solving most of the top face. You can do it corners first, edges first, or both simultaneously. Usually, I look for a face that looks mostly solved already and solve it. However, I'd already decided, for this sequence of photos, to have red be the top face.

Cubes aren't standardized in terms of what the color pattern is. Most cubes have, as colors, red, orange, white, yellow, green, and blue. Of the three cubes I can easily put my hands on right now, all three have red and orange on opposite faces, but two have green opposite blue and the other has green opposite yellow. The two with green opposite blue are mirror images of each other. Not all cubes use those 6 colors. There's one cube I'm interested in getting which effectively uses six shades of gray. This lack of standardization means that cubists can't reliably talk about the red side or the orange side. Instead, they tend to refer to the faces as up, down, left, right, front, and back. Sometimes you'll see top and bottom, but up and down means you can abbreviate the faces as UDLRFB without confusion. With top/bottom, you'd have two B's.

A few turns of this cube gets me to here

Solving a cube top-bottom-middle 2/7

I've solved the corners of the up face. An important aspect of this is that, fundamentally, I don't think of the cube as having 6 sides, each with 4 corners and 4 edges. Each corner is shared by three sides, each edge is shared by 2 sides. There are 8 corners, and each is unique in terms of the colors on it. A corner isn't solved until it matches all three sides it's on. As you can see here, they do.

Actually, I went a bit farther in this picture than I needed to. One important aspect of solving the cube in stages is that in a given stage you don't need to care about things which are easy to deal with at a later stage. In this case, what's important is getting the up corners solved relative to each other. As long as I can turn just the up side to fully solve them, I don't need to fully solve them now. This gives me the freedom to turn the up side as needed without feeling like I'm backtracking.

The second half of the first stage gets me to here

Solving a cube top-bottom-middle 3/7

Now three of the four up edges are solved, at least relative to the four up corners. It's clearer here that I've ceased to care about keeping the up side aligned with the FLBR sides. The unsolved edge isn't a mistake. It's really a time-saver for later. One of the patterns I use later can be a lot shorter if I don't have to worry about keeping all the edges in the up face solved. The short form messes up one up face edge. If I arrange for the unsolved up face edge to be the one messed up, then I haven't backtracked.

When cubists want to communicate and write about their methods and patterns, they typically can't use pictures, animations, etc. Rather, they've come up with a number of written notations. The notation which seems to be standard now uses the initials of the faces, U, D, L, R, F, and B, to indicate turning that face a quarter-turn clockwise, as viewed facing the side. Also standard is using a prime marker, ', to indicate the inverse turn (i.e. a quarter-turn counterclockwise). To indicate a half-turn, a ² is used. Turns in a sequence are written left-to-right in the order they are done. So RL'F²R'LD² would mean to turn the right face clockwise, then the left face anticlockwise, then the front face a half-turn, then the right face anticlockwise, then the left face clockwise, then the down face a half-turn. This, or something similar, is one of the patterns I use in a later stage of this solution.

At this point, I turn my attention to the bottom corners

Solving a cube top-bottom-middle 4/7

When solving the top face, I can be rather free about how I turn the cube; there's little structure I'm trying to preserve. When working on the bottom, I have to be careful to not destroy the top face in the process. This is where patterns come in, and where the stages get more fine-tuned. Whereas the first stage of this solution was to virtually solve the entire up face of the cube, the second stage is merely to get the four corners in the down face to have the down color on the down side. There's a single pattern I use to accomplish that goal, but I sometimes use it twice, and I sometimes use it backwards. It's often handy to know patterns both forwards and backwards. If I only knew it forwards, I'd have to use it twice a lot more often.

Sometimes you get lucky when solving by stages. Sometimes, by chance, the next stage is already solved for you. That happened here. Stage 3 is rearranging the down corners so that they are solved relative to each other. This time, when I got the corners twisted right, they were already solved relative to each other. So there's no other picture, and we can go right on to....

Solving a cube top-bottom-middle 5/7

...solving the bottom edges, and finishing off the final top edge. The simple pattern for moving an edge from the center slice (i.e, the four edges which aren't on the top or bottom faces) to the bottom edge also moves an edge from the bottom edge to the top edge. Leaving that hole in the top face allowed me to successively move edges into the bottom where they belonged, pushing the edge that was already there into the hole in the top, where I don't care about it. After solving three of the bottom edges that way, I just have to put the top edge into the final bottom spot before I put the final bottom edge in place. Doing so puts the final two top/bottom edges into place simultaneously.

I find solving the cube to be a very tactile endeavor, and the standard notation to be...not so much. To me, R and L' (and conversely R' and L) feel much more similar than their notation implies. Sure, one is clockwise and the other is counterclockwise, but they both are turning the top of a side away from me. I'm not always big on left/right differences, and I find clockwise/anticlockwise to be the same sort of hard-to-internalize distinction. When I'm trying to work a sequence from notation, I find I have to be very careful or I'll screw it up. I don't really know a pattern until it's committed to muscle memory.

At this point, the only non-trivial part of this cube is the four edges in the center slice, so let's do something about them

Solving a cube top-bottom-middle 6/7

There are two things which can be wrong with an edge in the center slice: it can be in the wrong place, and it can be twisted. Normally, I'd proceed by putting them all in the right place as stage 5, and reorienting them as stage 6. Once again, I got lucky, and didn't have to do one of these stages. Specifically, after putting all the edges in the right place, they happened to be properly twisted, so I didn't have to do anything.

One of the things which makes patterns easy to discover (assuming you know how to solve a cube and therefor are willing to play with a solved cube without fear of permanently messing it up) is that a pattern, repeated several times, will eventually get the cube back to where it started. Playing with the cube and finding nifty patterns can also lead to neat designs on the cube. For instance, R²L²U²D²F²B² is a well-known pattern that puts an X on all 6 faces. A lesser-known pattern I like is RLFBRLFBRLFB, which puts a zig-zag stripe across four sides.

This cube is now trivially easy to solve, as it's just a series of U and D turns away from solved.

Solving a cube top-bottom-middle 7/7

I don't even tend to count that as a stage.

There's more to say about the cube, but it can wait for another posting (I have more pictures to upload to Flickr, as well).

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Have you seen the electronic, touch sensitive ones?

heh heh heh. He said boobs!!

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