Making Sense of Rubik's Cube™

by Billy Gard

Terms and Definitions
Move Strings
Solving the Cube
Rubikian
Glossary
Other Cubological Puzzles

I remember the first time I tried to solve the cube. When I was told that you were to turn the layers until every face was a solid color, my reaction was “You got to be kidding”. It was all I could do to get one face solid. Conclusion: It was a neat toy to play with, but I sure wouldn't want to own one.

Sure enough, Christmas came, and I had already descibed this puzzle to dad, and he knew he saw it before. And wouldn't you know it - I got the cube from Santa that year. I knew it was my sentence to figure the cube out. Regretably I didn't really figure it out on my own; I “tapped into some resources”.

Anyone who can solve the cube by just tinkering around with it is either autistic or some kind of genious, a “blockhead” if you please. They are the kind of people who you can give any date in the past and they can flash you the day of the week it fell on. But a highly analytical person may be able to figure it out using concepts like the ones I will be giving below.


Terms and Definitions

The cube is composed of three kinds of cubies: six center cubies, each displaying one tile in the center of a cube face; twelve edge cubies, each displaying two tiles in the middle of an edge; eight corner cubies, each displaying three tiles on a corner. The center cubies are named after the face they are on (e.g. top center), the edge cubies after the two adjoining faces (e.g. the front-right edge), and the corner cubies after the three adjoining faces (e.g. top-right-front corner). Center tiles are labeled after their face (e.g. the top center tile), edge tiles after the face they occupy and the adjoining face (e.g. the top edge tile of the right face), and corner tiles after the face they occupy and the two adjoining faces (e.g. the top-front corner tile of the right face).

center cubies edge cubies corner cubies

Move Strings

Probably the greatest facilitator to cube solving is the development of a notation for sequences of moves called strings. For instance, R2M-1R2M is a string that means "a half turn of the right face, a counterclockwise quarter turn of the middle layer, another half turn of the right face, and a clockwise quarter turn of the middle layer." The effect of this string is to cycle three edge cubies without upsetting the rest of the cube. The 7 fundamental moves are:

U = a quarter clockwise turn of the top (up) layer,
D = a quarter clockwise turn of the bottom (down) layer,
F = a quarter clockwise turn of the front layer,
B = a quarter clockwise turn of the back layer,
L = a quarter clockwise turn of the left layer,
R = a quarter clockwise turn of the right layer, and
M = a quarter clockwise (viewed from above) turn of the middle horizontal layer.

Putting -1 after a move indicates doing it backwards. E.g. R-1 means a quarter counterclockwise turn of the right layer. Putting 2 after a move indicates doing it twice. E.g. F2 means a half turn of the front layer. You can also put -1 or 2 after a variable representing a move string or after a move string in parentheses, indicating to do the whole string backwards or twice.

If you tried the 3-cycle string given above and messed up your entire cube, maybe by trying to do it backwards, that means you will now have to really learn to read strings, because we will use them to solve the cube.


Solving the Cube

Turn the cube so the white center tile is on top. Turn the sides so all the edge cubies with a white tile are in the top layer, Whether the white tiles are on the sides or on the top doesn't matter for now.

Next is to move the corner cubies with white tiles into the top layer. But you want to do it without disturbing the edges which you have just placed. To do this, turn a side face to move a top corner without a white tile to the bottom layer, then turn the bottom face to move a white tile into that corner, then turn the first face the other way to bring that corner to the top. (e.g. R-1DR). This is the simplest example of a conjugate string, which consists of a string preceded by a preparatory string and followed by the same preparatory string backwards - in short, any string of the form xyx-1. Its chief purpose is to move cubies to a position where they can be operated on, and then move them back again. You should end up with all 9 cubies on the top layer having a white tile.

Now we want to twist the corner cubies so their white tiles are all on top. To twist the top-front-right corner cubie clockwise do R-1DR to move the corner cubie to the bottom and FDF-1 to bring it back up the other way. To twist it counterclockwise do it the other way, FD-1FR-1D-1R. Repeat until all four top corner cubies are untwisted and a solid white X appears on the top face.

Next we want to flip the edge cubies so their white tiles are on top. To flip the top-right edge cubie do R-1MR to move the edge cubie to the middle layer and RM2R-1 to bring it back up the other way. Repeat until all four top edge cubies are unflipped and the entire top face is white.

Next we want to switch around the four top corners so they all "agree", i.e. all four sides of the top layer to have a matching pair of corner tiles. If no corner tile pairs match, then swapping any diagonally opposite pair will fix it. Do LR to move two opposite top corners to the bottom, D2 to swap them, and R-1L-1 to move them back up. Now all four sides should have matching corner tiles. If the two corner tiles match on only one side, then the two corners on the other side from them need to be swapped. Face these to the front. Do LU-1R to put these two corners into the bottom in diagonally opposite positions, D2 to dwap them, and R-1UL to bring them back up. Now the four top corners should agree.

Next we want to move the four top edges so that they agree with the corners. You can easily do this with two strings. To swap opposite edge cubies hold the cube so they are at the top-left and top-right. Do R-1L-1 to move them to the middle layer, M2 to swap them, and LR to bring them back up. To swap two adjacent edge cubies hold the cube so they are at the top-front and top-right cubicles, do R-1UL-1 to move these to diagonally opposite positions in the middle layer, M2 to swap them, and LU-1R to bring them back up. Keep swapping edge cubies until they match the corner tiles and the whole top layer is solved.

Turn the solved layer to the bottom. We now want to solve the middle layer by placing the four edge cubies so they match the the center tiles on the 4 sides. Look for any edge cubie on the top layer that doesn't have the color of the top center tile, i.e. belongs in the middle layer. Turn the cube so that the front and right center tiles are the two colors of that edge cubie. If the edge cubie's top tile matches the right center tile, turn the top layer so that the edge cubie is at the top-right cubicle. Do RU2B to move the front-right edge to the top layer without moving any other middle or bottom layer cubies there. Do U-1 to replace it with the right cubie, and B-1U2R-1 to bring it back down. Now the two center tiles should have the correct edge between them. If the edge cubie's top tile matches the front center tile, turn the top layer so the edge cubie is at the top-back cubicle and do RU2BUB-1U2R-1, which is the above string backwards. If all the middle layer edges are in the middle layer, but any are not placed right, hold the cube so an edge that is in wrong is in the front-right cubie and do either of the above moves to pull it out of the center layer, then follow the above instructions for placing the correct one there. Repeat as necessary. By now the middle layer should be solved. You can now turn the two solved layers so they are in agreement.

Look at the top center tile. This is the color for your ramaining face. First you want to untwist the top corner cubies so that all four corners have the tile matching the top center tile turned to the top, i.e. you have a solid X on the top face. A corner will either be twisted a 1/3 revolution clockwise, be twisted a 1/3 revolution counterclockwise, or will already have the correct color turned up. You will either have an even number of twisted corners with the same number in each direction, 3 twisted corners all in the same direction, or no twisted corners. Hold the cube so a corner cubie that needs to be untwisted clockwise is at the top-front-right (or if none exist, any corner that needs to be untwisted will do). Then do R-1DRFDF-1, the same string that twisted a corner in the first layer. This twists the the corner cubie clockwise, but totally messes up the bottom two layers. But now turn just the top layer to bring a corner cubie to the top-front-right cubicle which needs to be untwisted counterclockwise (or if none exist, any corner that needs to be untwisted will do). Now do FD-1F-1R-1D-1R. Now since you did one string to twist a corner cubie one way and did the same string backwards to twist another corner cubie the opposite way, whatever you did on the bottom two-thirds of the cube was undone and is all fine now. Notice that? This is an example of a commutator string, which consists of a string followed by an adjustment string, then the first string done backwards and finally the adjustment done backwards as well - in short, a string of the form xyx-1y-1. Its chief purpose is to do any operation on a part and it's opposite on another part, so that two opposing operations cancel out each other's effects on the rest of the cube. Repeat the above twisting procedure until all four corner cubies are untwisted.

Next you want to unflip edge cubies so the top face is solid. There are always an even number. Hold the cube so an edge cubie that needs to be unflipped is in the top-right cubicle. Do R-1MR2M2R-1, the same move that flipped an edge in the first layer. The edge cubie is unflipped, but the bottom layers are also messed up. But turn the top layer to bring another edge cubie that needs to be unflipped to the top-right cubicle. Do RM2R2M-1R, the same string backwards. Repeat this until the edge cubies are all unflipped and the top face solid.

Next we want to bring the corners into agreement. It is similar to the way it was done for the first four corners, but involving longer strings to preserve the bottom two layers. If only one side has a matching pair of corner tiles, then cycling three corners will fix them. We will do this by swapping two pairs of adjacent corners. Hold the cube so that they are on the right side and do LU-1RD2R-1UL-1, the move used to swap two adjacent corners in the first layer. Rest of cube messed up? Well then do a U to bring the right two corners to the front and go LU-1RD2R-1UL-1, the same string forward or backward. Now the corners should agree. If there are no pairs of matching corner tiles, do LU-1RD2R-1UL-1, then U2 to bring the back two corners to the front, then LU-1RD2R-1UL-1 again. Now the corners should agree.

Finally we have to move the edge cubies into place so they agree with the corners. If there are three non-conforming edge cubies, hold the cube so that the middle edge of that threesome and the side it belongs on occupy the front and right faces. Do R-1UL-1M2LU-1R, the same move used to swap two adjacent edges in the first layer. To fix the mess, turn the top layer to bring the remaining pair of nonconforming edges to the front and do the R-1UL-1M2LU-1R to fix them and the rest of the cube. If two pairs of adjacent edges need to be swapped, do R-1UL-1M2LU-1R to swap one pair, U2 to turn turn the other pair into place, and R-1UL-1M2LU-1R to fix them too. If two pairs of opposite edges need to be swapped, do R-1L-1M2LR to swap the left and right edges, then U to rotate in the last pair and repeat. That this point you have only to turn the last layer into conformity and the cube is solved.


Rubikian

There is now an actual spoken language for depicting cube moves. Each syllable stands for a turn and begins with a consonant depicting which face: F=front, B=back, L=left, R=right, N=top (north), and S=bottom (south). The consonant is followed by a vowel: I=clockwise quarter turn (inward), E=counterclockwise quarter turn (ex, or outward), and U=half turn (U-turn). An M after the vowel indicates that the turn applies to the inside layer adjacent to the specified face. Try the following moves:

Semremsimrim spotted my cube and I'll get him.
Sumnumlumrumfumbum is a checkerboard I presume.


Glossary

center
One of the 6 one-tiled cubies or associated cubicles; the tile located thereon.
commutator
A move string having the form xyx-1y-1, where x and y are substrings.
conjugate
A move string having the form xyx-1, where x and y are substrings.
corner
One of the 8 three-tiled cubies or associated cubicles; a tile located thereon.
cubicle
One of the 27 positions in the cube occupied by cubies.
cubie
One of the 27 little cubes making up the the cube, to include 8 corner cubies, 12 edge cubies, 6 center cubies, and 1 twenty-seventh (and actually nonexistant) cubie deep inside.
cubist
Someone who can solve the cube.
edge
One of the 12 two-tiled cubies or associated cubicles; a tile located thereon.
flip
To reorient an edge cubie within the same cubicle.
flippancy
(of an edge) A binary value that indicates whether the edge is flipped, 1 if it is flipped, or 0 if it isn't. (of a cube) the sum of the flippancy of all edges modulus 2. It is always zero, meaning that the total number of flipped edges is always even.
grand master cubist
Someone who can solve the 4 x 4 x 4 version of the cube.
layer
Any horizontal 1 x 3 x 3 slice of the cube.
Rubikian
A speakable language for describing cube moves.
cube
A generic term for the Rubik's Cube™ puzzle, consisting of a cube sliced into a 3 x 3 x 3 solid, and colored tiles on the outer surface, with each face its own color. The layers can be rotated to scramble the colors. The object of the puzzle is to turn the layers so that every face is restored to a solid color.
string
A written seqence of cube moves, using a letter to depict a clockwise quarter turn of a particular layer, followed by a -1 to depict a counterclockwise quarter turn, or a 2 to depict a half turn.
tile
any one of the colored squares on the cube, each face consisting of 4 corner tiles, 4 edge tiles, and 1 center tile.
twist
v. To reorient a corner cubie within the same cubicle. n. (of a corner) a ternary value for the corner's twistedness: 1 if it is twisted clockwise, 2 if twisted counterclockwise, or 0 if not twisted. (of a cube) the sum of the twist of all corners modulus 3. It is always zero, meaning that the number of clockwise twist minus the number of counterclockwise twists is always a multiple of 3.

Other Cubological Puzzles

Rubik's Mini Cube™, though far simpler to solve, is in a way more confusing because of the absence of center tiles to define face colors. Solutions for the regular cube that start by fixing just the corners are the total solution for this cube.

Anyone who's been introduced to the Mini Cube is bound to have thought of this one already: Rubik's Revenge™. Anyone beguiled by the regular cube is bound to be scared under the table by this one.

The octagonal barrel is in reality a regular cube that has had the four parallel edges truncated. That I have made a couple of these from cubes myself proves this point. The number of combinations is reduced, and there are twenty-four solved states rather than one, since the vertical columns of color can be arranged in different ways and still be considered solved. What really makes the barrel spectacular is how it looks when you scramble it.

The pyramid is based on the same idea as the cube. As beguiling as it may look, it is considerably simpler than the cube puzzle. If you are familiar with the concept of conjugates and commutators you may figure this one out fast.

This dodecahedron, beguiling as it may appear, has a structure similar to the cube in that it has one cut through it for each face, and like the cube it has corner, edge, and center "cubies" and can thus be solved using similar strings.

Rubik's Professor Cube™ is designed for those who are a glutton for punishment, and essentially goes to the structual limit of how many cubies you can cram in and still have a puzzle that stays in one piece.

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