Rubik's cube and a modest solution

Appendix (in separate window) - Tips for steps 1 and 2: the top layer

Introduction

The Cube is an absolutely amazing invention - or should I call it a discovery? - made by the Hungarian professor Ernö Rubik in the mid-1970s. It is a cube where each face is divided into 3x3 squares, and where each of the six faces can be twisted at will.

Conceptually, the Cube consists of 3*3*3 = 27 subcubes. However, the central "subcube" does not exist, and the other "subcubes" only show three faces at most. There are 8 corner "subcubes" with three faces visible, 12 edge "subcubes" with two faces visible, and 6 central "subcubes" (actually squares) with only one face visible. Contrary to the initial impression that everything can be changed, the six central squares remain fixed with respect to each other. They are the only stable points and are vital for orientation!

The first mystery is this: How in the world can the cube hang together? Why does it not fall into pieces? The other one is: How is it possible that something so simple can be so devilishly difficult to solve? In fact, from the mathematician's perspective it remains an unsolved puzzle. No one knows "God's algorithm", i.e. the recipe for solving the cube in as few twists as possible from any starting position. No one knows what the minimum number of moves is from the (unknown) "worst" position.

For myself, the greater mystery is probably the mechanical engineering aspect. Although I am an engineer by training, I am by no means a practical man. My training was in physics, mathematics, theoretical stuff. To me it would "intuitively" seem obvious that the cube is impossible. Or at least that it would take an elaborate set of miniaturised cog wheels and ball bearings to allow such freedom of movement without losing structural integrity. But in fact the inner working of the cube is very simple once you open it (but I do need advice both to disassemble and assemble it). Actually, this raises a third mystery: Why was the cube not invented thousands of years ago, or at least, say, 500 years ago, when watchmakers started developing great mechanical skills?

I was given a cube by a lady friend in 1980, luckily during the summer vacation, and managed to solve it in a week. The difficulty, of course, was to develop a "tool kit" of useful "operators", i.e. series of twists that manipulate a subset of the small "subcubes" that make up the Cube without affecting all the other "subcubes". Predictably, the first thing I did was to inadvertently get lost, so that I could no longer return to the cube's virgin state. This made it a whole lot more difficult to study the effects of different operators. A second difficulty was that my "recipes" tended to be long and complicated series of twists. This, together with my cube being "sticky", made it all too easy to lose track and be forced to start again from "square one". Of course, I wrote down my solution, but I lost that paper a long time ago.

Last year my then eight-year old daughter used Alexander's approach to deceive me: she stripped off the colored tape bits and rearranged them to "restore" the cube to its virgin state. Smart girl!

I have since looked at other people's solutions both in books, magazines and on the Web. There you can find a large library of algorithms/tools/recipes/strategies. In fact, by combining a large memorized library of series of twists, a lot of exercise and a well lubricated "racing cube", it is possible to solve the cube with astonishing speed: something like 15 seconds. It probably helps to have a highly developed talent for spatial visualization: anything beyond two or three twists is beyond my mental horizon!

To amuse myself during lazy days on the beach, I have compiled an extremely limited set of operators that will allow anybody to solve the cube in an easy-to-remember step-by-step fashion. Only five sequences of length between 7 up to 13 twists are needed to solve the cube. These sequences all seem to have their own flavour and "personalities", which makes it feasible even for me to remember them and to avoid mixing them up. The drawback, of course, is that I need to use these series repetitively. I will use many more moves than necessary just to avoid having to memorize a confusing plethora of similar, closely related sequences that would exploit the symmetries and specifics of the situation.

Notation

To describe the moves, a simple convention is proposed: F for Front for the face facing you. B for the Back, L for the Left face, R for the Right face, U for the Upper face, D for the Down face. Each clockwise twist (as seen from outside the cube looking at the face in question) is symbolised by the corresponding letter. A counterclockwise twist is symbolised by F-1, B-1 etc. Please note in particular, that the B twist is in the opposite direction to the F twist, and the D twist is in the opposite direction to the U twist, just as screws twisted into a wall from opposite sides turn in opposite directions.

To clarify:

 Starting from a solved cube, with red facing you and white on the Up side... ...this is the result of "F"... ...and this of "R-1".

Two similar twists in a row are symbolised e.g. "F2".

For each sequence of moves any side can be selected to be the "front side", irrespective of its color, but throughout that sequence the orientation is frozen. It is recommended not to shift the cube around too much during a sequence, as it is easy to lose its orientation and start confusing F turns with R turns etc.

Summary of the solution

Select one color for the "top floor". I always pick white, as it is easy to spot, but any of the six colors will do. Steps 1 and 2 do not require any memorized sequences, just a few tricks. They are described separately.

The views were chosen to show what the operations do. Be sure to face the correct side of the cube when applying the sequences given below. They are described in detail later in the next section.

The sequences may look intimidating, but when you execute them, you will discover that they are not as random as they look. Moreover, the last two turns in steps 4 through 7 become obvious, as they restore what you have already achieved in previous steps.

Each step may have to performed several times with different orientations of the cube.

 Step Operation Method Sequence of turns Resulting position 1 Align top edges with same-color central squares See appendix See appendix 2 Position top corners in correct orientation See appendix See appendix 3 Position mid-deck edges in correct orientation Facing the right side in the drawing on the left: R-1DRD FD-1F-1 4 Position lower deck corners D-1LD FD-1F-1 L-1FD 5 Orient lower deck corners Facing the left side in the drawing on the left: R2B-1R2B RB-1RB 6 Position lower deck edges L2B-1 R-1LF2 RL-1U2 BL2 7 Orient lower deck edges Facing the left side in the drawing on the left: B-1R2B2 RB-1R-1B-1 R2 FDB D-1F-1

Details of step 3 - middle layer edges positioned and oriented

The objective is to position and orient the mid-floor edge subcubes without destroying the top floor. The method is to move the edge subcube on the bottom left side to the right edge of the front side:

Note that the view is from the left side here, in order to show where the edge subcube comes from (a) and where it is going (b). The "green" face is considered to be the front face here.

To work properly, the left-facing side of the (a) subcube must have the same color as the central square of the front and the down-facing side of (a) must have the same color as the central square of the right side. This is clearly not the case here. The left-facing side of (a) is orange, while the central square of the front is green. The hidden right side is yellow (i.e. its central square is yellow), so we want to place the green/yellow edge subcube in the (b) position. First we have to find it and place it in its correct orientation in the (a) position. If it is in the middle layer, it first has to be moved to the bottom layer. This is achieved by orienting the Cube so that the green/yellow edge is in the front/right edge position and then applying the step 3 algorithm.

Once the green/yellow (in this case) subcube is in the bottom layer, we rotate it into the (a) position by twisting the bottom, i.e. by performing as many D (or D-1) twists as necessary. Two cases may arise. If it is in the correct orientation, we go ahead and apply the step 3 algorithm. This will bring the green/yellow subcube to its correct position in the correct orientation in the middle layer. If it is in the incorrect orientation, this can be fixed by first rotating the subcube to the front side (a D twist) and then applying the step 3 algorithm. This will place the subcube in its correct (a) position and orientation, so all we have to do is to apply the algorithm a second time to move it to (b) in the correct orientation.

The sequence (algorithm) has 7 twists. (It may help to count to seven when you execute it.)

R-1DRD FD-1F-1.

The first four twists involve the right side and the down side. The final three concern the front side and the down side, of which the last two become obvious in order to restore the (white) top layer.

The same procedure is repeated, changing front side as necessary, until all the edges in the middle layer are correctly placed and oriented. Two layers have now been completed!

Details of step 4 - bottom corners positioned

From now on, we need to manipulate the bottom layer without upsetting the top and mid layers, so at the end of each of the remaining four steps, the first two layers should be intact.

To prepare for step 4, we start by twisting one of the four bottom corners into its correct position. Then we can always move the remaining three corners into position by exchanging adjacent corner "subcubes" using the following method once or several times as needed. The Cube is oriented so that its bottom layer faces us and thus becomes the Front side, and so that the corners to be exchanged are placed at the lower and upper right positions.

The exchange is achieved by the following algorithm which has 9 twists:

D-1LD FD-1F-1 L-1FD

It is conveniently grouped into three subgroups. It only involves the Down, Left and Front faces. As before, the last two twists become obvious in order to restore the top layer.

The result will look something like this. Unless we have been lucky, several corners still need to be rotated to be oriented correctly.

Details of step 5 - bottom corners oriented

This can be achieved more efficiently by choosing a sequence that matches the specific situation, but I have found that the repeated application of a single sequence that rotates three of the four corners clockwise by 120 degrees is sufficient to cover all situations, and that the sequence is easy to remember, as it only involves two sides of the Cube.

In the illustration, we look at the right side of the Cube. The upper right corner of the front side (green here) is left unchanged, while the remaining three corners of the right side are rotated. However, the left side (green in this case) is considered to be the front face.

The sequence has 8 steps (counting double turns as a single move):

R2B-1R2B RB-1RB

It only involves the right and back sides of the cube in a repetitive pattern. This is the sequence that I find easiest to remember out of the five needed, which is welcome as it may have to be executed repeatedly to reach the penultimate state, where one corner is correctly oriented and the other three each require a single 120 degree rotation.

The result is that the corners of the bottom layer are correctly placed and oriented. All that remains is to place and orient the bottom layer edges.

Details of step 6 - bottom edges positioned

Again, this is achieved by repeatedly applying a single sequence, rather than using a specific tool for each situation that can arise. The tool I have chosen looks like this:

As we look at the bottom layer of the Cube, the algorithm moves the edge "subcubes" around in a triangular fashion. This is enough to handle all situations through repeated application.

The sequence has 10 steps:

L2B-1 R-1LF2 RL-1U2 BL2

I have arranged it into four groups. The first and the last group mirror each other. The two central groups in effect twist the central vertical layer (between the left and right sides) followed by a double frontal twist and double top twist, respectively. Folks, this procedure has individuality!

Upon completion, the Cube looks almost finished, but appearances can deceive. The first time I solved the cube back in 1980, this is where I spent most of my effort. It is not easy to fix the last wrongly oriented edges without negating all the previous progress!

Details of step 7 - bottom edges oriented

The algorithm inverts two adjacent edge "subcubes". Repeated application will solve all cases.

The sequence is long and difficult, 13 steps. This should not come as a surprise, as all side effects now have to be negated along the way. It is easy to go wrong, and this is especially frustrating so close to the goal. It is recommended to concentrate on the matter at hand!

The two edges to be inverted are placed on the right and down side of the right face. The left side (green in this case) is considered to be the front face. The algorithm then is:

B-1R2B2 RB-1R-1B-1 R2 FDB D-1F-1

How to memorize all of this? Well, for the first 8 moves, only the back and right sides are involved. The first group of three moves has to be learned by rote, then the next four moves seem more "natural". I have put the "R2" in a separate group, because it is unexpected and easy to forget. Then the FDB group has to be learned by rote. The last two moves are obvious, once you get there.

The result is...

Bliss!

Applets to do the cube online

I do not normally do the cube on the computer, as it seems easier to inspect it when you hold it in your hand, but there are some "cool" Java applets that allow you to play with the cube on the screen. It also offers a big advantage for anyone who wants to explore move sequences and needs to return to a "virgin cube" frequently.

I used this applet from 1996 as a template for the graphics.

Here is a better applet that makes it easy to turn the cube for inspection.