Introduction
Notation
Summary of the solution
Details of step 3  middle layer edges positioned and
oriented
Details of step 4  bottom corners positioned
Details of step 5  bottom corners oriented
Details of step 6  bottom edges positioned
Details
of step 7  bottom edges oriented
Applets to do the cube online
Links
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 mid1970s. 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 eightyear 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 easytoremember stepbystep 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. "F^{2}".
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 samecolor central squares 
See appendix 
See appendix 

2

Position top corners in correct orientation 
See appendix 
See appendix 

3

Position middeck edges in correct orientation 

Facing the right side in the drawing on the left:
R^{1}DRD FD^{1}F^{1}


4

Position lower deck corners 

D^{1}LD FD^{1}F^{1} L^{1}FD 

5

Orient lower deck corners


Facing the left side in the drawing on the left:
R^{2}B^{1}R^{2}B RB^{1}RB


6

Position lower deck edges 

L^{2}B^{1 }R^{1}LF^{2}
RL^{1}U^{2} BL^{2} 

7

Orient lower deck edges 

B^{1}R^{2}B^{2} RB^{1}R^{1}B^{1}
R^{2} FDB D^{1}F^{1} 

Details
of step 3  middle layer edges positioned and oriented
The
objective is to position and orient the midfloor 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 leftfacing side of the (a) subcube must
have the same color as the central square of the front and the
downfacing side of (a) must have the same color as the central
square of the right side. This is clearly not the case here. The leftfacing
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^{1}DRD FD^{1}F^{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^{1}LD FD^{1}F^{1} L^{1}FD
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.
The sequence has 8 steps (counting double turns as a single move):
R^{2}B^{1}R^{2}B RB^{1}RB
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:
L^{2}B^{1 }R^{1}LF^{2} RL^{1}U^{2}
BL^{2}
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 algorithm then is:
B^{1}R^{2}B^{2} RB^{1}R^{1}B^{1}
R^{2} FDB D^{1}F^{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 "R^{2}" 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.
Links
Just "googling" for Rubik on the web results in an abundance
of hits. Here are a few interesting links:
Chris Hardwick's
Rubik's cube page with many further links
The
mathematics of Rubik's cube