I've promised this at the very
beginning of this blog and I will now honor this promise: let me
explain the method I use for solving the (3×3×3) Rubik's cube. I hope I don't need
to describe what this devilish puzzle of Ernő Rubik's consists
of: unless you've been spending the last thirty[#] years on a desert island
somewhere, you must surely have seen one of the artefacts in question
(often it is encountered for the first time lying on a shelf in the
house of friends or relatives — generally in a scambled state
because another visitor was there before). I hold the Magic Cube to
be one of the most mind-boggingly amazing (if useless) inventions of
all times, and, to my eyes, the *real puzzle* was that of
solving the engineering problem of getting the cube to hold together
and just “work” much more than unscrambling it. But,
anyway.

Obviously there are many methods to solve the puzzle (here's
one, here's
another, here's yet
another and here's a (very
concise) fourth one). Some people start by getting the corners in
place, and then they handle the edges; others start by doing one side,
then the middle slice, and then the last third; yet others have more
“global” solutions, or more “orthogonal” ones.
Every method has its advantages and disadvantages: a tradeoff must
somehow be made between efficiency and simplicity, between appeals to
geometry and symetry, and so on. What I like about the solution I use
is that it is *simple to remember* (a good thing for people,
like me, who have a bad memory for motions in space) and relatively
simple to describe, and that it relies very little upon geometric
insight; as a counterpart, it is not very efficient, whether in terms
of speed or in terms of number of moves. But when there are thee
objects that must be put in place (say), I find it simpler to randomly
permute them until they fall in place than to try to figure out how to
achieve the right permutation in a single stroke.

### How I learned to solve the Rubik's cube

I first “solved” the cube somewhere around '92, after
reading an article[#2] about it
in a (then already) old issue of Scientific American, but
I solved it in a completely theoretical (and mathematical) fashion: I
did it all on paper, by writing simple computations of
group-theoretical nature, without ever looking at a real cube and
without appealing to my sense of 3D vision (I have none
whatsoever), and the solution consisted of a list of
“recipes” (operations, or explicitely presented group
elements) for exchanging two edges or two corners, flipping two edges
or rotating three corners. In *theory*, knowing these recipes
is sufficient to get the cube from any accessible configuration to any
other (and, in particular, to unscramble the puzzle), but in practice,
following without error the very long sequence of moves that served
to, say, rotate three corners (*without altering anything else in
the cube*) turned out to be very difficult, remembering (or
reading from the abstruse mathematical notation) which three corners
would be rotated and in which direction was also a pain, and, quite
generally, solving the puzzle using these computations was more than a
little unpractical. I did it once — it took me hours —
and I abandoned the cube, satisfied that I had defeated the beast.
The idiocy in my way (which reminds one of many mathematical jokes) is
that it is quite useless to have an operation that, for example,
rotates three corners without altering anything else in the cube: if
one decides to solve the cube “corners first”, then it
doesn't matter if the operation used to rotate three corners also
messes up with the edges, since they will be dealt with later: now
that simplifies the task considerably. And, quite generally, at every
step of the resolution except the very last, it is permissible for the
operations to mess up with some parts of the cube which will be put in
place later on. So my solution had the interesting
“feature” that it could be applied in any order (start
with the edges, or with the corners, or with the top, or with the
bottom, it doesn't matter since every operation does a very simple and
precise job). But that was also the reason why the operations were so
long and complicated. And why there were so many of them.

Then around '96 I encountered (a French translation of) a little
German book — or pamphlet, perhaps I should call it — by a
Prof. Dr. Kurt Endl, which claimed to describe the fastest
elementary solution, without mathematics

. Well, that was
something I needed: I bought the book and learned the method. So it
is from this Kurt Endl that I know the tricks I am about to describe.
I then totally forgot about the cube for years, until, in early 2003,
someone put one into my hands and asked me to solve it. Well, it took
me a whole evening of twiddling with it before I came up with a
solution that was similar to the one I had learned although quite
different in some parts. Since the way I rediscovered it is not as
good as the way I had learned about it in the first place, and since I
found again the book that I had lost, I will stick to the original
method.

Anyway, on to business. I will not explain all the details, because it would take a tremendous number of words to do so (or a fair number of pictures, and producing pictures is a pain), and it is probably best to let the reader discover them by himself. The basic strategy is: first do one side, call it the top one just to settle things, then do the middle slice (except possibly for one edge), then do the bottom corners, then the missing edge in the middle slice, then place the bottom edges, and finally orient them correctly.

I will not explain how to do the top slice: doing one side is
something that people discover very well by themselves (and although
some are naïve and do the side without caring about the adjacent
colors of the cubies — that won't do, of course: a full
*slice* has to be done, not a mere side — it isn't really
any harder to do the job properly). Essentially, start by doing the
edges, and then the corners. It isn't difficult (though the exact
details would be exceedingly tedious to write down), because one
doesn't care how the rest of the cube becomes scrambled, so one can
move things in place rather freely (if something already in place gets
in the way of something that needs to be put in place, just
“move it out of the way, perform the motion, and then move it
back”). So let's assume that the top slice (red, say) is
done.

After that, one needs to learn just three basic operations, which I
will call Operation
One

, Operation
Two

, and Rubik's
Operation

(because, apparently, Rubik himself discovered the
double-edge-flipper in question). Leave aside Rubik's Operation for
the moment. Operation One and Operation Two leave the top slice
intact; Operation One leaves the middle slice intact except for just
one edge (the front-right cubie[#3], say) while Operation Two leaves
the middle slice entirely intact. Both Operation One and Operation
Two start by moving a corner cubie (the top-front-right corner cubie)
out of place in the same manner, and then putting it back in place
(since the top slice must be preserved at the end of the operation) in
one of two different ways. Now let us describe these operations in
detail.

### Operation One

The image on the left (read
from left to right) graphically displays the sequence of seven moves
that make up Operation One, starting on a pristine cube. The cubie to
keep track of is the top-front-right (red-white-green) corner cubie.
The first three moves (common with Operation Two) move that cubie away
from its natural position, while the rest of the operation gets it
back in place. For those who prefer instructions in words rather than
in pictures, Operation One is: Right³ · Bottom³ · Right · Bottom ·
Front · Bottom · Front³, where Right

means turn the right
cube face one quarter turn clockwise (when looking at the face in
question)

, Right³

means turn the right cube face one
quarter turn counterclockwise (or, equivalently, three quarters turn
clockwise)

, and so on, and the operations are to be performed in
the order given (note that I have written Front·Bottom·Front³ whereas
the picture shows a movement of the back and middle slices instead,
but up to rotation of the cube this is equivalent, it's just that what
is Bottom

in one case should be Right

if we are moving
the back and middle slices: or Front·Bottom·Front³ =
(Back&Middle)·Right·(Back&Middle)³ if you wish, it's all a
matter of viewpoint).

The picture on the right, above, shows the result performing Operation One on a pristine cube. Two things should be noticed: first, only one edge of the middle slice has been displaced (the front-right edge cubie), and the cubie that got in its place is the one that was previously the bottom-left edge cubie; second, the two right bottom corners got exchanged, whereas the two left bottom corners stayed in place.

Of course, Operation One can be performed with right and left
exchanged, in a perfectly symmetrical fashion. To keep track of this,
we can call the one described here Operation Right-One

, and its
symmetric counterpart Operation Left-One

.

### Operation Two

The image on the left (read
from left to right) graphically displays the sequence of seven moves
that make up Operation Two, starting on a pristine cube. Again, the
cubie to keep track of is the top-front-right (red-white-green) corner
cubie. The first three moves (common with Operation One) move that
cubie away from its natural position, while the rest of the operation
gets it back in place, but in a way different from Operation One. For
those who prefer instructions in words rather than in pictures,
Operation One is: Right³ · Bottom³ · Right · Bottom³ · Right³ ·
Bottom² · Right, and obviously Bottom²

means turn the bottom
cube face one

(to make this clear
on the picture, a smaller cube has been added, showing the position
after a quarter turn counterclockwise).*half* turn (either way)

The picture on the right, above, shows the result performing Operation Two on a pristine cube. The top and middle slices are unaffected by the operation. The bottom corners are exchanged in opposite pairs (that is irrelevant, really: a half-turn rotation of the bottom face will get them back in place if necessary — although we do not include this in Operation Two) and three of them are rotated; and the bottom edges are permuted in some way.

Of course, just as Operation One, Operation Two can be performed
with right and left exchanged, in a perfectly symmetrical fashion. To
keep track of this, we can call the one described here Operation
Right-Two

, and its symmetric counterpart Operation
Left-Two

.

Well, that's about all one needs to remember. Here's how to use
these two operations to solve the puzzle, starting from the top slice
already in place. First get all the edges of the middle slice in
place using Operation One: this is easy because Operation One will
move one edge from the bottom slice to the middle slice, so by
positioning the edge in the appropriate place and then using Operation
One we can always get it to go where it should be (and if we choose
intelligently between Operation Left-One and Operation Right-One we
can even get it properly oriented: it's not all that important,
because Rubik's Operation is a universal edge-flipper, but we might as
well get things right when we can). Only put three of the middle
slice edges in place, ignore the fourth one for the moment, it will be
displaced in the next step. Second, get the bottom corners in place
(but possibly badly oriented), again using Operation One: since the
latter exchanges the two (bottom) right corners, this is easy (use the
“unsolved” middle slice edge as a sort of “work
space” which will get continually modified by Operation One, but
so far we don't care about it); one rotation (of the bottom slice) and
at most two exchanges of pairs will always position do the trick. The
third step is to get that last edge from the middle slice in place,
without disarranging the corners: the trick to do this is to use
Operation One *twice* in a row (without rotating the cube or
anything in between), either Right-One twice or Left-One twice,
because that will do a double exchange of (the same) corners, which
doesn't hurt, and still move the edges around; basically, doing
Operation Right-One twice will get the bottom right edge in the front
right position (again, choosing intelligently between Right-One and
Left-One avoids an annoying flip later on).

Almost done! We must still rotate the bottom corners. This is
done with Operation Two: since Operation Two rotates three corners (in
the same direction) and since there is a total rotation invariant over
all corners, it can be done — just remember that Operation
Right-Two (followed, if need be, by a half-turn rotation of the bottom
slice) rotates (counterclockwise) every bottom corner cubie except the
bottom front left one. The penultimate step is to get the bottom
edges right (except possibly for orientation): for this, we just have
to know how to permute three cyclically; the trick is to use Operation
Right-Two followed, after a rotation of the full cube, by Operation
Left-Two, where the rotation is chosen (clockwise when looking from
the top) so that both Operations Two are performed on the same corner
cubie (performed on

means that that cubie is the one from the
top slice which gets shifted out and back in place); this combination
will cyclically permute three edges from the bottom slice, so we are
done.

Only one thing might be amiss now: some edges might be incorrectly oriented (always an even number: this is another invariant). To solve this problem, we introduce the third operation, Rubik's Operation, which is rather longer than Operations One and Two, but it is also very easy to remember because of its elegant structure. Here it is:

### Rubik's Operation

The image on the left (read
from left to right) graphically displays the sequence of fourteen
moves that make up Rubik's Operation, starting on a pristine cube.
It's simplicity will become evident if we write it as follows: calling
Down the opertion which consists of lowering the *central*
slice of the front face (so that the top center cubie goes to front,
front center goes to bottom, bottom center goes to back, and back
center goes to top, and of course the top front edge goes to bottom
front and so on), and Up the inverse operation, then Rubik's operation
is: Down · Bottom · Down · Bottom · Down · Bottom · Bottom · Up ·
Bottom · Up · Bottom · Up · Bottom · Bottom (I chose to write
Bottom·Bottom rather than Bottom² so as to make the pattern more
apparent; perhaps it is even more apparent if we move the last Bottom
at the start of the sequence, which merely changes which edges get
flipped in the final result).

The picture on the right, above, shows the result performing Rubik's Operation on a pristine cube: this is merely a double edge flipper.

To get everything in place, take a pair of edges that has to be flipped, get them in place for Rubik's Operation (this might require one or two rotations to get the edges in opposite position if they are not so initially), apply Rubik's Operation to flip the edges, and then if necessary move them back where they were. Repeat until every edge has been flipped to its correct state. The cube is solved!

[#] I don't know the exact date, but the Rubik's cube turned thirty quite recently.

[#2] By Martin Gardner, or Douglas Hofstadter, or Ian Stewart, or another one of these mathematical chroniclers for the monthly.

[#3] The word
cubie

refers to one of the 3×3×3=27 little cubes which form the
big cube. Of these, 8 are corner cubies, 12 are edge cubies, 6 are
face-center cubies (which determine the color of the face), and one is
they hidden (and utterly irrelevant) central cubie.