How do you solve the NxNxN cubes: the Pocket Cube (2x2x2), Rubik's Cube (3x3x3), Rubik's Revenge (4x4x4)...

                                  NxNxN Cubes

Rubik's Pocket Cube (2x2x2)

Mechanism: 6 armed spider
Combinations: 88,179,840
Difficulty: Easy

Rubik's Cube (3x3x3)

Mechanism: 6 armed spider
Combinations: 43,252,003,274,489,000
Difficulty: Average

Rubik's Revenge (4x4x4)

Mechanism: grooved sphere
Combinations: 1.776257242*10^47
Difficulty: Painful

Mastermind/Ultimate (5x5x5)

Mechanism: both (best guess)
Combinations: 5.405747889*10^82
Difficulty: Be afraid. Be VERY afraid.

To put that last number into perspective, some scientists believe that
there's "only" about 10^79 protons, neutrons, and electrons in the
entire UNIVERSE.  Even so, once you get the method down pat, which is
VERY easy to learn, you should be able to solve each of these puzzles
in 1, 3, 15 and 30 minutes respectively.  I've decided to group these
puzzles together because in spite of the fact that they have radically
different numbers of combinations and sizes (the Ultimate puzzle being
about the size of a Borg cube, and even more intimidating), their
methods of solving are remarkably similar.  In fact, you can adapt (or
is that assimilate?) these methods to solve ANY NxNxN cube they decide
to come out with in the future.

NOTATION

I'll be using 2 different notations.  First is the standard notation
used in just about every solution manual to date, which is T(op),
B(ottom), L(eft), R(ight), U(p), and D(own), followed by a +
(clockwise quarter turn), - (counter clockwise quarter turn), or a 2
(half turn).  The other notation was used in "Rubik's Revenge: The
Simplest Solution", whose author escapes me at the moment, which is in
the format x D y.  x and y are vertical columns on the front face,
starting at 1 on the left and N on the right.  D is the starting
direction to turn the bottom face, which is ALWAYS a quarter turn,
R(ight) for clockwise and L(eft) for counterclockwise.  For example, 1
L 2's move sequence would be column 1 DOWN, bottom LEFT, coloumn 2
DOWN, bottom RIGHT, column 1 UP, bottom LEFT, column 2 UP, bottom
RIGHT.  To summarise, the columns ALWAYS alternate with the bottom
face, the first 2 column moves are ALWAYS down, the last 2 are ALWAYS
up, and the bottom face ALWAYS alternates directions, ie. it's either
L, R, L, R, OR R, L, R, L.  Confused yet? :-) OK, if you are, try this
pattern now:

1 L 2:
1 DOWN bottom LEFT 2 DOWN bottom RIGHT 1 UP bottom LEFT 2 UP bottom
RIGHT.

To get back to where you were before, perform the above pattern twice
more.  If you're still uncomfortable with the pattern, try a few more,
eg. 1 R 2.

Each sequence of moves in this notation represents the (x, y)
coordinates of the piece we're focusing on, with (1, 1) in the bottom
left corner.  For those of you not familiar with coordinate geometry
that this means is that the bottom left corner is (1, 1), the piece to
its right is (2, 1), the piece just above it is (1, 2), the
centre/corner piece beside both of these is (2, 2), etc.  Or, put more
simply, for coordinates (x, y) means count off x pieces to the right
and y pieces up, counting the bottom left piece as 1 in both cases.  If
this is confusing the heck out of you, here's a diagram to help you
out, using the Mastermind/Ultimate/Borg cube (obviously not done to
scale).

          +-----+-----+-----+-----+-----+
         5|(1,5)|(2,5)|(3,5)|(4,5)|(5,5)|
          +-----+-----+-----+-----+-----+
         4|(1,4)|(2,4)|(3,4)|(4,4)|(5,4)|
          +-----+-----+-----+-----+-----+
         3|(1,3)|(2,3)|(3,3)|(4,3)|(5,3)|
          +-----+-----+-----+-----+-----+
         2|(1,2)|(2,2)|(3,2)|(4,2)|(5,2)|
          +-----+-----+-----+-----+-----+
         1|(1,1)|(2,1)|(3,1)|(4,1)|(5,1)|
         y+-----+-----+-----+-----+-----+
         /x  1     2     3     4     5

Along the way, I may refer to coordinates or a pattern with x or y
instead of a number.  Merely stick the needed number in place of the
letter.  For example, x L 1 could refer to 1 L 1, 2 L 1, 3 L 1, 4 L 1,
or 5 L 1.  Similary, 2 R y could refer to 2 R 1, 2 R 2, 2 R 3, 2 R 4,
or 2 R 5.  It just depends on what piece you need to move.  N, as
you've probably guessed, is merely the size of your cube.  This heavily
mathematical representation may seem to overcomplicate things, but
trust me... you'll end up thanking me later.

THE SOLUTION!

During this solution process, you will undoubtedly come across several
sections which simply don't apply to your cube, unless, of course,
you've decided to take the plunge and engage the Borg.  Simply skip
over those sections until you come to something relevant.

THE TOP LAYER

This should be easy enough to solve on your own by kludging at it for a
while, but if not, I'll give you a few basic methods to kludge it
together on your own.  If you have centre pieces on your cube, it's
easier to keep everything straight, but if not, don't worry.  Just pick
a corner to define as "correct" and take it from there.

The Corners

If the corner you need to fix is on the bottom side of the cube, then
move it so it's directly below where it needs to be.  If the proper
colour is on the:

1. left side, hold the cube so the proper colour is on the left side
   and perform L+ D+ L-.
2. Right side, hold the cube so the proper colour is on the right side
   and perform R- D- R+.
3. bottom side, hold the cube so the proper cubelet is on the
   left side: perform L+ D- L-
   right side: perform R- D+ R+
   and then perform 1 or 2 as needed.

If the cube in question is in the right place, but oriented wrong, then
perform 1 or 2 as appropriate (the same restrictions apply), reorient,
and perform 1 or 2 again.

Pocket Cube users, you're almost done!  Skip down (WAAAAAAAY down), to
THE BOTTOM LAYER, and leave the rest of us to suffer through the
remainder of the solution.  Right now, you should have a solid ring of
colour around the edges, ie. the top corners on each side should have
the same colour.  If not, just pretend any errant pieces are oriented
incorrectly and go back to 4.  This warning serves for the rest of you,
too.  If the colours aren't matched up, then go back to 4.  Original
Cube users, drop down to The Top Edges.

[ Editor: the page has no step 4 in this section, and no section titled
THE BOTTOM LAYER; the Pocket Cube resumes at BOTTOM CORNERS below. ]

The Top Centres

1. If the centre piece you need is on one of the sides, then rotate the
whole cube until you come across a "gap" that is in the same relative
position, ie. if the centre is in the top left corner, rotate the cube
until the gap is also in the top left corner of the top layer.  Then,
move the needed centre piece AWAY from the front side (left or right),
move the needed top slice DOWN, move the needed corner back in to fill
the gap, and move the top slice back UP where it belongs.
2. If the needed centre piece is on the bottom, then position it so
that if you could look straight through the cube, the centres would
match up.  Then move the gap DOWN, move the needed center AWAY (again,
left or right), and then move the gap back UP.  Now go to step 1.

The Top Edges

1. If the edge you need is on a SIDE EDGE of the cube, then determine
where it belongs and make that the front side.  Move your edge piece to
the BACK side of the cube, rotate the FRONT side left or right as
needed so the gap is on the same layer as the edge piece.  Move the
needed piece around to fill the gap, and rotate the front slice back.
CAREFUL!  If you rotate the front side wrong, the edge piece may go in
backwards!  If you do this by accident, move the edge piece out of
there, do an F2, and move it back in.

2. If the edge piece you need is on a BOTTOM edge of the cube, with the
proper colour on the SIDE of the cube, then move the edge piece so it's
just below where it should go.  Then do an F+/- (your choice), move the
edge piece in to fill the gap, and do an F-/+.

3. If the piece you need is on a BOTTOM edge, with the proper colour on
the BOTTOM side, then move it so it's directly below a gap (any gap).
Move the gap down, move the edge piece away, and move the gap back up.
Now, go to 2.

The top side is now complete, and if you've done it right, you should
have a solid ring of colour around the top edges.  If not, move any
errant edge pieces out of their current position and move them to where
they should be, using the above tips.

Even numbered cube users, skip the next step and go onto
THE MIDDLE CENTRES

If your cube is and odd numbered cube and thus has a true centre on
each side, then rotate the middle layer of the cube until the centres
match the ring around the top layer.  If you don't do this, you'll have
plenty of headaches later when you come across "impossible"
configurations and wonder how it could have possibly gone so horribly
wrong.

THE MIDDLE LAYERS - Part 1 - The Edge Pieces

Flip your cube around now so that the bottom side becomes the front
side.  From here on in, this solution will become FAR less verbose.

Find a piece you need to move and rotate the front side until its
non-front side matches the colour of an adjacent side.  Now rotate the
cube until the edge you want to move it to is on the bottom.  If the
edge piece is on the:

1. Left side: 1 L y
2. Right side: N R y

See?  I told you this was going to get less verbose! :-)

BOTTOM CORNERS

Welcome back, Pocket Cubers!  For those of you who skipped over all the
intervening stuff like I suggested, then rotate your cube until the
bottom layer is now the front layer.

Unless the 4 corners are in the proper positions already or will be
simply by rotating the front face (if they can be, do so now), then if
possible, rotate the front face until there is only one cubie in the
right position.  Then, if the corners need to be rotated as:

1. <----
   | +-+_
   | |  /|  1 R N
   v + /

2. ----->
   _+-+ |
  |\  | |   N L 1
    \ + V

Those of you who are unlucky enough to get either 0 pieces or 2 pieces
in the proper place no matter how you rotate the front layer, simply
perform either of the above.  Then you will be able to get only 1 piece
in the proper place.

If you combine these two, rotating the cube to gain the correct
alignment before performing the second part, DOES NOT end back with
what you bagan with as you might expect.  Instead, it rotates 2 corner
pieces.  Either way, it rotates the top 2 corners by moving the top
colour to the front.  If this seems too cumbersome then move the front
side back to the bottom again, and R- B- R+ B- R- B2 R+ B2 rotates all
but the front left corner counterclockwise, B2 R- B2 R+ B+ R- B+ R+
rotates them clockwise.  All other pieces are untouched.  At most, you
will need to execute this pattern 3 times to orient them all correctly.

Pocket Cubers, YOU'RE DONE!  Jump down to the bottom for a special
message.

BOTTOM EDGES

Of course, they're not really the bottom now, but that's beside the
point.  OK, here we go...

1. Rotate (1, x), (x, 1), (x, N) clockwise: x L 1.
2. mirror: x R N.

Eg. if you're using a Rubik's Revenge and x=2:

1. Rotate (2, 1), (1, 2), (2, 4) clockwise: 2 L 1.

The mirror image would be:

2. Rotate (3, 1), (4, 2), (3, 4) counterclockwise: 3 R 4.

To rotate the bottom and side centre edges, perform one of the
patterns, flip the cube around so that the top is now the bottom, and
then perform the other.  For bottom and left centres, start with 1.
For bottom and right, start with 2.

ALL combinations of edge pieces can be solved with these 2 basic
patterns.  If you're using the Rubik's Revenge or above and you've got
2 edge pieces left, then you truly know the meaning of the words
"Rubik's Revenge".  To solve for this special case, rotate one of the
non-centred, middle layer slices a quarter turn, then solve for these
middle layer edge pieces again.  If they ever come out with a 6x6x6 or
above, make sure you rotate the proper layer!

Original Cubers, YOU'RE DONE!  Join your Pocket Cubing companions at
the end for a special message.

FINAL STAGE - The Centres

If you encountered Rubik's ultimate revenge, then you know why this
step was saved for last.  If not, your time WILL come... trust me.

Corner Centres

Rotate the front slice so that the necessary centre is over the a gap
in the same place in the bottom layer.  If you rotate the cube upwards
so the bottom is now the front, the position should be identical.  Now,
if the corner is in the:

1. top left: 2 R N-1.
2. top right: N-1 L 2.
3. bottom left: 2 L N-1.
4. bottom right: N-1 R 2.

Don't feel that you're forced to maintain the same front side at all
times.  If it's more convenient to use a different face as the front,
then by all means, reorient the front and move the cube around as much
as you want!
Rubik's Revengers: REVENGE IS YOURS!  Join your fellow Cubists and
Pocket Cubists at the end for a special message.

Edge Centres

The same thing applies here.  If another side would be more convenient,
then reorient and MOVE IT AROUND!  OK, people, we're into the final
stretch.  Again, get your edge centres lined up with an appropriate
gap, then if it's at the:

1. top: 3 L 2 OR 3 R 4.
2. left: 2 R 3 OR 2 L 3.
3. right: 4 L 3 OR 4 R 3.
4. bottom: 3 R 2 OR 3 L 4.

Whew!  It's been a long haul, but you've just become THE ULTIMATE
CUBIST!  Join your fellow cubers who passed before you for a special
message, and then invite them all over to party hearty!

                      C O N G R A T U L A T I O N S !

You've just joined the ranks of the elite, and completed the ultimate
revenge!  Give yourself a pat on the back.  You deserve it!  Heck, for
completing these puzzles, you deserve a standing ovation!  Before I go,
here's a few patterns to try out.

For everyone: (L+ R+ F+ B+)^3

For original cubers and above:

L2 R2 U2 D2 F2 B2
U+ (R2 F2 B2)^2 U- (R2 F2 B2)^2

Try making your own, completely original patterns, or combine the ones
above.  On the Rubik's cube or above, you can make all letters in the
Roman alphabet.  On the Ultimate puzzle, you're able to make all of the
Japanese katakana.  Try spelling your name on the cube, or perhaps
spelling the name of your favourite anime.  By picking up your cube,
you've opened up a door to a wondrous paradise of patterns that you've
never experience before.  The only limit to what you can create is your
imagination.