Solving Rubik's cube and variants.

APPEARANCE:
The Rubik cube is a representation of  a stack of smaller cubes making one larger cube.  The original Rubik cube looks like a stack of 27 cubes arranged in 3 layers, each layer being 3 lines of 3 cubes.  In reality, this is not the case as only the initially showing outer faces all round the cube are ever visible.  The 'internal' faces are not proper faces, but shaped as part of the mechanism which holds the whole object together.

SIZE:
This does not mean the physical size of the whole cube, but the number of 'elements' (i.e. apparent smaller cubes) making up each dimension of the cube.  The original Rubik cube was size 3 (3x3x3 in appearance).  Then came a smaller 2-cube.  Rubik launched a 4-cube (Rubik's Revenge) and, later, a 5-cube (Professor Cube).  Now there are even larger cubes available called V-Cubes.  Cube sizes 2, 4, etc are even-sized cubes, and 3,5, etc are odd-sized cubes.

ELEMENTS:
The individual smaller 'cubes' are called elements and for solving purposes are of 4 types:
Corners: every size cube has exactly 8 corners and on each corner there are 3 faces showing.
Edges: these are elements between adjacent corner cubes and have 2 faces showing.  The number varies with the size of cube.
Centres: these are elements in the exact centre of a face and have only 1 face showing.  Even cubes do not have centre cubes while odd cubes have exactly 6
Middles: these are elements with only 1 face showing but which are not in the exact centre of a face.
Centres do not need to be solved as they effectively stay in the same place relative to each other.  Therefore, for cubes with centres, the centres show what colour each face has to be.
The 2-cube has 8 corners and nothing else.
The 3-cube has 8 corners, 12 edges, 6 centres, and 0 middles.
The 4-cube has 8 corners, 24 edges, 0 centres, and 24 middles.
The 5-cube has 8 corners, 36 edges, 6 centres, and 48 middles.

FACES:

F = Front face - the face straight in front of you as you look at the cube
B = Back face - the opposite to the Front face
L = Left face - the face on your left as you look at the cube
R = Right face - the face on your right as you look at the cube
U = Up face - the face on the top as you look at the cube
D = Down face - the opposite to the Up face

For some solutions the move is from a position in one face - the Source - to a position in another face - the Target.  These are called S and T and will vary depending on circumstances.

MOVES:

Just naming one of the Faces by its letter means a quarter turn clockwise of the relevant Face.  Clockwise means in the direction the hands of a clock would move if affixed to the face in question.  Thus, for example, a clockwise move of B would be physically in the opposite direction to a clockwise move of F: that is to say that F requires the top of the Front Face to move to the right as you look at the cube from in front whereas B requires the top of the Back face to move to the left as you still look at the cube from the front.

For an anti-clockwise move, the letter is followed by a negative sign.  Thus L- means an anti-clockwise move of the Left Face.

For a half-turn, which is the same whether clockwise or anti=clockwise, the letter is followed by number 2 (effectively meaning 2 quarter turns).  Thus U2 would be a half turn, or 180 degree turn, of the Up face.

In sequence patterns the move directions may be separate from the Face identities.  In these cases, clockwise moves are identified as + or as 1 while anti-clockwise moves are - and half-turn moves are 2.

For the S and T Faces, these will sometimes be inner slices - i.e. not one of the 6 standard Faces.  If the slice is between U and D, turn the same way as D would turn, and if between F and B, turn the same way as B would turn.

LOGIC MOVES:

For some of the earlier stages of the solution you are expected to use your own logic.  If you cannot work out these simpler stages for yourself, solving the cube becomes just a memory process with no real understanding of what is happening.  These notes on solving the cube are aimed at those who can get so far with their own logic and need help with the later stages.

SEQUENCES WITH DEFINED MOVES AND FACES:

There are sequences of moves - perhaps 4, 8, or even more, when some short method of remembering them will help.  For these, other letters are used.  For example, P means a series of 4 moves being 2 clockwise turns followed by 2 anti-clockwise turns.  This can be written as ++-- and as P stands for Positive this helps to remember that the 2 positive (+ = clockwise) moves come first.  We add the relevant Faces after the P, and if there only 2 Faces identified we repeat the Face pattern until we have 4 moves.  Thus PUR means U+ R+ U- R-.   In general, if fewer Faces are identified than the number of moves required, just repeat all or as much of the Faces in order to reach the required number of moves (and repeat several times if necessary).
Similarly, therefore, N (for Negative) means --++ and NUF would be U- F- U+ F+.

This method of using a letter to define a series of + - or even 2 moves helps to remember quite complex sequences.

My full list is:

P = + + - -
N = - - + +
Q = + + - - + - - +
W = 2 - 2 + 1 - 1 +
G = 2 - 2 - 2 2 2
Z = - + - 2 + - - 2 2
J = - - - + + - + + -
{note: J is - followed by -(Q)}

AAHH...SEQUENCE

This is a sequence operating on the top corner cubes to twist them so thast the correct colour is uppermost (on the U Face).  If a corner is not correct, it is manipulated from a position at top right (junction of U F and R).  A turn of either F or R will place the correct colour on U.  This is called A (Away), and moving it back again is called H (Home).  There is a corresponding A and H being the same type of move on the other face (i.e. if the first A is applied to F, the next is applied to R and vice versa.  The AAHH pattern, therefore, is either R F- R- F  or  F- R F R- depending on the required start move as explained above.

The full AAHH... process is:

1.  with first corner needing twist in top right position, do AAHH twice
2.  move U so that the next corner needing twist is at top right, and do AAHH twice
Continue until all corners needing twist have been done.

IN AWAY... SEQUENCE

An edge which needs to go onto U from F can be moved simply by first moving the slice which is parallel to U and which contains the edge so that the edge is out of the way ( i.e. no longer on F), then bring the destination edge place down with either F or F-,  move the slice back again so that the required edge is returned to its former position, and finally performing F- or F to undo the previous F move.  This is the logical process described below in doing the Horseshoe and its opposite face.  BUT this disrupts an edge in the bottom layer, so a special sequence is needed to protect that bottom edge, as follows:

IN:                Move required edge onto U but ensure it is the wrong way round (F or F-)
(This brings the bottom cube which needs protection to the space previously occupied by the relevant edge)
AWAY:        Move the slice with the protected element sideways out of the way
BACK:         Return the required edge to its start position
AWAY:        Now move that slice to the side again
TURN:         Turn F again to bring the ultimate home of the required edge down to the receiving slice position
INSERT:       Move the slice with both the relevant and the protected edges back to the front
RECOVER:   Turn F to get desired result

CUBE SOLUTION - Standard 3-Cube (3x3x3), LAYER BY LAYER

1  Bottom layer: Logic

2  Middle layer
a) Logic method to place edge from top front centre edge to relevant middle edge:
B2  D2 to put bottom layer safe
F or F- as required
D2  B2 to recover
b) If this would disrupt already positioned edge:
PUR NUF (top front centre to right middle)
NUF PUR (top right centre to right middle)

3  Top edges orientation
Flip opposite edges (back centre and front centre) with B PLU B-
Flip adjacent edges (bach centre and left centre) with B PUL B-

4  Top edges position
Leave left centre still and rotate other three clockwise with WUR

5  Top corners
Position with QURUL (leaves front right and moves others anti-clockwise)
Twist with AAHH...

CUBE SOLUTIONS: All Sizes: CORNERS - EDGES - MIDDLES

1  Bottom corners (oriented to Centre colours for odd sizes): Logic

2  Top corners
a) position:  QURUL  (U face)
b1) twist: one by one with AAHH...
OR
b2) twist 3 clockwise (ULB junction unchanged) with WUR
b3) twist 3 anti-clockwise (ULF junction unchanged) with (-W)UR
b4) if need to twist 1 clockwise and 1 anticlockwise, do b2 with the corner needing anti-clockwise in the safe postion, and then do b3

3  Top edges (start)
Horseshoe pattern: Use logic to complete 3 of the 4 edge lines on U

4  Bottom edges
Turn cube over so that D face becomes U face and the incomplete edge from what was U and is now D is facing forward.
Use same Horseshoe logic but make sure the incomplete D (ex-U) edge is always kept facing forward as it will get disrupted.

5  Finish Top/Bottom edges
Turn cube over again so that D (with one edge side outstanding) reverts to U: Use IN AWAY ... Sequence

6  Middle edges
1) Central line as Source and Target
a) to flip right side edge on F use GRS
b) to move 3 edges round cyclically use ZRSRF
2) Other (non-central) line
a) move right side on F from lower position (Source) to upper position (Target - same distance from central line): GRS
b) move 3 edges round cyclically on Source: ZRSRF

7  Remaining middles of just one face: Logic

8  Remaining middles of all other faces (or can use this throughout if not able to complete 7 above);
a) determine Source and Target to move one middle from U to F.  The start and end positions of the identified middle must be equidistant from the UF junction and line up across the centre point of the UF junction either diagonally or straight
b) move U clockwise to give a revised Source
c) the S (Source) and T(Target) slices are those running left to right in their relevant planes, thus S is a slice parallel to F and T is a slice parallel to U
d) JUSTS will move relevant middle element from revised S position to Target position

SIAMESE CUBE SOLUTION - based on Standard 3-Cube, LAYER BY LAYER

The Siamese Cube is fully joined across one whole junction - not the staggered variety.

It is necessary first to establish which colour face goes where.  By counting edge cubes you will find two that are 'short' (because they are excluded by the junction).  Arrange the Siamese so that the 'short' colours are positioned correctly relative to both the junction AND the conjoined corners.

1  Bottom (con-joined) layer: logic

2  Middle layer
a) get 2 adjacent edges in postion by the 'put safe' logical method of the standard cube
b) the other two edges must be put in together by the same method.
To achieve this, flip and/or move the edges while on the top layer (with the conjunction at the front) to get the desired relative positions and orientations
Flip using:
B D PLU PLU D- B- to get same efect as standard B PUL B-
B D PUL PUL D- B- to get same effect as standard B PLU B-
note: doing PLU twice or PUL twice to achieve the other's result is because doing just once moves one of the bottom corners.  By doing twice, the corner is repositioned again, although it does get twisted: ignore this because it gets sorted out later)

Move using WUR (with conjunction at left)

3  Top edges orientation
Flip as above

4  Top edges position
First look at the top corners.  These will be correctly positioned relative to each other, but may need twisting.  Using the positions of the corners to determine the correct relative positions of the edges, use WUR as above, moving U as necessary if repeats are needed.  It may be necessary to do this several times so as to get one of the edges correct first and then complete the remainder

5  Remaining corners.
Corners will now all be in their correct positions but some may neeed twisting.  These may be in pairs (one needing a clockwise twist and the other anti-clockwise), or trios (all needing twisting in the same direction) or a combination of these.  With the Siamese positioned so that the junction is through the L of the upper section and the R of the lower section, the move described below will twist the pair of adjacent corners located at ULF and ULB of the upper section.  ULF will twist clockwise while ULB will twist anticlockwise.  If necessary, moves corners to get them in the correct starting position, do the twisting move, and restore to original positions.  If corners need to be twisted the same way, do one of them by this method, then pair the other up with a different corner and do it again.  And so on...
The twisting sequence is:
WUR followed by (-W)UR (i.e change all + or 1 to - and all - to + on second complete pass)