# Mag's 3x3 Rubik's Cube Solution

This is a small collection of operations for the 3x3 Rubik's Cube that more or less reflect the way I solve the cube. I tried to keep it minimal, so it assumes that you already know how to get one entire face solved; this means not only having all of one color on one side, but also having those 8 pieces in the proper position with respect to the other sides.
The basic approach to my solution is to get one face (the "top") solved, then get the middle slice solved, then get the bottom solved. Some of the operations below (e.g., the first) assume that parts of the cube have not yet been solved, so they may affect already-disturbed pieces other than the particular ones being manipulated. Other operations, on the other hand (e.g., the last), affect only the pieces being manipulated; these operations (and many others) are often useful in other situations.
Note: Since I've put this up on the Web, I occasionally get queries for more online information about the cube. I have a small list of cube-related URLs, and a list of possible sources for cube-related puzzles. I don't really know anything else.

## Notation

Cubelet
There are 27 of these, one of which is hidden in the middle of the cube, and 6 more of which (the centers of the faces) are probably never referred to at all. The rest of them move about the cube freely, to our great consternation.
Corner
A cubelet at the intersection of three faces. May be referred to by the names of the three faces that meet there. For example, UFR and DBL are corners.
Direction
(of a turn). There are 3 directions. + means a quarter-turn clockwise, - means a quarter-turn counterclockwise, and * means a half-turn. If the turn is not specified in a move, it means a quarter-turn clockwise.
Edge
The three cubelets comprising the intersection of two faces. May be called by the names of the two intersecting faces; for example, UF and FR are edges. May also refer to the edge cubelet.
Edge cubelet
The middle cubelet of the three on an edge. May also be called the edge.
Face
One of the six surfaces of the cube. The faces are called U (top), D (down), R (right), L (left), F (front), and B (back), relative to the orientation of the cube in your hand.
Flip
Turn a single cubelet -- usually an edge cubelet -- in place. See also rotate. Do not confuse with swap.
Inverse
(of an operation). The operation that exactly reverses the moves involved in the operation. Take each turn backwards, and take all of the turns in reverse order. For example, the inverse of (R2- D R2 D* R2- D R2) is (R2- D- R2 D* R2- D- R2).
Move
A single turn of one or more slices of the cube together. A move consists of a face, a set of slices (usually just one for the 3x3 cube, but often more for bigger ones), and a direction.
Operation
A sequence of moves.
Orientation
The way a cubelet is situated in its position. Corner cubelets may have 3 different orientations; edge cubelets may have 2.
Position
Where a cubelet is located.
Rotate
Turn a single cubelet, probably a corner, in place. See also flip. Do not confuse with swap.
Slice
Also layer or slab. Nine cubelets. The 3x3 cube is 3 slices deep any way you look at it. The slices are numbered 1, 2 and 3, relative to a face. If slice(s) are not specified in a move, it means slice 1, i.e., the face itself.
Swap
Switch the positions of 2 or 3 cubelets. Do not confuse with rotate.

## Operations (My Solution)

### Move an edge cubelet from the bottom slice onto the middle slice without disturbing the top slice

Hold the cube so that the destination spot is at FR (front right).
Turn the bottom so that the edge cubelet that is to be moved up is at FD.
If the color showing on the front of the about-to-be-moved edge cubelet matches the color on the (center of the) front face, do:
D- R- D R D F D- F-
Otherwise, do:
U123 D* L D- L- D- F- D F
Note that that step 3b is really just "turn the cube sideways and do the mirror image of step 3a".

### Get bottom corner cubelets in place, ignoring their rotation in place, without disturbing the top two slices

It is always possible to turn the bottom face so that there are two corner cubelets in the right place. Do this.
If the two out-of-place corners are adjacent to each other, hold the cube with them at DFL and DFR, and do:
R- D- R F D F- R- D R D*
Otherwise, hold one of the out-of-place ones at DFR (and the other one at DBL) and do:
R- D- R F D* F- R- D R D-

### Rotate bottom corner cubelets in place, without disturbing the top two slices

See how many cubelets need to be rotated. An easy way is to look at the bottom of the cube and see how many corners' colors match the center cubelet.
If two corners need fixing, use the operation to rotate two corner cubelets.
If three corners need fixing, hold the one that is already fixed at DFL and do:
R- D- R D- R- D* R D*
or its inverse. Which one you do depends on whether the corners need to be rotated clockwise or counterclockwise. In practice, I can never remember which is which, but it turns out that doing the above operation twice is the same as doing its inverse. So in practice I just apply the above operation, and if they're still not fixed, I do it again.
If all four corners need fixing, hold the cube so that none of the "bottom color" is showing on the front face. (There will be either one or two such sides.) Now do the operation given in the previous paragraph. This will fix one corner. Now do that operation again as described in that paragraph.

### Rotate two corners, without disturbing any other cubelets

If the corners are not adjacent, do a single turn to make them so. Remember which face you turned, so you can do its reverse later.
Hold the two adjacent corners so that the one needing to be turned clockwise is at UFR and the other is at UFL, and do:
R- D R F D F- U- F D- F- R- D- R U
Undo the move you did in step 1.

### Put bottom edge cubelets in place, ignoring their orientation

The key operation is:
R2- D R2 D* R2- D R2
(or its inverse). This operation moves the three edge cubelets at DL, DB, and DR around each other clockwise. (Its inverse moves them counterclockwise.)
If there is one edge cubelet in place, put it at DF and do either the above operation or its inverse, as appropriate.
Otherwise, all four edge cubelets are disturbed. It doesn't matter which one is in front. The above operation or its inverse will make it so that one cubelet is in place. Do that, then go to the previous step.

### Flip two edge cubelets in place

If the two cubelets are on adjacent edges, hold them at UF and UR and do:
R- D2- R* D2* R- U- R D2* R* D2 R U
If the two cubelets are on edges across from each other but on the same face, hold them at UR and UL and do:
R- D2- R* D2* R- U* R D2* R* D2 R U*
If the two edges are not on the same face, put them there with a single turn (remembering it), then go to one of the first two steps as appropriate. After that, undo the single turn you did.
Note: Despite my clarification of what "D2-" means, I still get occasional emails asking for an explanation of this operation, which I have now finally written:

## "2" Moves in Mag's Cube Solution

Ever since I published my cube solution, with my own unique notation for representing operations, I've been hearing something like this about once a week:
> I have a question on your notation regarding the moves
>
> D2-, D2*, and D2.
>
> What do these mean? Any help would be appreciated because I keep
> getting stuck on flipping two adjacent bottom edges for the final
> sequence. Thanks.

What's up, everybody? They make perfect sense to me. :)
The "2" refers to the 2nd level deep, relative to the face mentioned. I use this notation because it extends to arbitrary cube sizes (4x4x4, 5x5x5, and larger). On the 3x3x3 cube, it is simply the middle slab of the cube. When the face is D (or U), it refers to the middle horizontal layer.
The -, *, or + (or lack thereof) still refer to how far to turn the slice: quarter-turn counterclockwise, half-turn, or quarter-turn clockwise, respectively.
D2- has the same net effect as doing {U- D} and then turning the whole cube a quarter-turn counterclockwise relative to the bottom (which I would denote as D123-, by the way). As you can see, this is a much more concise way to write that, in addition to the extensibility advantage that I mentioned above..
`   D2 (also U2):           R2 (also L2):      +---+---+---+          +---+---+---+     /   /   /   /|         /   /###/   /|    +---+---+---+ +        +---+---+---+ +   /   /   /   /|/|       /   /###/   /|/|  +---+---+---+ +#+      +---+---+---+ + + /   /   /   /|/|/|     /   /###/   /|/|/|+---+---+---+ +#+ +    +---+---+---+ + + + |   |   |   |/|/|/     |   |###|   |/|/|/+---+---+---+#+ +      +---+---+---+ + +|###|###|###|/|/       |   |###|   |/|/+---+---+---+ +        +---+---+---+ +|   |   |   |/         |   |###|   |/+---+---+---+          +---+---+---+`

Tom Magliery
tom@magliery.com