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How to solve the Rubik's Cube

The following essay, "How to solve the Rubik's Cube", describes one method to complete the popular Rubik's Cube. This is not the only of such methods, simply one of them.

Table of contents

Background

The following is one of many solutions to the Rubik's Cube. This solution was developed by David Singmaster[?], a British mathematician.

Before we get started, we must work out a method for describing the various moves that will be made. There are six faces, with the following notations:

  • Upper, or top face = U
  • Down, or bottom face = D
  • Left face = L
  • Right face = R
  • Front face = F
  • Back face = B

We can turn each face either clockwise or counter-clockwise, with respect to the center (i.e. a move that may be clockwise to you, when looking at the cube, may not be clockwise for that face, in relation to the middle of the cube). The names for the different kind of moves (I'll use the U face as an example) are:

  • A 90-degree turn clockwise on a face, is denoted by U.
  • A 90-degree turn counter-clockwise on a face is denoted by U' ("U prime") (Also note this is the same as U, done three times).
  • A 180-degree turn either clockwise or counter-clockwise on a face, is denoted by U2 ("U squared") and is the same as two clockwise turns, or two counter-clockwise turns.

We can refer to individual pieces by a two-letter (for edges) or three-letter (for corners) combination. For example, the piece in the upper right front corner is called URF, and the edge piece to the down and left of the cube is called DL.
Also, these notations refer to the piece that is in that place at that time, not the piece that should go there.

Also, note that during any sequence of moves the position of the center pieces with respect to one another is unchanged.

Step 1 - Top edge pieces

So your cube is scrambled right now. The first thing to do is to chose a color, say white (it tends to stand out from the other colors on the cube). It's also a very good idea to always do a specific color first, since you will begin to learn which colors are adjacent, which speeds up things considerably.

The first step is to form a cross on the top face of the cube. Orient the cube so that the white center piece is on top. You want to get the correct pieces in the UL, UB, UR and UF locations. So, you will need to do some of the following moves: (be sure to do those in the first step first.

  • If a white-other color (OC) piece is on the U face:
    • If white is in the U position, simply rotate the U face until the OC is lined up with its center.
    • If OC is in the U position, rotate the U face so that the piece is at an adjacent edge location to its desired location. Hold your cube so that white is the U center and OC is the F center. Now rotate U so that the white-OC piece is in the UR position. Now apply R' F'.
    • If a white-OC piece is in the middle slice of the cube (the middle third), then hold the cube so that white is still on the U face, but this white-OC piece is in the FR location. Now, you should notice that you will be able to move it to the U face by applying F' (if the white face on the R side) or R (if the white face is on the F side). Find the spot where that white-OC piece should go. Rotate U until you can apply either F' or R to move the white-OC piece in the correct spot, so that the white face will move to the top. Examples: You want to move the piece in FR, with white being the R face, to its home location at UL, so apply U' F' U. You want to move FR, with white being the F face, to its home location at UL, so apply U2 R U2. See how you simply move U, then bring the edge piece up to the U face, then move U back to restore the original position, plus the piece you just moved.
  • If a white-OC piece is on the bottom slice of the cube:
    • If the white is on the D face, simply rotate D until the OC is directly underneath its center, and apply F2 (assuming the piece is at the FD position) to put it in the correct location.
    • If the OC is on the D face, hold the cube so white is the U center, and OC is the F center. Rotate D so that the white-OC piece is in the RD position, and apply R F' R' (you do not need R' if the UR piece had not been placed correctly yet).

You should now have a white cross formed on the top of your cube. Also you should be developing an intuition about these moves. What you will learn to do after a few times through this, is just think how the edge pieces are located relative to one another. This should speed things up.

Step 2 - U face corner pieces

The second step is to correctly position three of the U face corner pieces. The reason that you will only put three of them and not four into place is that this method uses a "working space" which greatly simplifies the later steps.

There are three basic possibilities for putting corner pieces into place:

  • The piece is on the D slice, with the white side not on the D side. In this case, rotate the D face so that it is directly underneath the location that it should go to. Now, hold the cube so that the piece is in the DRF spot, and the intended location is the UFR spot.
  • If white is on the R side of the corner piece in DRF, apply R' D' R.
  • If white is on the F side of the corner piece in DRF, apply F D F'.
  • The corner piece is on the D slice, but the white face is on the D side. Rotate the D face so that the corner piece is in the DRF spot, and the intended location is the URF spot. Now apply R' D2 R D R' D' R. Note that you are doing R' D2 R to move the white side off the bottom of the cube, so that you can use one of the moves in the previous section. Also note that equivalent to this is: F D2 F' D' F D F'. I would imagine if you are left-handed this would make things easier. Speaking of which, if you are left-handed, I would be interested if you naturally use a particular one of these processes. Personally I am right-handed and do the "R' D2 R..." move, without really thinking about it.
  • The corner piece in question is in the right spot but incorrectly rotated. Therefore, we must rotate it. Hold the cube so that it is in the URF location. Now,
  • If the white side is on the R face, apply R' D' R D R' D' R.
  • If the white side is on the F face, apply F D F' D' F D F'.

So now you should be done one side, except for one corner piece. This location will be used to swap corner pieces in and out, greatly simplifying later processes. The moves in the first two steps are really quite intuitive. After only a few repetitions, you should find them simple and natural to do.

Step 3 - Middle edge pieces

This step involves correctly placing three of the four edge pieces on the "middle" layer of the cube. For these moves you will need to hold your cube so that the white face is on the bottom. The only middle layer edge piece that you do not position is the one right above the corner piece that you did not position correctly in step 2.

First of all, make sure the white side is on the bottom, and the "empty" (i.e. incorrect) corner piece on the white side is in the DRF location. The middle layer edge pieces will all be positioned in this step, except for the FR one.

To move a piece into position, rotate the cube about its vertical axis, so that the intended location is the FR location. (For example, you want to put the FL piece in place. Rotate the cube a quarter turn counter-clockwise). Now rotate the bottom slice so that the incorrect corner piece is in the DRF location. (So in the previous example -- for the FL piece -- you would first turn the cube, then apply D').

Now you are ready to do the move. The move to put the new edge piece into place can only be done if it is on the U slice. If it is, note which side is NOT on the U face. You will need to apply either F' or R, depending on the orientation of the edge piece you want to move. Then, apply U until the piece you want to move is in the UF or UR (depending on your previous move) location, and then F or R, to get it back to normal. I'd better give an example...
Yellow is the F center. Orange is the R center. You wish to position the Yellow-Orange edge piece, to the FR position. You have already rotated the D face so that the DRF location does not contain a white corner piece. You see the Yellow-Orange piece in the UB location. You note that Orange is the U side, and Yellow is the B side. Thus, you apply F' U2 F. All that description for three easy moves :-).

To continue, simply keep rotating D or D' and moving the cube to set up the same position, with an "empty" corner in DRF, the intended location at RF, and the piece you want to move in the U slice. Note that in some cases the piece may already be in the correct location, but orientated incorrectly. In this case you will have to take it out first (i.e. put any edge piece with the color whose center is opposite white on your cube into that location) and then put it back in that spot. In other words, with the DRF corner "empty" and the offending piece in the FR spot, apply F' U' F U R U' R'.

Now you should be done 2/3 of the cube, less two pieces: a middle layer edge piece and its adjacent corner piece, that appears to take a chunk out of the bottom (white) layer. Note that it is possible for the "empty" corner piece on the bottom layer to get solved by accident. If so, just ignore it, and pretend that it is unsolved.

Step 4 - Solve remaining edge pieces

This is the only step that requires any actual memorization. I think you'll find that the moves from the other steps become very natural after a short time. There are two basic parts to this step, as follows. The goal of the whole step is to solve all of the 5 remaining edge pieces. The first part is to solve three of these (UF, UL, UB), and the second part is to solve the other two together.

First of all, hold the cube so that the "empty" edge piece is in the BR position, and thus the "empty" corner piece is in the RDB position. To do moves in this part, you first of all move a piece into the BR location, then move it to the U face, to one of those UF, UL, or UB positions. The move is as follows. First, optionally rotate U. Then, apply R' or B. Then rotate U the desired amount. Then do R or B' (to undo the first part of this move). Pretty simple huh?! An example may be in order. Let's say the Blue-Yellow piece is in the BR location. Furthermore, Blue is the U color, and Yellow is the L color. You would thus apply U [to put the UL location (the destination) in the right spot] B U' B'. However, when actually trying to solve the cube quickly, before applying U' in the previous move, you should look to find the next edge piece that you wish to put in the right location. So rotate U until it is in the UB location, and then by applying B' you return the cube to a stable position. Then, you will need to rotate U some amount to get the UL piece (Blue-Yellow in the example) back to the right place. There is a tremendous amount of freedom in this sequence of moves. In fact, you do not need to return the edge pieces to the correct spots in between repetitions of this move. Simply realize how the pieces go with respect to one another, and then finally align them, when all three (UF, UL, UB) are done.

Now, there are four possibilities. The remaining edge pieces are the BR piece and the UR piece. Do the following:

  • Luckily, the pieces are correct. Move to the next step and smile at your good fortune.
  • The pieces are in the correct locations, but incorrectly oriented. Apply B U' B' U R' U R U'.
  • Both edge pieces (BR and UR) have the same color on the R side of the piece, which is the same color as the R center. Apply U' R' U' R U' R' U' R U'.
  • The other case (the UR piece has the R color on its U side, and B color on its R side, and the BR piece has the U color on its R side, and the R color on its B side). Apply B U B' U B U B' U2.

If you want to reduce memorization at the expense of some speed, two of these moves suffice. In other words, if you apply all three of these moves in any sequence to an all-edges correct cube, you will get back an all-edges correct cube.

Step 5 - Position corner pieces

In this step you want to move the remaining unsolved corner pieces to their correct locations, irrespective of orientation. Hold the cube so that the "empty" corner piece on the bottom (white) face is in the DRB location. Rotate the U face so that the piece that you want to position is in the UFL location. Apply L D2 L'. Now, rotate the U face so that the location (with respect to the top-layer edge pieces) of the corner piece you are working on is in the UFL location. Now apply L D2 L' again. Rotate U so that everything lines up. Repeat this (up to 3 times) until all of the corner pieces are in the right location.

This step is a little confusing at first. First of all, make sure your DRB piece is that "empty" (unsolved...not missing :-) ) corner piece. Say the UFL piece is Blue-Yellow-Orange. But that piece should go in the URB location. You would do the following moves: L D2 L' [move the piece in question out of the way (to the DRB location, if you're interested)] U' [move the correct location to the UFL spot] L D2 L' [move the piece question back to the U slice] U [undo the U twist you did earlier]. One thing to note when doing this move, make sure the original UFL piece does not contain the color of the bottom face (white in my ongoing example). Also note that you are free to rotate the U face before the move so that you can move a particular corner piece that you want to position into the UFL location so that you can work with it. The only (slight) difference will be that you will need to rotate U at the end to make up for that. Note that these U-rotations should be very obvious. You can simply line up the top-layer edge pieces with their respective centers.

Step 6 - Orient corner pieces correctly

Corner pieces must be rotated in pairs -- one clockwise and one counter-clockwise. Find two incorrectly rotated corner pieces that are on the same slice. Hold the cube so that one of the pieces in the UFL position and the other is somewhere on the U slice.

  • To rotate a piece clockwise, apply L D2 L' F' D2 F.
  • To rotate a piece counter-clockwise, apply F' D2 F L D2 L'.

Note that after orienting the first corner piece, apply U until the other corner piece goes in the UFL location. You will then need to turn U to undo the previous twisting (this should be fairly obvious). Here's an explicit example -- the UFL piece needs rotating counter-clockwise, and the UFR piece needs rotating clockwise. The full sequence would be as follows: F' D2 F L D2 L' [orient UFL piece] U [position other corner] L D2 L' F' D2 F [orient original UFR piece] U' [undoes rotation of U that was done earlier].

You may need to apply this pattern up to three times. Note that you can only do one clockwise and one counter-clockwise twist. You cannot twist three corner pieces all clockwise. If the two remaining corner pieces are diametrically opposed (e.g. at UFL and DRB), then you can apply R2 (in this case) to bring both of them onto the U slice. Then, do the sequence. The apply R2 again to get to the original configuration.

see also other HowTos



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