How to solve the Rubik's Cube
|
|
This page describes one method of solving the popular Rubik's Cube puzzle. It is not the only method.
| 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 starting, a method is required 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
Each face can be turned either clockwise or counter-clockwise, with respect to its center (i.e. a move that may be clockwise to the viewer 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 (the U face will be used 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.
The individual pieces can be referred to 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. These notations refer to the piece that is in that place at that time, not the piece that should go there.
These six moves (U D L R F B), and any sequence of moves, do not move the position of the 6 center pieces. (They rotate the center pieces. This solution assumes that the rotation of these center pieces is unimportant.)
Step 1: Top edge pieces
The cube is assumed to be scrambled. The first thing to do is to choose 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 it will soon be learnt which colors are adjacent, which speeds things up 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. The aim is to get the correct pieces in the UL, UB, UR and UF locations. So, some of the following moves are needed: (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 the 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):
- Hold the cube so that white is still on the U face, but this white-OC piece is in the FR location. Find the spot where that white-OC piece should go. Rotate U until either F' or R can be applied to move the white-OC piece in the correct spot, so that the white face will move to the top. Now, move it to the U face by applying F' (if the white face is on the R side) or R (if the white face is on the F side). Example 1: The piece in FR has to be moved, with white being the R face, to its home location at UL, so apply U' F' U. Example 2: FR has to be moved, with white being the F face, to its home location at UL, so apply U² R U². Notice how U is simply moved, then bring the edge piece up to the U face, then move U back to restore the original position, plus the piece 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' (R' is not needed if the UR piece has not been placed correctly yet).
There should now be a white cross formed on the top of the cube. By now, it will be possible to think how the edge pieces are located relative to one another, which should speed things up.
After only a few repetitions, they will be 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 hold the cube so that the white face is on the bottom. The only middle layer edge piece that is not to be positioned is the one right above the corner piece that was not positioned 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, the FL piece is to be put 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—first turn the cube, then apply D').
Now all is prepared for 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. Either F' or R will need to be applied, depending on the orientation of the edge piece to be moved. Now, apply U until the piece to be moved is in the UF or UR (depending on the previous move) location, and then F or R, to get it back to normal.
Here is an example: Yellow is the F center. Orange is the R center. The Yellow-Orange edge piece is to be positioned, to the FR position. The D face has already been rotated so that the DRF location does not contain a white corner piece. The Yellow-Orange piece is in the UB location. The Orange is the U side, and Yellow is the B side. Thus, apply F' U² F.
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 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 take it out first (i.e. put any edge piece with the color whose center is opposite white on the 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 two thirds of the cube should be done, 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. The moves from the other steps should 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, 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).
An example: Say the Blue-Yellow piece is in the BR location. Furthermore, Blue is the U color, and Yellow is the L color. The order would be: 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, look to find the next edge piece that is required to put in the right location. So rotate U until it is in the UB location, and then by applying B' return the cube to a stable position. Then, 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, there is no 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.
- 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.
To reduce memorization at the expense of some speed, two of these moves suffice. In other words, apply all three of these moves in any sequence to an all-edges correct cube, and the result will be an all-edges correct cube.
Step 5: Position corner pieces
In this step it is required 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 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 being worked 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 the 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. Do the following moves: L D² L' [move the piece in question out of the way (to the DRB location)] U' [move the correct location to the UFL spot] L D² L' [move the piece in question back to the U slice] U [undo the U twist done 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 the ongoing example). Also note that it is satisfactory to rotate the U face before the move so that a particular corner piece can be moved into the UFL location so that it can be worked on. The only (slight) difference will be a need to rotate U at the end to make up for that. Note that these U-rotations should be very obvious. 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. Then 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].
This pattern may need to be applied up to three times. Note that with this method only one clockwise and one counter-clockwise twist can be done; other methods twist 3 corners but have side-effects on edges. If the two remaining corner pieces are diametrically opposed (e.g. at UFL and DRB), then apply R² (in this case) to bring both of them onto the U slice. Then, do the sequence. Then apply R² again to get to the original configuration.
Alternative method
While the above method may be good for a beginner, it is too slow to be used in speedcubing. Speedcubists tend to use a method whereby a 2x2x2 corner is correctly made (like a Pocket Cube) and expanded to a 2x2x3 cuboid (or rectangular parallelepiped). Then 2x3x3 (the equivalent of solving the top and middle layer above). Finally the last layer is solved using scores of algorithms (that must be memorized) based on the positions of the remaining cubies.