THE ULTIMATE SOLUTION TO RUBIK'S POCKET CUBE

The Ultimate Solution to Rubik's Pocket Cube utilizes the methods given in The Ultimate Solution to Rubik's Cube.  Obviously the Edge Piece Series is not used here but you must first master the Corner Piece Series and The End Game before attempting this method of solving Rubik's Pocket Cube.

Solving a Rubik's Pocket Cube (2x2x2) is not the same thing as solving a Rubik's Cube (3x3x3) after all of the edge pieces have been properly placed. Rather, it is simpler and, at the same time, more complex.

It is simpler because the corner pieces of the first face can be placed with one or two moves of cube faces or, at the most, a three turn series. In fact, since there are no center or edge pieces to match, the first corner piece is in place even before you begin!

But it is more complex because of a fundamental difference between the 3x3x3 and 2x2x2 cubes. In the former cube when only three corner pieces remain you will find them either all in position or all out of position. You cannot, for example, find that all pieces are in place except two corner pieces which are out of position. But with the Pocket Cube you can find exactly that, six corner pieces are in position and properly aligned and two are out of position. That creates a new problem.

In addition, the lack of center and edge pieces means that you have no aids to remind you of the proper orientation of the cube, of the turns you have made or the turn you must make next. Concentration is of critical importance, particularly in a complex End Game.

THE FIRST FACE

Since there are no center or edge pieces with which the corner pieces must be aligned, the first corner piece is in place simply because you say it is. Choose a color, any color. This is the color of the first (top) face. Choose the piece which you will proclaim to be in place with its top face color on top of the cube. Although it is possible that a second corner piece will be in place relative to the first corner piece, we will assume that it is not.

As was true in The Ultimate Solution to Rubik's Cube, all pocket cube images shown here will depict the front, top and right sides of a cube. If an entire cube is rotated by, say, 180o the new image will be described in terms of its new front, top and right faces.

The cube in Fig. 1a shows the yellow/orange/blue corner piece in place at right/front/top. The easiest corner piece to deal with at this point is one with its top color (blue in this case) on the bottom of the cube. If we rotate our cube by 180o we get the cube in Fig. 1b. In this figure the corner piece at right/front/bottom has its blue face on the bottom. A simple rotation of the right face by 180o (either direction) will move this corner piece into place at right/back/top (Fig. 1c).  Its orange face will be next to the orange face of the first corner piece.

Fig. 1a           Fig. 1b            Fig. 1c           Fig. 1d

It is preferable that the second corner piece goes into place in one of the positions adjacent to the first corner piece (as we have just done). This will leave a vertical cube face which may be rotated freely (in this case the front face). If, instead, a corner piece is moved into place in the opposite corner from the first corner piece it would then be impossible to move a third corner piece into place with a single move (since that would move one of the first two pieces out of place).

Another blue corner piece may be seen at right/front/top in Fig. 1c but the red/yellow/blue piece is at left/back/bottom and has its blue color on the bottom surface of the cube.  A  180o rotation of the bottom face and then a 180o rotation of the front face moves the third corner piece into place at left/front/top (Fig. 1d).

In trying to place the second or third corner piece you may find that the blue color of the target corner piece is on a vertical face of the cube. You may be able to see how a face turn will move the corner piece into place. Or a rotation of the bottom face may move your target corner piece into such a position. But such moves are frequently blocked (by corner pieces already in place).  If such a move is not obvious then rotate a face so as to move the blue color of a corner piece to the bottom of the cube and proceed as above.

Once you have placed three pieces by this method you must use a three turn series to place the fourth (or the third and fourth if the first two are diagonal to each other). The target corner piece must be in the bottom half of the cube and have its blue color on a vertical face. The bottom half of the cube is rotated so that the target corner piece is below its appropriate place on the cube as in Fig. 2a. Rotate the front face one turn clockwise (Fig. 2b); rotate the bottom face one turn clockwise (Fig. 2c); rotate the front face one turn counterclockwise (Fig. 2d).

Fig. 2a           Fig. 2b            Fig. 2c           Fig. 2d

If the blue color is in the right face rather than the front face (Fig. 3a) then the series is: right face one turn counterclockwise (Fig. 3b): bottom face one turn counterclockwise (Fig. 3c); right face one turn clockwise (Fig. 3d) completing the top face. This series is seen to be a mirror image of the first series.

Fig. 3a            Fig. 3b           Fig. 3c            Fig. 3d

The target corner piece may be in position but not properly oriented. In this case replace it with another corner piece using one of the two series just described. Which one? The one which brings the target corner piece down to the lower half of the cube with its blue color on a vertical face.

You might complete the placement of the first three corner pieces and find that the fourth has its blue color on the bottom of the cube. In this case rotate the bottom face of the cube until the target piece is below the place where it belongs. Rotate the right face counterclockwise, the bottom face clockwise and the right face clockwise. This is similar to the first series. It differs in that the second move is a clockwise rotation of the bottom face rather than a counterclockwise rotation. It moves the target corner piece to the left/back/bottom corner of the cube and puts the blue face on a vertical face of the cube. Now a 180o rotation of the bottom face moves the target corner piece to the right/front/bottom corner and it is ready for the series which will place it properly on the top of the cube.

THE SECOND FACE

When the first face is complete turn the cube over. White (in this example) will now be the top color. There are many possible arrangements of the pieces on this face. The pieces could be in different positions relative to each other and each one could have any of three different colors on top of the cube.

What you want to find (or create) is the situation in which the top face is ready for the End Game (the same one you learned in the solution to Rubik's Cube). In this case one piece will be in place (proper position and alignment) and the other three will all be out of position.

Suppose that, when you turn the cube over you find two pieces with their white color on top of the cube. Select one of them and rotate the top face until this piece is in position. Now, let us assume that when you examine the cube you find two adjacent pieces in proper position and two which are out of position. One of the two pieces out of position has its white color on top of the cube (Fig. 4a). Rotate the top face one turn to place this piece in proper position and alignment (Fig. 4b). Surprise! You now have the End Game arrangement.

Fig. 4a            Fig. 4b

If we consider only the order of the pieces on the top face, and not the possible color combinations, we find there are only three possibilities. The number of possible arrangements increases when we consider the colors of the corner pieces on the top surface. However, it develops that the only time color is significant is when we ask whether or not a given piece has its top color on top of the cube.

The following are the significant arrangements:

1) All four pieces are in the correct order. Color is immaterial.

2) Two pieces opposite each other are in the correct order. The other two pieces are also in the correct order relative to each other but are out of order relative to the first two pieces. Color is immaterial.

3) Two pieces adjacent to each other are in order and the other two are out of order. None of the out-of-order pieces has its top color on top of the cube.

4) Two pieces adjacent to each other are in order and the other two are out of order. At least one of the out-of-order pieces has its top color on top.

Note that there are only four different combinations of the top corner pieces, including colors, which are significant. One of the four (arrangement 4) may be converted to the End Game by a single turn of the top face. Each of the other three may be converted to arrangement 4 by a single Corner Piece Series.

Although it is not necessary, many will find it easier to identify the arrangement of pieces when at least one of the pieces is in position. If, when you turn the cube over after completing the first face, at least one corner piece has its common color on top, turn the top face to place that piece in proper position/orientation. If none of the pieces has its common color on top then choose any piece and put it in position (if it is not already there).

1. You might have all pieces in position. This is the first arrangement. Apply a Corner Piece Series to any three pieces such that the piece which moves across the diagonal of the top face has its common color come to the top of the cube. Move the piece which moved across the diagonal into place (rotate the top face by two turns) and you will have the End Game arrangement.

2. Suppose you find two opposite pieces in position. The other two are out of position (relative to the first two). This is the second arrangement. Apply the Corner Piece Series to any three pieces. Again you must make sure that the common color of one of the pieces comes to the top. However, this time it must be one of the pieces which moves along an edge of the cube. Either one will do. No matter which three pieces you choose, nor which piece (moving along an edge) has its common color on top at the end of the series, you will always be one move away from the End Game arrangement.

3. You may have the third arrangement: two pieces adjacent to each other are in place and neither of the other pieces has its top color on the top face.  Again you will convert this arrangement to the End Game by applying a single Corner Piece Series.   One of the out-of-order pieces must move across the diagonal and its common color must come to the top.  This piece is then moved into place giving the End Game arrangement of corner pieces. However, this time you do not have complete freedom as to which corner pieces to include. You may include only one of the two adjacent corner pieces which are in position relative to each other.

4) If you have the 4th arrangement, two adjacent corner pieces in place relative to each other and at least one of the out-of-order corner pieces with its common color on top, this is, in fact, the End Game. You may find it as the End Game itself or you may find it with two pieces in place and two out of place.   This will become the End Game upon a 90o turn.

The third arrangement is shown in Fig. 5a. The white/red/green piece (at right/front/top) and the yellow/white/red piece (at right/back/top) are both in position. The other two top corner pieces are out of position. None of these pieces has the top color on the top face of the cube. Only one of the pieces in position (either one) may be involved in the required Corner Piece Series. One of the out-of-position pieces must move across the diagonal and have its white color come to the top. We will send the white/yellow/orange corner piece (at left/front/top) across the diagonal with its white color replacing red. The result of this series is shown in Fig. 5b.

Fig. 5a           Fig. 5b            Fig. 5c

Two of the corner pieces in Fig. 5b are in place and adjacent to each other. The other two are out of place. If we rotate the top counterclockwise one turn we will obtain Fig. 5c. The orange/yellow/white corner piece (at left/back/top) is in position and aligned. The other three top pieces are out of place and ready for the End Game.

In any of these situations you may have a problem getting the common color of the appropriate corner piece to come to the top during a Corner Piece Series. The cube in Fig. 6a has two opposite pieces in position. How can you move the green/red/white corner piece (at left/front/top) along a cube edge to left/back/top and have its white face on the top face of the cube.

Fig. 6a           Fig. 6b

You have seen this kind of problem before in the solution to Rubik's Cube. If you rotate the front face by 90o clockwise you will get the cube in Fig. 6b. Now the white/red/green piece can move across the diagonal of this cube and its white face will be on the top face of the cube. Following the Corner Piece Series turn the front face back and you are ready for the End Game.

The End Game is exactly the same as it was with Rubik's Cube. However, the lack of center and edge pieces creates an orientation problem. You must be careful to keep the orientation of the cube in mind and to keep track of the turns you have made. Remind yourself repeatedly of the turn you will make when the Corner Piece Series is completed. The process is not too difficult but you cannot let your mind wander.

Rubik's Pocket Cube is not well suited to racing. It is far less solidly built than Rubik's Cube and, in particular, you cannot spin faces rapidly or the cube may come apart. While it may not be broken this will certainly spoil the race.

Once you have learned the solution you may want to try to reduce the number of moves required to restore a scrambled cube. The best I have ever done with a Pocket Cube is six moves and I have done this more than once. It should be clear that, with a number this low, when the first face was done the second face was found to be complete as well. The average of 100 trials was 22 moves, less than half of the average number required for any other solution I have seen.

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