STEP FIVE: THE END GAME
With only three corner pieces remaining out of position all must be involved in the final Corner Piece Series. Further, each must move simultaneously into its correct position and with the right orientation.
It is possible that the three target corner pieces are on the same face and will move into place via a Corner Piece Series without any preliminary moves. But that would be rare indeed. We won't even think of it. Rather, the first thing for you to do is to look for a corner piece which can move along a cube edge into its correct position while rolling over into its correct orientation. Surprising as it may seem, in about half of all instances you will find that such a piece exists. About one-fourth of the time you will even find two such pieces.
The corner piece at back/top/right in Fig. 14a is just such a piece. (Its color on the back side of the cube is blue.) The proper Corner Piece Series will cause it to move along the top/right edge to the front/top/right corner of the cube where it will be in the correct position/orientation. There are two different cube faces which may be used as the front face in the Corner Piece Series and two different triangles about which the corner pieces may move.
In one such case, the blue/red/yellow piece at front/top/right will move to the back/top/left corner (i.e. the yellow/blue/red corner) with its yellow face on top of the cube. In that series, the green face is front and the first move is a rotation of the blue (top) face to the left.
In the other case, the front face is red (the left side in Fig. 14a) and the first turn is again a rotation of the blue (top) face to the left. In this case the blue/red/yellow piece will move to the corner now occupied by the blue/green/red corner piece. Again the yellow face of the blue/red/yellow corner piece will come to the top of the cube.
You should note that the first two corner pieces will be on the same face and will be adjacent to each other. The first will go into position/orientation replacing the second. The second will go to either of the two remaining corners on the top face with the color which is on the right side of the cube at the start of the Corner Piece Series coming to the top. We will refer to the third and fourth corners as the "target positions".
Naturally the second corner piece must replace the third corner piece with the third replacing the first. But where is the third corner piece? A little thought should convince you that, in this case, the third piece is somewhere in the yellow face of the cube. Fig. 14b (the entire cube in Fig. 14a has been rotated 90o clockwise on an axis from top to bottom) shows that the third corner piece is in the yellow/blue/red corner. It should be clear that if the yellow face of the second corner piece replaces the orange face of the third corner piece (the face which is in the yellow face of the cube) the second piece would be in the right position and have the right orientation.
Fig. 14b Fig. 14c
How can we make that happen? We must bring that orange face to the top (blue) face of the cube in one of the target positions. This can be done with a 90o clockwise rotation of the red face as shown in Fig. 14c. Then, with red as the front face, a Corner Piece Series beginning with a turn of the blue face to the left will cause these three corner pieces to move clockwise about the front/top/right; back/top/left; back/top/right triangle.
The yellow face of the yellow/red/blue corner piece replaces the orange face of the orange/blue/yellow corner piece and when the red face is rotated 90o counterclockwise (the preliminary turn is reversed) each of the corner pieces will be in its correct position and orientation.
But, you might say,..."You made sure that each of the first two pieces went to the right place but you did nothing about the third". You don't have to worry about the third corner piece for, when all other pieces on the cube have been properly placed, the last piece must be in position and must be properly oriented as well. It is physically impossible for it to be otherwise. Thus you must ensure that the first two pieces are correctly placed and the third one will take care of itself.
As indicated above, when you reach the End Game, half the time at least one of the corner pieces will be in position such that a Corner Piece Series will cause it to move along a cube edge and into position/orientation. If you don't find such a corner piece you must create one. About 70% of the time you will do so with a single 90o rotation of a cube face. And about 30% of the time you will do so with a 180o rotation of a cube face.
The cube in Fig. 15a does not have a corner piece which can be moved along a cube edge and into place. But the red/white/yellow corner piece is diagonally across the yellow face from the corner where it belongs. It could be made part of a triangle which would send it across the diagonal to its proper place. A corner piece in such a position may be moved to an adjacent corner so as to create a corner piece which can be moved along a cube edge and into place. You can do this with a 90o rotation of either vertical face in which the corner piece is located. For example, the orange face could be rotated 90o clockwise. Fig. 15b shows the resulting cube with white as the front face.
Fig. 15a Fig. 15b Fig. 16
The cube in Fig. 16 does not have an appropriate corner piece adjacent to its correct position. Neither does it have a corner piece which could be moved across a face diagonal to its proper place. But it does have a corner piece which can be rotated 180o and be in a position from which it can move along a cube face and into proper position/orientation. That corner piece in Fig. 16 is the green/red/white piece in the back/top/left corner of the cube (all you can see is the green face).
You can turn either of the faces, which (of the three remaining out-of-position corner pieces) contain only the green/red/white corner piece, by 180o and that piece will be in position such that the proper Corner Piece Series will move it along a cube edge and into place. Turning the third face of the cube (yellow in this case) will not work.
With this understanding we can construct the rules for successfully completing the End Game:
1. Find a corner piece which may be moved along a cube edge and into place. If none can be found create one.
2. Note the face color (of the second corner piece) which will come to the top face of the cube in a target position during a Corner Piece Series.
3. Locate the third corner piece in the cube face of this same color and note the color of the third corner piece face which is in that cube face.
4. Bring the third corner piece to the top of the cube, by various face turns, such that that color is in the top face of the cube.
5. Perform the Corner Piece Series so that the target color of the second corner piece replaces this color (named in 4 above).
6. Reverse the preliminary face turns indicated in 4 above.
This process is much easier to carry out than it is to describe. Several examples will be given below.
The white/yellow/orange corner piece at front/right/top of the cube in Fig. 17a may be moved along the white/yellow edge and into place. It will replace the red/green/blue piece. The latter will move to either of the target positions with its blue face coming to the top. We examine the blue face of the cube and find the yellow face of the white/red/yellow corner piece in that blue face. Hence we need to get the yellow face of the third corner piece into one of the two target positions. This is done by a 180o rotation of the green face giving the cube in Fig. 17b (the entire cube has been rotated 180o so you can see all three faces of the third corner piece).
Fig. 17a Fig. 17b
Now hold the cube with the green face in front and begin the Corner Piece Series with a turn of the top (white) face to the right. The third corner piece goes across the diagonal of the white face (with its white face coming to the top) and into its proper position and orientation.
You do not always have to examine the face of the cube which has the target color. When the white/yellow/red piece in Fig. 18a moves along the white/yellow edge of the cube the green face of the second corner piece comes to the top in either of the target positions. We need to move the face of the piece at the lower left of the cube which is now in the green face (we see only its white face) into one of the target positions.
Fig. 18a Fig. 18b
Rotate the bottom face by 90o counterclockwise and then the left face 90o clockwise. This gives Fig. 18b. The face we were looking for is yellow (we would have seen that if we had turned the cube to reveal the green face). The first turn of the top face is a rotation to the top left. When the series is completed reverse the original two turns and the cube is complete.
The corner piece at back/top/left in Fig. 19a will move along the top/left edge of the cube and into place. The green face of the white/orange/green corner piece will come to the top in a target position. The yellow face of the third corner piece is in the green face of the cube. We can bring that yellow face to a target position by the three turn sequence we learned in Step Four. It is: green counterclockwise; blue (bottom) clockwise; and green counterclockwise.
Fig. 19a Fig. 19b
There is the yellow color on the top face in the back/right corner of the cube (Fig. 19b). Apply the Corner Piece Series and reverse the three preliminary turns. The cube is restored.
The cube in Fig. 20a has two corner pieces which can move along a cube edge and into position. We could use either one of them as the first corner piece. Note that all three of the final corner pieces have an orange face and, in each case it is in the orange face of the cube.
Fig. 20a Fig. 20b
We will use the yellow/white/orange piece as the first corner piece. Let us turn the entire cube 90o clockwise on an axis from top to bottom. Now turn the bottom (green) face 90o clockwise and then the red face also 90o clockwise. This gives the cube in Fig. 20b. (Again we chose this view so you could see all three faces of the third corner piece.) With red as front, turn the top face to the right and complete the Corner Piece Series. Reverse the two preliminary turns and the cube will be done.
In Fig. 21a none of the pieces is adjacent to any other. As a general rule this should be avoided but it is not much of a problem in this case because any one of the three corner pieces could be turned 90o and thereby create a corner piece which could be moved along a cube edge and into its proper place. For example, turn the orange face 90o counterclockwise (Fig. 21b shows the result except that the entire cube is turned clockwise in order to show all three faces of the green/white/red corner piece) or turn the red face 90o clockwise.
Fig. 21a Fig. 21b
But if the orientation of each were to be changed so as to put a corner piece face into the cube face of the same color then things would change drastically.
(For example, the yellow color of the red/blue/yellow piece is moved to the yellow face of the cube at front/top/left, the red face of red/white/green is moved to the red face of the cube and the white face of orange/white/yellow is moved to the white face of the cube.) Although this arrangement is rare, it is the most difficult you will encounter.
I will leave this for you to solve for yourself. But I'll give you a hint. You will have to move one of the corner pieces in two different directions.
If you have one of those cubes with symbols on four of the faces then go to Rubik's 4th Dimension Cube.
If you are having trouble
following this description you may be interested in a little help. A videotape is
available showing a 53 minute description of this solution. Or you may simply wish to give
the videotape to someone else.