In the Ultimate Solution to the 3D Rubik?s Cube, we work by first placing the edge pieces, and then the corner pieces, effectively working from the centers outward. We are going to apply precisely the same idea here. Working from the centers outward on a 4D cube means solving the two-colored face pieces first, then the three-colored edges, and finally the four-colored corners.
Solving the two-color face pieces on a 4D cube is very much like solving the two-color edge pieces on a 3D cube. And as you will see, the sequences for two-colored pieces are the same in both cases. Similarly, solving the three-color edge pieces on a 4D cube is very much like solving the three-color corner pieces of a 3D cube. This means you can get through the first two-thirds of the solution without a great deal of new information. To solve the corners of a 4D cube, we just need to introduce a few new sequences. The ideas behind using these corner sequences are the same ones for solving the faces and edges.
We are going to start out, as we do with a 3D cube, by creating a cross on one of the faces. A 4D cross is slightly different in that it is composed of a centerpiece surrounded by 6 face pieces (two on each side of the center piece in three directions).
Lets not worry about pure efficiency yet, and start with the top face first. After a full scramble, the cube will look something like this.
In my scramble, the Top face was the blue face, but this may not be the case for you. If you want to follow this solution with face colors in the same positions, you can move the blue face to the Top using 4D view rotations. You can perform 4D view rotations by holding down control key while clicking stickers.
Although there are more ways to twist a hypercube, for this portion of the solution, you may want to restrict yourself to doing only ?face? twists. That is, twists done by clicking on the face pieces. You will not need the more complex twists available yet.
As in the 3D cube, when we are solving the cross, we need to place all the face pieces with blue in them such that their blue color is on the top face and their second (non-blue) colors line up with the corresponding center pieces in the center section.
It is difficult to see without being able to move the view around with the ?3D rotations? of the program, but the only face piece with blue in it currently on the Top face is the blue/purple piece (labeled 1). The blue sticker of the piece is on the Top face, and the purple sticker of this piece is on the Upper (cyan) face. We can get this piece to its proper position and orientation with the following twist
We just place our first piece! Only 71 more to go. The cube now looks like this (with the piece we placed still labeled 1).
Continue to place pieces composing the blue cross. You can see I?ve labeled two more face pieces with blue in them that jumped out at me to be solved. #2 can be placed correctly with this move:
#3 is a little more complex, but can be solved with these moves:
Note that moving the #2 and #3 pieces into their proper positions and orientations did not unsolve any of the pieces placed previously to them. If moving these pieces had displaced others, we would need to put the displaced pieces back into their proper positions and orientations. After executing the moves above, we now have 3/6 of the face pieces on the cross solved.
I won?t continue the move descriptions because every situation is different. This is one of the ?thinking? portions of solving the cube because you are not blindly carrying out a sequence of moves. It takes some practice to see which face clicks to execute to get the pieces where you want them (Do you remember the first time you tried to make a cross on the 3D cube?) But after a while, it will not be difficult. When you are done with the cross, your cube should look like this.
All 6 face pieces have been successfully placed in their position and orientation. This also means the second (non-blue) colors of these face pieces all line up with the corresponding center pieces in the center section.