? Step Four

# The Ultimate Solution to a 3x3x3x3

## Step Four:  Three-Color Edge Pieces

To place the edge pieces on the hypercube, we are going to have as tools 3 Three-Color Series that are very similar to the 3D cube?s Corner Series.

The First Three-Color Series

The Second Three-Color Series

The Third Three-Color Series

Simple enough naming scheme.  There are 32 edge pieces that we need to solve using these series.  On the 3D cube, there were 8.  Not significantly harder, just longer.

Like the 3D cube (do I say that too much?), the key is moving pieces into their correct position and orientation by execution of one of the sequences.  The First and Second Series will shuffle around 3 edge pieces, and so these are the two series you will use most often.  An interesting phenomenon of the 4D cube is that we can reorient only one three-color piece at a time.  That is what The Third Three-Color Series will do.

What if no pieces will move into their correct position and orientation by executing one of the sequences?  This is actually the norm in this puzzle because there are so many more three-color pieces.  Rarely will you find an edge piece that will move into its correct position and orientation only by executing the sequence.  The answer then, as in the 3D case, is that you must do preliminary moves.  Do a set of moves such that an edge piece will move into its correct position with one of the sequences, perform the sequence, and then undo your preliminary moves.

An Example:

After solving a few of the edge pieces, my cube looks like this.  Lets go through solving the next edge pieces together.

I am going to use the Second Three-Colored Series to place the piece I have labeled #1.  You can see that the colors of this piece are cyan/pink/green.  Upon execution of the Second Series, the pieces will go to the following positions on the cube.

 1a moves to: Top, 18 1b moves to: Lower, 12 1c moves to: Right, 16

Of course this piece won?t be solved if we just execute the Second Series with the cube as shown.  Can you find where the #1 piece actually belongs?  When you do, you will see that the yellow/blue/orange edge piece is occupying the space where we want the #1 piece to be.

Our goal, then, is to do preliminary moves so that when the #1 piece moves into its new position according to the Second Series, it will be moving into the yellow/blue/orange piece.  In other words, we want our preliminary moves to put

yellow into the Top, 18 sticker position,

blue into the Lower, 12 sticker position, and

orange into the Right, 16 sticker position.

After executing the sequence and then undoing the preliminary moves, the #1 piece will be solved.

One possible set of preliminary moves is:

 Back, 15, Left Back, 15, Left Back, 11, Left Back, 11, Left Lower, 15, Left

Making the cube look like this.

You can see that this put the yellow/blue/orange piece where we wanted it.

Note also that our preliminary moves did not move the third edge piece altered by the Second Series (in this case the blue/red/pink piece).  If it had, that is ok, but we would need to make sure that the new piece in that position was not one we had already solved.  If it was, we would be undoing our previous work.

Executing the Second Series and undoing our preliminary moves yields this.  Note the cyan/pink/green piece is now solved.

1. This step will probably require liberal use of 3D rotations.  In addition, you will need to do some 4D rotations (using cntrl-click) to see the edge pieces that are currently part of the bottom face.

1. You can use both the sequences and the reverse sequences to help you, but I tend to use only the Second Series (and the Third if required).  The reason is that I find it is easier to do preliminary moves without disturbing the pieces in the sequence I do not want to move.

1. It is also possible to ?flip? a three-color piece on a 4D cube.  If you happen to do this, you will have one of the three colors lines up, with the other two flipped.  This is impossible on a 3D cube.  To fix this scenario, you will have to use one of the first two Series to get the piece out, then an additional series to put it back in without being flipped.

2. After placing each edge piece, quickly check that all visible face pieces are still correct. If some are wrong, undo your last set of moves and find which preliminary move you reversed incorrectly.

When you have only 3 edge pieces left, you will need to solve them all at the same time just like you have to solve the last 3 corner pieces at the same time on a 3D cube.  This means you will most likely need to do preliminary moves for two of the pieces instead of just one (as I did in the example).

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