Katsuhiko Okamoto's Void Cube

Void Cube

When I saw this cube, I just had to have one. It is a very clever piece of design and engineering. It still works like a regular cube but the centre pieces are missing with a void that runs right through the puzzle. It is one Katsuhiko Okamoto's inventions.

Turning is not as smooth as a regular cube and corner cutting is out of the question, due to the way it is constructed.

The puzzle is slightly more difficult than a regular cube due to the lack of a centre piece to act as a reference but this is not much of a hurdle. The more interesting obstacle that this cube introduces is the possibility of a parity issue. If, after solving the top layer, the middle layer is solved next with the bottom layer done last, there is a 50% chance that the middle layer is rotated by 90°. This will prevent the final layer from being solvable.

Parity

To show more clearly the parity issue on the void cube, and the concept in general, please see the picture of the regular cube below. The centres on the middle layer are deliberately mis-solved by 90°, i.e. a qurter turn, out of alignment...

If we then try to solve the yellow face, we discover that it is impossible. To solve this layer, we need to swap two edge pieces...

Regardless of what algorithms are used to move the edges around, at least two will always be in the wrong place. Hence, we have an odd parity case. We can infer from this, that performing a quarter turn on the E-layer is an odd number of edge swaps. Counting them confirms this.

It is obvious on a regular cube but the lack of centres on a void cube makes this potential displacement of the centres invisible.

Solution

To avoid the possibility of an odd parity case rearing its ugly head, I solve this puzzle following the steps below. Obviously, the algorithms that work with a normal cube are applicable here, too.

Step 1 - Solve the White Face

This is an intuitive step.

 

Step 2 - Solve the Yellow Corners

This can be solved with one only algorithm or at the most two.

Step  3 - Solve the Yellow Edges

Again, nothing special about doing this.

You could solve three edges and then use the fourth one to rotate the middle layer's edges but you will need to ensure that you are rotating them to the correct orientation.

Step 4 - Solve the Middle Layer

Use the gap in the yellow layer to facilitate the process, permuting the middle pieces at the same time if possible.

This is a fun puzzle. It is possible solve using any method used for a standard cube but using an extra algorithm to solve an odd-case parity if it arises. I prefer to solve in such a way to avoid the parity issue.