- A little history
- The middle layer
- Odd and even permutations
- Getting back to the Cube-shape
- Cube-shape preserving motions
- More...

A little history

The main reason I want to talk about it now is that after this you can (almost) ignore it, i.e. theoretically. The reason being that the two shapes can be obtained without disturbing the rest of the layers.

*Proof* :

- Irregular hexagon to square: Do the following two moves three times

- Turn the right half of your 'cube'
- Rotate the upper layer 180 degrees

- Square to hexagon: do the inverse of 1.

In the mean time read the lecture notes of W.D. Joyner, in particular about permutations and the definition of odd and even permutations.

Since we don't want to loose the Cube-shape at this stage... there are only a few

- 1. , and
- 2.

First we glue a few pieces together (well not really, let's just pretend).
Then it looks (inside our head(s)) as:

Note that we can also do this in the mirror image. So now we have a "2x2x2
cube" (we are still ignoring the middle layer).

We don't have to worry about orientations. Then we see, by using the 2x2x2-permutation
group (S8), that the glued blocks can be permutated as we please. Then note
that for all (basic and hence all cube-preserving) moves we have that:

*The permutations of the narrow and wide parts are either both odd or both even.*

Hence we are done (you have to check all this of course).

After some thought I finally found a solution. I will however show here only that it is indeed possible to obtain a single swap of two wide parts (check that this also implies that we can obtain a single swap of two narrow parts). For this look at the following shape (obtainable for instance by using my list)

By turning the eye (left) a 180 degrees. We see that we get four single
swaps of narrow parts and a single swap of two wide parts. So if we had
at this moment the Cube-shape and went to the shape above, turned the eye
180 degrees and then going exactely the same way back. Then we have obtained
an even permutation of narrow parts and a single swap of two wide parts.
So by using our knowledge of the previous sections we can 'cancel' the
even permutation of narrow parts with the help of Cube-preserving motions
and hence have obtained a single (odd) permutation/swap.

Of course this is not the most simple way to achieve a single swap of two
wide (resp. narrow) parts. But it does prove that they are possible. A shorter
sequence of moves to achieve this is left to the reader (you that is). If
you find a nice (short) one, please let me
know. I plan to make a short list of usefull moves.

Author: Christian Eggermont

Last updated: 13 March 1997