The first steps...
Once upon a time I bought a Square One. It took me some time to live happily
everafter. This is a much abbreviated account (I left out (most?) of my
mistakes) of my search for happiness, i.e. a solution. Since I don't want
to spoil all your fun, I have only written down a few ideas (the important
ones), the basic concepts and leave to you to fill in some of the details.
I hope this will stimulate you to invent your own solution and also be very
A little history
Happy reading and hunting !
The middle layer
The middle layer consists of two pieces which form either a square or a
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.
- 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.
Odds and ends
This section will (in time) explain some group theoretical important concepts
(for really understanding any solution). At this time I am still searching
what (and how) to include here and if it should be placed here or in a more
general context (and hence in a seperate file). Come back later to see if
I've made my mind up. (permutations, odd/even, conjugation, Theorem of Lagrange....
In the mean time read the lecture
notes of W.D.
Joyner, in particular about permutations and the definition of odd and
Getting back into shape
Well, the nice thing about this section is that it is short. Since I redirect
you to my list where you can get your mangled
'Cube' back into shape.
Poëtry in motion
So, I presume we have managed to get into the Cube-shape. Most people are
now terrified to lose this dull, but sometimes long searched for, shape.
So did I at a time. So I started with finding nice moves. Of course I already
had some but this time I wanted nice moves for Square One.
Since we don't want to loose the Cube-shape at this stage... there are only
a few basic moves we admit. They are all obtained by the following
two moves (and 90 degrees turns/rotation of the top and bottom layer):
So what is the group that we get ? Well, by looking at it from a slightly
different angle I saw it immediately. Of course I had to check it. For those
which already solved the 2x2x2-cube let me sketch the way I knew what this
- 1. , and
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:
And by using (if you want to both) 2x2x2-cube-group(s) we can easily obtain
- 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).
To be totally honest. Finding the results of the former paragraaf, I thought
that I solved Square One. So when I turned and turned ... Oeps! I suddenly
was left with only a single swap of two (narrow) parts. Which of course
posed me with a problem. Since we know that the Cube-preserving motions
are all even permutations, I had to look at the total group to solve this
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