# Group theory and Rubik's cube

1. Jul 19, 2005

### Antiphon

I've always been fascinated by Rubik's cube. I have developed solutions for it and
all the related cubes 2x2, 3x3, 4x4, 5x5. For me the cube it is to group theory
(of a partcular type of group) what a slide rule is to real arithmetic. Even "laboratory"
might not be too stong a label for it.

For example it's immediately obvious how $$xy \neq yx$$. If you turn the front of the cube
and then the right you get a very different set of faces than the right followed by front.
Also, you can discover marvelous "operators" (my terminology) by doing some random
series of twists (abc) followed by a particular twist (Z) then undoing the first
twists (via cba), that is: abcZcba where the letters stand for some particular oriented
twist. What happens is that most of the cube is unperturbed except for some
marvelous little permutation like a twisted corner in place or three swapped edges.

My solutions then consist of applying these "operators" in sequence by inspection
to see which one is "needed" next.

Alas however, I am not formally trained in group theory and I would like
to know: How would one go about using GT to develop a more effective
or efficient solution to something like Rubik's cube? I know it has been
done, but my question is very specifically: Can anyone explain to the group theory novice
(but Rubik's cube expert) how one would actually go about using GT to
devise (more) efficient solutions to such a puzzle?

2. Jul 19, 2005

### matt grime

find the book by dik winter.

3. Jul 19, 2005

### Antiphon

Thanks....

4. Jul 19, 2005

### Edgardo

Hello Antiphoton,

you could contact Chris Hardwick, he is a speedcuber and interested in math too. Go to
www.speedcubing.com > Chris Hardwick's Corner > at the bottom is his e-mail.

Also try the Yahoo Speedcubing group. I'm sure there are also some math interested people there:
http://games.groups.yahoo.com/group/speedsolvingrubikscube/

P.S. By the way, what's your 3x3 average time?

Last edited: Jul 19, 2005
5. Jul 19, 2005

### Antiphon

Never really measured it, but I think maybe 1+ minutes. I'm more interested
in optimality (number of turns) and coming up with novel
operators (i.e. combinations of turns which do something interesting.)

6. Jul 19, 2005