# Message #433

From: David Vanderschel <DvdS@Austin.RR.com>

Subject: Re: [MC4D] Doing a class presentation on Rubik’s Cube and Group Theory… suggestions?

Date: Wed, 14 Nov 2007 10:27:02 +0000

On Tuesday, November 13, "iatkotep" <iatkotep@gmail.com> wrote:

>I’ve never solved a rubik’s cube. the idea for my

>presentation is to take the techniques that I’m

>learning in my abstract algebra class, and use them

>to derive a solution to the cube.

Good luck! I would be surprised if you succeed in

this venture; but it will be an impressive achievement

if you do succeed.

Yes, Rubik’s Cube is a good example of a non-trivial

group.

You might want to start with a simpler permutation

puzzle - like the 2x2x2 analogue of Rubik’s Cube.

>I want to extend that to also deriving a solution to

>the 4D Magic Cube.

If you do succeed for the 3D puzzle, then extending

for the 4D puzzle should not be so hard. Aside from

the fact that there are a lot more pieces to fool

with, there is a sense in which manipulating the 4D

puzzle is actually easier than the 3D puzzle.

>I’m at the beginning now… I know what properties of

>the cube make it a mathematical group, but that’s as

>far as I’ve gotten. I have a strong feeling that

>jumping from the cube as a group to a full blown

>solution involves the study of subgroups, but I’m not

>really sure where to start.

There is plenty of information out there which

addresses the puzzle from the Group Theory point of

view. The source of this sort to which I have paid

the most attention is W. D. Joyner’s Web page here:

http://web.usna.navy.mil/~wdj/rubik_nts.htm

>do we have any math people in there that could kind

>of point me in the right direction?

The truth of the matter is that every method I have

ever seen for working Rubik’s Cube approaches it from

a rather empirical point of view. There are some

important facts about what you can and cannot achieve

that are implied by the theory, but you don’t really

need to know the theory to take advantage of the facts

themselves. (Indeed, the facts can eventually become

apparent even without having known about the theory

which implies them.)

I first laid my hands on a Rubik’s Cube in 1979. I

was actually pretty well trained in Group Theory at

the time, and I did realize that the puzzle could be

regarded as a representation of a group. However, my

knowledge of Group Theory played little role in my

figuring out how to work the puzzle. I suppose it did

lead me to try things like commutators and

conjugation; but I probably would have done so even if

I had not known what such operations were called in

Group Theory.

Regards,

David V.