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.