Message #436

From: david lawson <>
Subject: Re: [MC4D] Doing a class presentation on Rubik’s Cube and Group Theory… suggestions?
Date: Thu, 15 Nov 2007 09:06:04 -0700

I talked to my professor… he has seen the stuff I’m already doing… I’m
starting by categorizing all the possible twists of a hypercobe face into
it’s own group… and he said while what I had done so far was really great,
it’s a hell of a lot of work. and he said he was fine with me presenting
what I have learned of the cube from a group theory perspective and the
questions that I now have.

soooo… I’m open to modifying the presentation a bit. perhaps DERIVING
isn’t really a practical idea given my timeframe. maybe instead I could
take existing solution algorithms and point out their significance to group

On 14 Nov 2007 23:39:39 -0600, David Vanderschel <> wrote:

> On Wednesday, November 14, "Jenelle Levenstein" <
> <>> wrote:
> >If all you want to do is make a program solve a
> >rublix cube you could always solve it using the brute
> >force method where you try all possible combinations
> >of moves until the you find the solution.
> Note that dave explicitly stated that he wanted to use
> techniques he was learning in his abstract algebra
> class to _derive_ a solution.
> >This works on the 2^3 and maybe on the 3^3 and only
> >because the cube is always within about 15 moves of
> >being solved.
> Sorry, not for 3^3. The search tree expands way too
> rapidly. Not only is a solution within 15 moves of an
> arbitrary start, but so is every other possible
> configuration of the cube. The number (on the order
> of 10^20) is way too large to admit a brute force
> approach. Fortunately, there exist more intelligent
> approaches which do work. Indeed, Don Hatch has
> posted a general one (for nD) here:
> Brute force, when it works, can find the shortest
> solution. Don’s method does not claim to be optimal.
> Regards,
> David V.
> On 14 Nov 2007 10:27:02 +0000, David Vanderschel <<>>
> wrote:
> > On Tuesday, November 13, "iatkotep" <<>>
> 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:
> ><>
> > >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.