# Message #436

From: david lawson <iatkotep@gmail.com>

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

theory?

On 14 Nov 2007 23:39:39 -0600, David Vanderschel <DvdS@austin.rr.com> wrote:

> On Wednesday, November 14, "Jenelle Levenstein" <

> jenelle.levenstein@gmail.com <jenelle.levenstein%40gmail.com>> 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:

> http://www.plunk.org/~hatch/MagicCubeNdSolve/<http://www.plunk.org/%7Ehatch/MagicCubeNdSolve/>

>

> 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 <DvdS@austin.rr.com<DvdS%40austin.rr.com>>

> wrote:

> > On Tuesday, November 13, "iatkotep" <iatkotep@gmail.com<iatkotep%40gmail.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<http://web.usna.navy.mil/%7Ewdj/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.

>

>

>