# Group theory and Rubik's cube

#### 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?

Related Linear and Abstract Algebra News on Phys.org

#### matt grime

Homework Helper
find the book by dik winter.

Thanks....

#### 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?

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#### Antiphon

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

#### Edgardo

I googled and found this pdf, "Mathematics of the Rubik's Cube":
http://web.usna.navy.mil/~wdj/papers/rubik.pdf [Broken]
http://web.usna.navy.mil/~wdj/books.html [Broken]

And some websites:
http://www.geocities.com/c_w_tsai/cube/
http://lar5.com/cube/

Methods with so-called commutators (I haven't tried them out myself but it seems popular among cubers):
http://grrroux.free.fr/begin/Begin.html
http://www.progsoc.uts.edu.au/~rheise/cube/
http://web.usna.navy.mil/~wdj/book/node179.html [Broken]
http://www.geocities.com/jaapsch/puzzles/theory.htm

Check out the Fewest Move Challenge.
http://www.cubestation.co.uk/
http://games.groups.yahoo.com/group/fewestmoveschallenge/

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