How can group theory be applied to improve Rubik's cube solutions?

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Discussion Overview

The discussion revolves around the application of group theory to improve solutions for the Rubik's Cube and its variants. Participants explore how concepts from group theory can enhance the efficiency of solving these puzzles, with a focus on both theoretical and practical aspects.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant describes their experience with Rubik's Cube solutions and suggests that the cube serves as a practical example of group theory, highlighting the non-commutative nature of moves.
  • The same participant expresses a desire to learn how to formally apply group theory to develop more efficient solutions, seeking guidance for novices in the field.
  • Another participant recommends a book by Dik Winter as a resource for understanding the mathematical aspects of the Rubik's Cube.
  • Additional suggestions include contacting a speedcuber, Chris Hardwick, and joining online groups focused on speedcubing and mathematics.
  • One participant mentions their interest in the optimality of solutions and the creation of novel operators rather than focusing solely on speed.
  • Several links to resources, including PDFs and websites discussing the mathematics of the Rubik's Cube and methods involving commutators, are shared, although one participant notes they have not personally tried these methods.

Areas of Agreement / Disagreement

Participants express varying levels of familiarity with group theory and its application to the Rubik's Cube, with some sharing resources while others seek foundational understanding. No consensus is reached on specific methods or approaches to apply group theory effectively.

Contextual Notes

Limitations include the lack of formal training in group theory among some participants, which may affect their understanding and application of the concepts discussed. Additionally, the discussion includes various resources that may not have been evaluated for their effectiveness.

Who May Find This Useful

This discussion may be of interest to those involved in speedcubing, mathematics enthusiasts exploring group theory, and individuals seeking to enhance their problem-solving techniques with the Rubik's Cube.

Antiphon
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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?
 
Physics news on Phys.org
find the book by dik winter.
 
Thanks...
 
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/
(You have to sign up and join the group).

P.S. By the way, what's your 3x3 average time?
 
Last edited:
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.)
 
I googled and found this pdf, "Mathematics of the Rubik's Cube":
http://web.usna.navy.mil/~wdj/papers/rubik.pdf
http://web.usna.navy.mil/~wdj/books.html

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
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/
 
Last edited by a moderator:

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