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

In summary, Antiphoton is asking how he can improve his Rubik's Cube solving skills. He found several websites that offer solutions and methods, but he is not sure if they are more efficient than his own methods. He also asks if anyone can give him advice on how to use group theory to improve his Rubik's Cube solving skills.
  • #1
Antiphon
1,686
4
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 [tex] xy \neq yx [/tex]. 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?
 
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  • #2
find the book by dik winter.
 
  • #3
Thanks...
 
  • #4
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?
 
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  • #5
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.)
 
  • #6
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:

1. What is "Group theory" and how does it relate to Rubik's cube?

Group theory is a branch of mathematics that studies the properties and structures of groups, which are sets of mathematical objects that can be combined together using specific operations. In the context of Rubik's cube, group theory is used to analyze the different ways in which the cube can be manipulated and how these manipulations can be combined to solve the puzzle.

2. How many possible combinations are there for a Rubik's cube?

There are approximately 43 quintillion (4.3 x 10^19) possible combinations for a Rubik's cube. This number is calculated by multiplying the total number of possible positions for each of the 26 cubelets on the cube.

3. Can group theory be used to find the most efficient solution for solving the Rubik's cube?

Yes, group theory has been instrumental in developing algorithms and techniques for solving the Rubik's cube in the most efficient manner. By applying group theory principles, it is possible to reduce the number of moves required to solve the cube from any given starting position.

4. What are the different types of symmetries present in a Rubik's cube?

There are three main types of symmetries in a Rubik's cube: rotation symmetries, reflection symmetries, and inversion symmetries. Rotation symmetries involve rotating the cube around one of its axes, while reflection symmetries involve reflecting the cube across a plane. Inversion symmetries involve flipping the cube inside out.

5. Are there any real-world applications of group theory and Rubik's cube?

Yes, group theory has been applied in various fields such as chemistry, physics, and computer science. For example, the principles of group theory are used in understanding the molecular structure of compounds and in developing efficient algorithms for data encryption. Additionally, the Rubik's cube has been used as a teaching tool to introduce students to the concepts of group theory and symmetry.

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