Question

Rubik's cube group is an algebraic group with elements corresponding to the cube operations on the toy called the Rubik's cube. Discuss the formal description of how Rubik's cube could correspond to an algebraic group and use the methods of group theory to determine the order of the group (or in this context, the number of ways to arrange the color squares on the Rubik's cube).

Answer 1

At my current level of knowledge, I'd probably try to somehow find the orbits and conjugacy classes of the Rubik's Cube. If we can find those, and the orders of these, then it's rather easy to find the total number of arrangements possible.

Answer 2

The hardest part of that idea, would probably be finding the conjugacy classes.

Answer 3

Would you like to discuss this with me in vyew? I have compiled some information that might help the both of us figure this out. I already know the total number of arrangements possible. I just need to know how they got it. I have plenty of information on it and I can piece it all together for you and maybe you can help me come up with a competent, logical explanation the links all of these elements together

Answer 4

I'm really sorry, but I actually have to leave now, but when I get back in about 3-4 hours I would be happy to.

Answer 5

Okay, looking forward to it.

Answer 6

I've already found a resource with the conjugacy classes already calcuated, so now, I just need help with using that to explain how to find the order of the group

Answer 7

@myininaya
@satellite73
@Zarkon
@JamesJ
@dumbcow
@FoolForMath
@asnaseer

Answer 8

@amistre64

Answer 9

@anyone

Answer 10

It took a little bit longer than expected, but I'm back now if you want to discuss this now.

BTW, where is that resource with the conjugacy classes enumerated?

Answer 11

Yeah, I have it but I think there's something wrong with it. Would you like to go to vyew to discuss this?

Answer 12

Sure. I need to sign up for that right?

Answer 13

No, I will post the link

Answer 14

I sent you the link

Answer 15

Hey @KingGeorge I found the section in that pdf document that talks about using cube moves as group elements. This is perfect!

Answer 16

I'm glad it was so helpful :P

Answer 17

I still don't have a good solution yet.

Answer 18

I'd like to look at your work if you don't mind. This seems interesting.

Answer 19

I don't have any useful work at the moment.

Answer 20

If I did, I wouldn't be up at 4am

Answer 21

Anything and everything that might keep my mind occupied?

Answer 22

Oh, I have plenty of related links to share if you think you can piece something together.

Answer 23

Sure - post em, I don't like your chances though.

Just heavily intrigued by the whole idea.

Answer 24

I don't know why you're intrigued by it. Honestly, this assignment disgusts me, but if you're as interested as you say you are, you have a better chance than me of piecing something together.

Answer 25

I will post some links soon

Answer 26

I'd even be curious to see your task sheet, if that is what is handed out for the assignment.

What questions are postulated and posed, etc, etc,

Answer 27

The question above is the only task. I don't know what you mean by a "task sheet"

Answer 28

Ah okay - for assessments I'm given a sheet with the criteria for marking, etc, etc.

Answer 29

There's no restrictions on a response to this by the way.

Answer 30

No format restrictions.

Answer 31

I will post some links now.

Answer 32

http://en.wikipedia.org/wiki/Rubik%27s_Cube#Permutations
http://en.wikipedia.org/wiki/Rubik%27s_Cube_group
http://www.gap-system.org/Doc/Examples/rubik.html
http://www.math.rwth-aachen.de/~Martin.Schoenert/Cube-Lovers/Dan_Hoey__The_real_size_of_cube_space.html

Answer 33

I will post one of the responses made by another person even though it isn't completely correct because she didn't show directly how the order can be derived from the operations of the Rubik Group.

Answer 34

Sample Response (although not completely correct):

In order for an algebraic structure to be a group, it must be closed under the operation, it must contain an identity element, every element must have an inverse, and it must be associative. By this definition, the possible moves of the Rubik’s cube can be represented as a group.
Let (G, *) be the group of possible moves. If M1 and M2 are both moves that can be made on the cube, then M1 * M2, which essentially means complete move one then complete move two, must also be a move. (G, *) does have an identity element called e which, in the case of the Rubik’s cube, represents an empty move. M * e is essentially the same things as completing move one and then doing nothing, so M * e = M. Logically, every possible move in (G, *) must have an inverse, or a move that can undo the original move. Completing the move and then undoing the move is the same thing as doing nothing, so M * M’ = e.
To show that (G, *) is associative,

First, let’s investigate what a sequence of two moves does to the cubie. If M1 and M2 are two moves, then M1 * M2 is the move where we first do M1 and then do M2. The move M1 moves C to the cubicle M1(C); the move M2 then moves it to M2(M1(C)). Therefore, (M1 * M2)(C) = M2(M1(C)). To show that * is associative, we need to show that (M1 * M2) * M3 = M1 * (M2 * M3) for any moves M1, M2, and M3. This is the same as showing that (M1 * M2) * M3 and M1 * (M2 * M3) do the same thing to every cubie. That is, we want to show that [(M1 * M2) * M3](C) = [M1 * (M2 * M3)](C) for any cubie C. We know from our above calculation that [(M1 * M2) * M3](C) = M3([M1 * M2](C)) =
M3(M2(M1(C))). On the other hand, [M1 * (M2 * M3)](C) = (M2 * M3)(M1(C)) = M3(M2(M1(C))). So, (M1 * M2) * M3 = M1 * (M2 * M3). Thus, * is associative. (Chen, n.d., p. 11)

Since (G, *) fits all the properties, it is a group. Although the order of this group is large, it is still finite. There are 2^(12)3^(8)8!12! possible ways to configure the Rubik’s cube. Not all of these configurations are considered valid because they cannot be achieved by a series of moves from a starting configuration (Chen, n.d.). There are 4 x 10^19 valid possible ways to configure the Rubik’s cube (Chen, n.d.).
References
Chen, J. (n.d.). Group theory and the Rubik’s cube. Retrieved from http://www.math.harvard.edu/~jjchen/docs/Group%20Theory%20and%20the%20Rubik's%20Cube.pdf

Answer 35

Correct me if I'm wrong but the only way you can prove associative is through isomorphism?

A bit of an amateur here. I may be wrong.

Answer 36

I wouldn't assume that isomorphism is the only way to prove associative.

Answer 37

If anyone else wants to help me out additionally, feel free.

Answer 38

this is easy

Answer 39

i could solve a rubik's cube blindfolded :P

Answer 40

@Tai so can I, but that's got absolutely nothing to do with the question. please keep your comments relevant.