Why should Rubik's cube get attention from mathematicians?

Question

I've seen a lot of math debate, calculations and other stuff related to Rubik's cube lately, but I don't really understand why is it important, why should anyone spend time asking and answering questions about Rubik's cube, and what good brings Rubik's cube to mathematics?

I have also found something that might be interesting, after a bit of google search. Numberphile have made a couple of videos about it, and they are quite nice:
Numberphile playlist on Rubik's cube

Yea I agree with user58512 - it's not important if you're not interested in those things :) — mathsnoob, Feb 24 '13 at 17:08
it is not important for those who don't want to really know it. — Yimin, Feb 24 '13 at 17:08
My question is why does it get attention from mathematicians — Mihai Bujanca, Feb 24 '13 at 17:11
Why is football soccer important? Much people spends time in it. — Gaston Burrull, Feb 24 '13 at 17:12
@IshanBanerjee But is it useful? Did math on Rubik's cube lead to new algorithms, or helped mathematicians find out something new? — Mihai Bujanca, Feb 24 '13 at 17:15
Because you can understand and solve Rubik's cube using group theory. — Mohan, Feb 24 '13 at 17:15
Coin tosses get a lot of attention, as do red & black balls in urns... — copper.hat, Feb 24 '13 at 17:16
@GastónBurrull i think football is not important from the same point of view. It's more of a social thing — Mihai Bujanca, Feb 24 '13 at 17:16
It's fun and having fun is important! (Mind you, not the only thing that is important ...) — Michael Joyce, Feb 24 '13 at 17:38
@MichaelJoyce I agree having fun is important, I was actually asking why should it get attention from mathematicians. I found some answers :) — Mihai Bujanca, Feb 24 '13 at 17:41
The mathematics developed relating to a particular game or hobby will often be used in fields far outside the scope of the original research. — zzzzBov, Feb 25 '13 at 01:56

Mohan · Answer 1 · 2013-02-24T17:53:26.877

16

The reason why Rubik's cube is important in mathematics is because set of all cube moves in a Rubik's cube form a group. So, it is an interesting example of finite group. You can find more information on this from this paper. Also, study of rubik's cube can lead to some interesting questions and whose solutions can in turn lead to some interesting mathematics.

edited Feb 24 '13 at 17:53

answered Feb 24 '13 at 17:22

Mohan

15,494

Dan Rust · Answer 2 · 2013-02-24T17:20:52.653

Often, mathematicians like to tackle a problem because it is either difficult or interesting. In the case of Rubik's cube, the associated mathematics can fall in to both categories. The necessary mathematics is also closely related to very important subjects such as combinatorics, computational complexity, and group theory, to name just three.

Also, the interplay between these subjects can often lead to interesting and sometimes surprising results. It is therefore a worthy cause to solve these problems, as the solutions can often shed light on problems in other areas that are much more applicable to real world situations (the often cited example of the role of number theory and algebraic geometry in cryptography is a good one).

Gaston Burrull · Answer 3 · 2013-03-20T14:14:21.223

Is a good pedagogical example for teaching and illustrating finite groups properties. And is a famous 3D game!. But is not more important than rest of examples in group theory and mathematics.

Rubik cube is isomorphic to the following finite high order group,

$$(\mathbb Z_3^7 \times \mathbb Z_2^{11}) \rtimes \,((A_8 \times A_{12}) \rtimes \mathbb Z_2)$$

Is not more important that others high order groups like monster group http://en.wikipedia.org/wiki/Monster_group, monster group is of greater mathematical interest that the rubik cube since the monster group is a simple group with too many elements.

The fact that the order of an element in a group divides the order of it, can easily illustrate with a rubik cube when you are teaching groups. Take a solved rubik cube and make any combination. For example moving two adjacent sides. Repeating this combination in theory should get to cube armed again since the group representing the rubik cube is finite. (it takes 63 repetitions. Try it!)

Here is a java animation showing a maximal order combination (order 1260)

http://www.rubikaz.com/ordmaxcomp.php

Note that your displayed equation doesn't completely specify what the Rubik group is, since semidirect products are not unique (even up to isomorphism). — hmakholm left over Monica, Mar 19 '13 at 17:11
Also, it is either 63 or 105 repetitions, depending on whether you're turning the two adjacent sides in the same or opposite directions. — hmakholm left over Monica, Mar 19 '13 at 17:14

Why should Rubik's cube get attention from mathematicians?

3 Answers3