Marianna Moncreiff

Math 522

Spring’16

Rubik’s Cube and Group Theory

When I was a kid in late 1980s, I used to play with the cube and was able to solve it without taking it apart. At that time to me, the Rubik’s Cube was just a toy, a puzzle, and a game. In my native language, I called it “Magicka kocka” (Magic Cube).  In my undergrad math program I learned that this wonderful Rubik’s Cube puzzle is made of beautiful and structured mathematical logarithms and properties.

Introducing the Rubik’s Cube:

Rubik’s Cube was invented in 1974 by Erno Rubik. In 1982, over 10 million cubes were sold in Hungary. The six faces  (3 layers) can be rotated. The puzzle is solved using rotations of the cubes.  Every sub cube has its own position; however, the number of different arrangements is 43,252,003,274,489,856,000. In spite of the enormous number of positions, Rubik’s Cube can be solved with approximately 22 rotations.

Let’s look a little bit closer on the 3x3x3 Rubik’s cube. The Rubik’s cube is composed of

6 center pieces

8 corner pieces

12 edge pieces or edges

How did the mathematicians came up with such a huge number of combinations?

Since there are 8 corners, for the corner pieces we have 8x7x6x5x4x3x2x1 = 8! (40,320) possible combinations. Taking in account the orientations of the 8 corner pieces; each corner piece has 3 sides. Therefore we have 3^8 possible orientations (6,561).

So for the corner pieces we have 8! X 3^8 combinations (264,539,520).

Similarly for the edge pieces: We have 12 edges, so possible combinations are 12! (479,001,600). Since the edges have two sides, the orientation is 2^12 (4096). However, if we twist one of the edges, we get a whole new set of combinations. Therefore for the edge pieces we have 12! x 2^12 possibilities (a huge number).

The 6 center pieces are fixed; therefore we don’t include them in our calculation.

But wait, if we take out one edge, it would give us whole new set of combinations; therefore we divide the whole calculation by 12 and get

(8! x 3^8 x 12! x 2^12 )/12 = 43,252, 003,274,489,856,000 combinations.

Labeling:

F = Front, B = Back, L = Left, R = Right, U = Up, D = Down (bottom)

Let C be a correctly oriented cubie (that means that it sits in right place consistent with the right colors).

Labeling Rotations of the Cube:

F = front 90degrees clockwise, F’= front 90degrees counterclockwise, B = back 90 degrees clockwise, B’ = back 90 degrees counterclockwise, R= right side 90 degrees clockwise (turn backward), R’ = right side 90 degrees counterclockwise (turn forward), L= left side clockwise (turn backward), L’ = left side counterclockwise (turn forward), U= upper sided 90 degrees clockwise, U’= upper side 90 degrees counterclockwise, D = down side 90 degrees clockwise, D’= down side 90 degrees counterclockwise.

I = Identity

R3=R’, F2 = 2x rotating, therefore 180 degrees

Let’s look at some Rubik’s Cube math. What I describe here is just a little fraction of the math in Rubik’s Cube. There are a lot of papers written with much more detailed and complicated algorithms.

Definition of a Group as we defined it in the class:

A group is a (nonempty) set G with an operation, satisfying the following properties:

a)      Operation is associative, closed for all a, b, c in G. We have (ab)c = a(bc)

b)      There is an identity element I in G such that for all a in G, Ia = aI = a

c)       For each element a in G, there is an element a^-1(a) = a(a^-1)=I.

d)      Closure

According to the definition learned in class; is Rubik’s Cube a Group?

To show whether the Rubik’s Cube is a Group we denote Rubik’s Cube into a Group as (RCG, *) closed under multiplication

Let M (move) be a rotation clockwise

Let M’ (move) be a rotation counterclockwise

M1 (C ) = M1 moves cubie C to the cubicle M1 ( C) and similarly M2, M3, etc…

For example the operation (or move) R puts upper right front cubie into back low right cubicle.

F(urf) = dlr

If M1 and M2 are 2 moves then M1*M2 is the same move as doing first M1 then M2.

Now we can show the properties of the group:
1) Identity (I): ( if we don’t do any operations to the cube) M*I = M = I*M

2) Inverse: M*M’ = I, so the M’ is inverse of M.  Also M’ * M = I. (right and left inverse)

3) Associativity: First we show what happened when performing two moves. M1 moves C to M1 (C ). Then M2 moves it to M2(M1 (C ). Therefore (M1*M2)(C ) = M2(M1 (C ).

To show associativity, we must show that (M1*M2)*M3 = M1*(M2*M3)

So [(M1*M2)*M3](C ) = [M1*(M2*M3)](C ) for all cubies C.

[(M1*M2)*M3](C ) = M3([M1*M2)](C )=M3(M2(M1(C ))).

Conversely [M1*(M2*M3)](C ) = (M2*M3)(M1 (C ))=M3(M2(M1(C ))).

Which shows that (M1*M2)*M3 = M1*(M2*M3).

4) Closure  - whichever moves are carried out, we still have all sets in the cube.

Therefore (RCG, *) is a Group under multiplication.

Show permutation on Rubik’s Cube

Let’s take (R U) operation and see how many cycles would it take to solve all the corners and edges permutation wise.

Starting with a cube that has Face - white, Front – Green, Right – Red, Left – Orange, Back- Blue, Down – Yellow.

1.       Make a move (RU) – then we will observe that: 

 

 WGR → GRW = WGR

We can see that we have a 5 permutation cycle; therefore we will repeat the algorithm RU five times: We will observe that even though the faces and edges are not oriented, the corners are permuted.

So (RU)5 = corners permuted but not correctly oriented

What about the edges?

Similarly, if we take a cube with Top – White, Front – Blue, Right – Orange, Left -  Red, Back – Green, Down – Yellow and perform R U, we will get this cycle permutation.

We can see that it takes 7 permutation cycle to get every edge permuted and correctly oriented edges, therefore we can say that (RU)7 is an algorithm that preserves the permutation and an orientation of the edges.

So  (RU)7=edges permuted and correctly oriented

But (RU)5 x (RU)7 = (RU)5x7= (RU)35 = corners are permuted but not correctly oriented, therefore (RU) 35 does not preserve  orientation of corners, rather it flips corners in one direction.

How do we get all the corners and edges correctly oriented? Since each of the edges are already correctly oriented and each corner has 3 sides, we have (RU)35 x 3 = (RU)105.

Yes, we have to perform (RU) 105 times to get back to all edges and corners correctly oriented. Basically we will get the whole cube solved again. So (RU)105=I (Identity).

Similarly also the operation RUR’U’ 6 times = (RUR’U’)6 = I (identity)

The other way we can solve some sides of the cube is using Commutators. One of the types of Commutators is Three-Cycles.

The commutators are in the form XYX’Y’ where X and Y are some algorithms, X’ and Y’ are inverses of X and Y.

For example we are given an algorithm:

X=R’UF2

X’=F2 U’R

Algorithm Y is algorithm that we need to put a cubie in the place that we want to. Mostly, it is just a single rotation 90 degrees.

Let Y = D

Y’ = D’

Noticed that F2=F2’ and R=R’’

We can use this algorithm on the cube if we see those conditions:

1)      Exactly two pieces of interest must be in the same layer

2)      Exactly one piece cannot be in the same layer

If we at some point, while solving the Cube, observe those conditions, we can perform the algorithm XYX’Y’, which could be one of the last steps in solving the cube. In some cases, we will need to repeat this algorithm again, or find a new algorithm to finish solving the cube.

The above mentioned are just very few examples out of many ways how Rubik’s cube relates to Group Theory.

I find the math behind the Rubik’s Cube quite fascinating. I also noticed that Rubik’s cube can be a great tool for teaching a Group Theory to the students. What I noted in this project is just little fraction of what can be observed and learned from the Rubik’s Cube regarding math. I’m definitely looking forward to learn more about the connection of a Group Theory and Rubik’s cube.

References:

Group Theory and the Rubik’s Cube, Janet Chen, http://www.math.harvard.edu/~jjchen/docs/Group%20Theory%20and%20the%20Rubik's%20Cube.pdf

The Home of Rubik’s Cube, https://rubiks.com/

Rubik’s Cube, Wikipedia the free encyclopedia https://en.wikipedia.org/wiki/Rubik's_Cube

How to solve a Rubik’s Cube http://how-to-solve-a-rubix-cube.com/

The Mathematics of the Rubik’s Cube, Introduction to Group Theory and Permutation Puzzles, March 17, 2009 http://web.mit.edu/sp.268/www/rubik.pdf

Group Theory and the Rubik’s Cube, Hannah Provenza, http://www.math.uchicago.edu/~may/VIGRE/VIGRE2009/REUPapers/Provenza.pdf