2014

A Mathematical Perspective

Kyle Botabara

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Invented in 1974 by a Hungarian architecture professor Ernὄ Rubik, the Rubik’s cube

puzzle gained its peak popularity in the 1980’s while winning an award for best puzzle of the

year. Although the peak of its popularity has passed, the cube still fascinates many puzzle

enthusiasts today. There are still numbers of communities who share the hobby of what is

known as “speedcubing” where the objective is to solve the cube in the fastest, most efficient

manner in competition with others.

However, aside from the obvious interest in solving the cube, there is another more

mathematical interest which intrigued many mathematicians whilst taking concepts of modern

algebra into consideration. Once analyzed further, it can be seen that the patterns of the

Rubik’s cube moves form group structure under composition.

Now, before any mathematical proofs or theorems can be formed, some definitions

must be established to create a working mathematical system. To do this, one must mentally

break down the idea of the Rubik’s cube to see what forms the base parts of the puzzle. In this

instance, these definitions will specifically apply to the 3x3x3 Rubik’s cube with no guaranteed

transfer to puzzles of other dimensional make up.

First of all, one would easily notice that there are six uniquely colored sides of the cube

which are divided into 9 sections: 4 corners, 4 edges, and 1 center piece. As the premise of the

cube itself is to solve it, it is easily verifiable that one can scramble the cube at random to reach

any pattern that is possible with the legal moves. However, specifically in the 3x3x3 cube, it is a

well-established fact that no matter how much the corners and edges are scrambled, one can

never scramble the center piece because they are structurally locked in place. Because of this

fact, one can differentiate between the faces by the color of the center pieces.

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Definition. A face of the cube is defined by the color of the center piece of any given side.

Now, using the definition of a face, one can create a definition to work with face

rotations that can easily differentiate between different faces.

Definition. A turn is defined as a rotation of any given face. Specifically, the clockwise 90-degree

rotation of any given face can be represented by a variable representing whichever face is being

rotated. For example, the clockwise 90-degree rotation of the green face can be represented

simply as “g” or for any face in general “f”. Furthermore, f∘ f = f 2 which represents the 90+90-

degree clockwise rotation or rather 180-degree turn. This generalizes to

f n∘ f m = f n+m.

From the definition of a turn, it will be shown that the set of turns on one particular

face, F, forms an abelian group under composition. To prove this, it will be shown that F is

isomorphic to ℤ4 under ⊕ which is already well established to be an abelian group.

First of all, one must show that there exists a one-to-one and onto mapping from F to ℤ4. Since

there are only 4 elements in each set, it can easily be done via listing. So, define…

Θ(f 0)=[0]4

Θ(f 1)=[1]4

Θ(f 2)=[2]4

Θ(f 3)=[3]4

We can therefore conclude that there exists a one-to-one and onto mapping.

Secondly, to prove isomorphism, one must show that the mapping preserves operations.

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So, to do this one must first define Θ(f n) = [n]4. So…

Θ(f n∘ f m)= Θ(f n+m)

= [n+m]4

= [n]4 ⊕ [m]4

= Θ(f n) ⊕ Θ(f m)

From this, it is confirmed that the preservation of operations is satisfied. Therefore, it is

concluded that the set of turns, F, is isomorphic to ℤ4 and is therefore an abelian group.

Definition. A move is defined as any combination of a finite number of turns. The set of all

possible moves will be represented by “M”. So, ∀m∈M, m=t0 … tn for some n∈ℕ where each t

represents a turn. Adding to that, the move that represents the combination of 0 turns will

specifically be represented by “z”. Furthermore, ∀m1,m2∈M, the operation m1 ∘ m2 is defined

as applying the two moves together in the order it is written.

Definition. Two moves m1,m2∈M are said to be equivalent if each move permutes to the same

pattern when applied to the same Rubik’s cube. Alternatively, it can be said that the two

moves’ combination of turns are congruent over all possible rotations. That is, if t is defined to

be a general rotation on a general face, given m1= ta∘… ∘ tb and m2= tc∘ … ∘ td,

∀m1,m2∈M, m1=m2 iff m1= ta∘ … ∘ tb ≡ tc∘ … ∘ td =m2 for a,b,c,d∈ℕ.

This can be done because the set of turns on a given face, as previously proven, is an

abelian group and thus each turn will have inverse turns that would effectively undo their

permutations while also having multiple congruent turns as well. So even if two moves are built

from a different combination of turns, they will be equivalent if and only if, through the use of

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congruences and inverses, the two moves can both be reduced to the same smallest

fundamental move to arrive to at the same permutation pattern.

Now that the definitions have been established along with the group structure of the

turns, it will be shown that the set of moves, M, forms a group under composition by proving

that M satisfies the properties of a group under ∘.

Closure. To prove closure, one must establish ∀m1,m2∈M, m1 ∘ m2 = m1m2 ∈M. So…

∀m1,m2∈M, m1 ∘ m2 = (ta∘ … ∘ tb)∘ (tc∘ … ∘ td) for some a,b,c,d∈ℕ =(ta ∘ … ∘ tb∘ tc ∘ … ∘ td)

= m1m2 ∈M

From here, one can see that m1m2 is a combination of a finite number of turns which fits the

definition for being in M. Therefore, closure has been satisfied.

Identity Element. To prove the existence of an identity element, one must show that there

exists an element that satisfies e ∘ m = m and m ∘ e = m, for e,m ∈M. It will be shown that the

zero-move, z, satisfies the requirement.

∀m∈M, z ∘ m = ( )∘ (ta∘ … ∘ tb) for some a,b∈ℕ =(ta ∘ … ∘ tb) = m

And…

∀m∈M, m ∘ z = (ta∘ … ∘ tb)∘ ( ) for some a,b∈ℕ

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=(ta ∘ … ∘ tb) = m

Therefore, it is confirmed that z satisfies the properties for an identity element.

Associativity. To prove associativity, one must show that ∀A,B,C∈M, (A∘B)∘ C = A ∘ (B ∘ C).

∀A,B,C∈M, (A∘B)∘ C = AB∘ C = (a1∘ … ∘ ax ∘ b1∘ … ∘ by) ∘ (c1∘ … ∘ cz) = (a1∘ … ∘ ax ∘ b1∘ … ∘ by ∘ c1∘ … ∘ cz) = abc

= (a1∘ … ∘ ax) ∘ (b1∘ … ∘ by ∘ c1∘ … ∘ cz) = A ∘ BC

= A ∘ (B ∘ C)

Therefore, it is concluded that associativity is verified.

Inverse Elements. To prove the existence of inverses, one must show that for any general

move, there is another move such that m ∘ m-1 = z. To prove this, it is first claimed that given

m= ta∘ … ∘ tb → m-1 = tb

-1∘ … ∘ ta-1. That is, the proposed inverse move consists of the inverse

turns in reverse order (which we know exist because of the abelian group structure of the

turns).

∀m∈M, m ∘ m-1 = (ta∘ … ∘ tb) ∘ (tb-1∘ … ∘ ta

-1)

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= (ta∘ … ∘ tb ∘ tb-1∘ … ∘ ta

-1)

= (ta∘ … ∘ tb-1 ∘ t0∘ tb-1-1∘ … ∘ ta

-1)

= (ta∘ … ∘ tb-1 ∘ tb-1-1∘ … ∘ ta

-1)

= …

= (ta∘ ta-1)

= (t0)

= ( ) = z

And…

∀m∈M, m-1 ∘ m = (tb-1∘ … ∘ ta

-1) ∘ (ta∘ … ∘ tb)

= (tb-1∘ … ∘ ta

-1 ∘ ta∘ … ∘ tb)

= (tb-1∘ … ∘ ta+1

-1 ∘ t0 ∘ ta+1∘ … ∘ tb)

= (tb-1∘ … ∘ ta+1

-1 ∘ ta+1∘ … ∘ tb)

= …

= (tb-1∘ tb)

= (t0)

= ( ) = z

Therefore, from the proof above, it becomes obvious that the inverse element property is

satisfied.

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Seeing that all 4 group properties is satisfied, it is concluded that the set of moves, M,

form a group under composition. This group, however, is not abelian because it can easily be

seen that commutativity is not satisfied. As a counter example, let m1=(b) and let m2=(r) in the

standard Rubik’s cube color position where b represents the clockwise 90-degree rotation of the

blue side and r for the red side (on non-standard color positions, pick two adjacent sides). One

may be able to see that after applying the moves to the same cube, (b)(r) is not equivalent to (r)

(b) by the definition of equivalence above.

As one can probably see, the Rubik’s cube is strongly connected to the world of modern

algebra and knowing that the set of moves form a group under composition, all theorems about

groups with similar properties automatically apply to the group of moves. One may wonder

how many elements are in the set M. At first glance, it just takes a simple observation to

conclude that the set of moves is infinite because there are an infinite number of ways to turn

the faces, but in such a perspective one may fail to realize that many of those moves are

equivalent. Putting congruences aside, the number of possible moves is essentially the number

of possible ways to scramble the patterns of the cube if one considers the definition of

equivalent moves mentioned earlier. Therefore, if one considers all equivalent moves as one

element of a modified set of moves M2, one could see that there are actually about 43

quintillion elements.

There are very many other mathematical properties that go beyond just group theory.

One could look at concepts such as permutations of the patterns under certain moves which

can easily apply to the algorithms that people use to solve the cube and are perhaps more

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accessible to the average puzzle enthusiast. However, it is clear that the Rubik’s cube is much

more than just a puzzle. The Rubik’s cube is a whole world of mathematics on its own.

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Works Cited

Chen, Janet. "Group Theory and the Rubik’s Cube." Texas State Honors Summer Math Camp,

n.d. Web. 5 June 2014.

Davis, Tom. "Teaching Mathematics with Rubik's Cube." The Two-Year College Mathematics

Journal 13.3 (1982): 178-85. Web.

Garron, Lucas. "An Introduction to Group Theory and the Rubik’s Cube." (2011): n. pag. Web.