Gallagher 1 Sean Gallagher Professor Ray Mest Sr. Seminar, MAT 492 May 2011 The Rubik’s Cube and its Group Theory Applications When the Rubik’s Cube was invented in 1974, it took the world by surprise. When the idea was first thought up by Ernő Rubik, he may not have known how big the puzzle was eventually going to be. The Rubik’s Cube has sold over 350 million copies, making it the best selling puzzle of all time; it is considered by many to be the best selling toy in history. A few years after its release in the market, nearly every child in every household owned the puzzle. It has even made its way into the Oxford English Dictionary. Introduction The Rubik’s Cube is unlike any other puzzle. It has many distinct and unique features that allow it to stand out among
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Gallagher 1
Sean Gallagher
Professor Ray Mest
Sr. Seminar, MAT 492
May 2011
The Rubik’s Cube and its Group Theory Applications
When the Rubik’s Cube was invented in 1974, it took the world by surprise. When the
idea was first thought up by Ernő Rubik, he may not have known how big the puzzle was
eventually going to be. The Rubik’s Cube has sold over 350 million copies, making it the best
selling puzzle of all time; it is considered by many to be the best selling toy in history. A few
years after its release in the market, nearly every child in every household owned the puzzle. It
has even made its way into the Oxford English Dictionary.
Introduction
The Rubik’s Cube is unlike any other puzzle. It has many distinct and unique features
that allow it to stand out among many other puzzles. A few key features that Ernő Rubik
pointed out himself are:
1) The cube (all of its pieces and all of its parts) stays together when being solved. Many
other puzzles that require moving parts have separating pieces.
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2) Several pieces move at once, in contrast to other puzzles that may only move one piece
at a time.
3) Most pieces of the cube have what is called “orientation.” This means that not only
does each piece have a correct “positioning,” but each piece has a correct orientation as
well. In other words, a single piece could be placed in the correct spot but could be
flipped (colorwise) the wrong way. Rubik says that the only other puzzles that have this
quality are assembly puzzles, which are very, very different types of puzzles as
compared to the Rubik’s Cube (viii).
4) The three-dimensionality of the cube is a unique characteristic trait. Three-dimensional
moving-piece puzzles are very rare. In Rubik’s eyes, this is a very important feature
(viii).
5) The cubicality of the cube. Simply put, the cube is a very satisfying shape to handle. It is
the most basic three-dimensional shape. On a cube it is easy to make specified turns
because everything is symmetrical and everything lines up nicely (Rubik viii)
6) The colors of the cube. It has great aesthetic appeal; some other puzzles lose their
appeal. Rubik put much thought into the colors of his puzzle. At first, he wished to
make opposite sides of the cube complementary colors. Later, he realized that he
wanted a white side to “brighten” the effect of the cube. So what he ended up doing
was separating colors on opposite sides by a factor of yellow. For example: yellow-
white, red-orange, and blue-green (Rubik viii).
7) The mechanism of the cube. This may be the most remarkable aspect of the puzzle.
When Ernő Rubik first proposed the idea of the Rubik’s Cube, people laughed at him and
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said that the puzzle was impossible to physically make. He ended up developing an
amazing core mechanism that fit together with each individual piece and allowed the
puzzle to exist.
8) The complexity of the cube. For such a simple looking puzzle, the complexity of the
cube is remarkable.
9) The mathematics of the cube. That is what this paper focused on. The Rubik’s Cube is a
great example of permutation groups and group theory.
10) The educational value of the cube. The ability to increase three-dimensional abilities of
children and adults.
Ernő Rubik first formulated the idea of the physical Rubik’s Cube in 1974 (Rubik 17). This
was when he thought up the final design and realized that it was actually possible for the cube
to be made. The cube is said to have been originally created for the purpose of illustrating
spatial moves. The cube became more than an illustration; it became a game with great
marketing possibilities. So in 1975 Rubik applied for a patent for the cube and began looking
for a mass manufacturer. Politechnika Cooperative eventually took the job (Rubik 17). The
Rubik’s Cube was put on the Hungarian market in 1977. It gained much self-propelled
popularity without too much advertisement. In 1980, over one million cubes were sold in
Hungary alone (Rubik 17).
In 1978, the Rubik’s Cube won a BNV prize at the International Budapest Fair (Rubik 17). In
1979, it won an award from the Hungarian Ministry for Cultural Affairs. In 1980, it won the
“Toy of the Year 1980” award in England, along with many other awards in other countries such
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as Germany and France. In 1981, the cube became included in the collection of design and
architecture for the New York Museum of Modern Art (Rubik 17).
Over 60 books have been published about the Rubik’s Cube (Rubik 17). The popularity of
the puzzle is enormous. There are dedicated fans and “cubers” around the world who spend
hours every day solving this amazing toy. There is even a term called “cubology,” with a big
focus on in depth analyses of the cube. Now let’s begin to get into the notation and
mathematics of the cube.
Notation
Let’s talk about the description and notation of the Rubik’s Cube. The original cube is a
3x3 cube puzzle made up of what appears to be 27 individual cubes. We will call each
individual cube piece a cubelet. If you dismantle the cube, you will see that the center cubelet
does not actually exist. In the center of all the cubelets is a core, rotating mechanism. This
mechanism attaches all six of the center face cubelets together. All of the other cubes interlock
with each other and are able to rotate in many different directions. The Rubik’s Cube as a
whole contains three basic types of cubelets. The first type is called a corner piece. There are
eight corner pieces on the cube, each of them at the eight different corners. These cubelets
have three different colors to them, considering they link a total of three sides. Each corner
piece can thus be oriented in three different ways. You can think of this by taking a starting
orientation of one of these cubelets and then rotating it clockwise. Rotating the cubelet
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clockwise three times will bring you back to the original orientation. The second type is called
an edge piece. There are 12 edge pieces on the cube, each of which surround the centers and
lie between the corner pieces. These cubelets link only two sides and they have two different
colors. Therefore, there are only two possible orientations for any given edge piece at any of
the specified 12 spots. The third type is called a center piece. There are six center pieces, each
obviously at the center of each of the six faces of the cube. These cubelets can never change
position with respect to another center cubelet. This is impossible because all six of the center
pieces are directly attached to the core mechanism on the inside of the cube. Considering a
center piece only consists of one color, it has only one possible orientation.
Say we set one corner cubelet into a designated spot on the cube; let’s call this spot a
cubicle. That leaves seven remaining cubicles for the seven remaining corner cubelets.
Following this pattern we have 8! possible positions for the corner pieces (“Group Theory” 10).
Using this same logic, there are 12! possible positions for the edge pieces. Since each corner
piece in its own cubicle can be oriented three ways, there are 3⁸ possible orientations for the
corner pieces. Following the same logic, there are 212 possible orientations for the edge pieces.
Therefore, there are 8!12!38212 possible configurations of the Rubik’s Cube (“Group Theory”
10)! This number is about 5.19 x 1020, or 519 quintillion! This number is a theoretical number
because not all of these configurations can actually be reached from a solved Rubik’s Cube.
There are some moves that just can’t be done. One example is flipping the orientation of just
one edge piece. The only way to reach all of these configurations would be to physically
dismantle the cube and put it back together, altering certain cubelets. In the Rubik’s Cube, only
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12 of the edge piece configurations match up with the original configuration of the cube. Only
13
of the corner piece configurations match up with the original configuration of the cube. Only 12
of the total possible configurations of the entire cube match up with the original configuration
of the cube. This claim can be shown by a specific theorem for the Rubik’s Cube:
Theorem: The total twist parity of the Rubik’s Cube cannot be changed (“Rubik’s Cube”).
To show this theorem, it’s important to understand a few more configurations that
cannot be obtained from twisting a solved cube. 1) As mentioned earlier, it is impossible to flip
the orientation of just one edge cubelet. 2) It is impossible to flip the orientation of just one
corner cubelet. 3) There is no process of moves that exchanges only two edge cubelets or two
corner cubelets (“Rubik’s Cube”). If we analyze each of these three smaller theorems, we find
out the following. Let’s say that we dismantle the cube and then put it back together; we will
obviously have choices on how we to reassemble the cube. For the first theorem there are two
possible choices: either flip the orientation of one edge cubelet or do not flip it. For the second
theorem there are three possible choices: the corner cubelet remains in its original orientation,
it is rotated once in the clockwise direction, or it is rotated once in the counterclockwise
direction. This is easy to see because each corner cubelet has three colors, allowing for three
possible orientations in its cubicle. For the third theorem there are two possible choices:
exchange two cubelets or do not exchange them. Therefore there are 2 x 2 x 3 = 12
independent ways of reassembling the cube. Only one of these 12 has the same parity of a
solvable cube (“Rubik’s Cube”).
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So it has been proven that 1 in 12 possible theoretical configurations can actually be
obtained from a solved Rubik’s Cube. Therefore, we take our theoretical number of 8!12!38212
and divide it by 12 to get 4.3252 x 1019 , or 43 quintillion, possible attainable configurations
(“Group Theory” 10)!
Now for the notation of the Rubik’s Cube. It is important when solving the cube that
specific faces are targeted. This is crucial because certain twists of the cube will need to be
done to certain faces. If we set the cube down on a flat surface, there are six different faces
that can be twisted. The face on top will be called “U” for “Up.” The face on the bottom,
touching the flat surface, will be called “D” for “Down.” The faces on the left and right will be
called “L” for “Left” and “R” for “Right,” respectively. The face that we are looking at is called
“F” for “Front.” The face that is facing away from us will be called “B” for “Back.” Here is a
diagram:
F
Front
R
Right
DDown
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This diagram illustrates the different faces of the Rubik’s Cube (“Mathematics of the
Rubik’s Cube” 1). Sometimes a specific cubelet may want to be addressed. Let’s say we are
looking at the F face and want to discuss the corner cubelet in the top right. This cubelet would
be called the “ufr” or “up, front, right” cubelet.
Now to discuss what it means to make a twist of one of the faces. To make a 90°
clockwise turn of the Front face is just denoted by “F.” This simply means to look at the F face
and twist it one quarter turn to the right (clockwise). The move “R” would mean to temporarily
look at the R face and make a 90° twist clockwise as well. This notation applies for all six faces
U
Up
L
Left
B
Back
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of the cube. The move “F’” (the tic after the F) means to rotate the F face 90°
counterclockwise. Therefore, the move FFR’R would mean F twist clockwise, F twist clockwise,
R twist counterclockwise, R twist clockwise. These sequences of moves, such as FFR’R, are
called algorithms. An algorithm simply means a sequence of specific moves designed to reach a
specific goal. The goal in the case of the Rubik’s Cube is usually to position or orientate a
specified number of cubelets. Solving the Rubik’s Cube involves executing an algorithm(s) to
restore the cube to its original state. One much asked question is “What is the minimal number
of twists to restore the cube from any mixed up state?” The answer to this question has been
referred to as “God’s Number” (“Mathematics of the Rubik’s Cube” 3). It has been proven to be
as low as 22. This means that from any mixed up state, the cube can be solved in 22 moves or
less!
Groups
First let’s define what a group is. A group, denoted by ¿, is a set G, closed under a binary
operation ¿ , such that the following axioms are satisfied:
1) For all a, b, c ∈ G, we have
(a ¿ b) ¿ c = a ¿ (b ¿ c). Associativity of ¿
2) There is an element e in G such that for all x ∈ G,
e ¿ x = x ¿ e = x. Identity element e for ¿
3) Corresponding to each a ∈ G, there is an element a’ in G such that
a ¿a’ = a’ ¿ a = e. Inverse a’ of a (Fraleigh 37-38)
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Note: A binary operation ¿ is a mapping from S x S into S (Fraleigh 20). (ex. +, /, …)
Now let’s prove that any sequence of moves performed on the Rubik’s Cube is a group.
Proof: To show this, we must show it satisfies the definition of a group as well as all three
axioms. If our group is ¿ on the Rubik’s Cube, then G and ¿ must first be defined. The set G on
the Rubik’s Cube is the set of all possible moves or sequence of moves. The binary operation ¿
is defined as M1 ¿ M2 where M1 is a move performed, followed by the execution of M2, where M2
is another move (“Group Theory” 11). It also must be shown that the set of moves is closed
under the binary operation. This is quite easy to see. If M1 is a move, and M2 is also a move,
then obviously M1 ¿ M2 is a move. Now that the set and binary operation are defined, let’s
prove the three axioms:
1) Associativity. It is important to note that a move can be defined by the change in the
configuration of the cube that it causes. So if we perform a move M1 on a cubelet, the
move puts the cubelet into another position M1(cubelet), that is the end result of the
move performed on the given cubelet. If a move M2 is then performed after M1, it puts
the cubelet into the position M2(M1(cubelet)), which equals (M1 ¿ M2)(cubelet) (“Group
Theory” 11).
For associativity, we want to show that [(M1 ¿ M2) ¿ M3](cubelet) = [M1 ¿ (M2 ¿ M3)]
(cubelet).
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From what we proved above, [(M1 ¿ M2) ¿ M3](cubelet) = M3[(M1 ¿ M2)(cubelet)] =