How to Find the Total Permutations of a Rubiks Cube
Content
Abstract
In my mathematics extended essay, I will attempt to solve the general case of a Rubik’s
Cube through the analysis of mathematical concepts and their constraints. The research
question at hand is, “how many different orientations does a Rubik’s cube have?” In
investigating this question, I first used my own initial approach (which was by examining the
differing orientations of each distinct piece), followed by expressing each orientation of a given
piece in an algebraic expression (i.e. 4! x 4) and finally forming an equation representing the
total “fair” and “unfair” orientations of the cube. In this context, fair represents a single, distinct
movement of four corner and edge pieces around a center noted as a layer while unfair
movements means any other form of movement not permitted in order to solve a Rubik’s cube
through suggested proper methods. The investigation included the usage of mathematical
concepts such as factorials (!), combinations (nCr) and permutations (nPr) to formulate a
working equation along with the use of diagrams provided by the Rubik’s branding company
which details proper movements, directional rotations and a solvers guide. The conclusion
reached by this evaluation is that given a corner has 3 orientations at 8 locations and an edge
piece has 2 orientations at n locations, as higher order cubes hold more edges, the total “unfair”
orientations of a Rubik’s cube equals 8! x 38 x n! x 2n. Conversely, the total “fair” permutations
are expressed as 8! x 37 x (n!/2) x 2(n-1) given that each for each orientation one piece remains
stationary. The centers of these cubes were not considered while forming these conclusions due
to their stationary positions upon the cube as well as their movements not being noticeable. The
conclusions were then validated through the use of several Rubik’s cubes ranging from a 2x2 to
5x5 dimensional cube and the permutation function on a TI-84 Plus graphing calculator enabling
my findings to overlap to find similarities between cube pieces. I finally looked at various other
solutions to solve for total permutations of larger order cubes holding greater number of edges.
Word Count: 350
1
Headings
Page
Introduction
3
Mechanism
4
Permutations
7
Initial Approach
8
Group Theory
9
Corners
11
Edges
12
Total Permutations
13
Further Cases
13
Bibliography
15
2
Introduction
The history of the Rubik’s cube came about by a young Professor of architecture in
Budapest. Born on July 13, 1944, Erno Rubik originally created the Rubik’s Cube, which at the
time was called the Magic Cube. The first working prototype of the Magic Cube was made in
1974 and distributed within Hungary by a toy making company named Politechnika. Due to
Hungary's communist environment, Erno needed a method in introducing his invention to the
world. By garnering the attention of mathematicians and Hungarian entrepreneurs, Erno’s
invention had reached various international conventions including the Nuremberg Toy Fair in
1979 where Tom Kremer, a toy specialist, agreed to sell and distribute Rubik’s Magic Cubes to
the rest of the world.
Kremer’s agreement enabled the Magic Cube to make its international debut in 1980.
Along the way, the Magic Cube became what is currently called the Rubik’s Cube.
Conventionally, the Rubik’s Cube can be challenging, but with proper understanding of the
cube’s fundamentals, it can be easily solved. Cubers, a term signifying someone who partakes
in solving Rubik’s Cubes, have created intuitive and complex algorithms1 that enable quick and
effective patterns that distill these fundamentals. Without these algorithmic patterns, it would be
more daunting to resolve these Rubik’s puzzles.
The primary focus of this Extended Essay will be in covering the total number of permutations2
that a Rubik’s Cube can have. My reason for choosing such a topic is that given how easy it is
1 A set of rules to be followed in calculations or other problem-solving operations
2 A way, especially one of several possible variations, in which a set or number of things can be arranged
depending on order
3
to solve a Rubik’s Cube with current forms of algorithms, many people forget how difficult it is to
be able to solve the puzzle from any given orientation. This being said, I find this topic enjoyable
due to my appreciation for simple and complex puzzles.
Finding a solution to the Rubik’s Cube came about from considering its mechanism,
permutations and factorials in order to create a working solution. Considering both legal and
illegal methods provided by the Rubik’s Company diagrams, I solved both standards of
permutations to enable calculations for total possible permutations and total fair permutations.
With these considerations, I will delve into permutations of each individual part of the Rubik’s
Cube. This analysis will begin with corner and edge permutations. Within this essay, Corners
are defined as cubies3 with three visible faces with differing colors which affects the outcome of
these Cubes due to having certain orientations requiring similar yet different algorithms to solve
while edges are cubies with two visible faces with differing colors. From understanding these
concepts of edges and corners introduces a new notion branded as layers4 which encompasses
both edges and corners. From the analysis of edges, corners and layers, I formulated an
equation to solve for total permutations of a Rubik’s Cube.
Please note that the beginning of this essay (and its solutions) will analyze the 3x3x3
dimensional Rubik’s Cube - however, I will explore further applications and solutions of
permutations in higher order cubes.
Mechanism
3 A cubical shape within the Rubik’s Cube that can be reoriented to form permutations
4 Consisting of 4 corners, 4 edges, and 1 center
4
The difficulty that arises within this investigation comes from inside the Rubik’s Cube. The cube
itself has 9 lateral faces for each of the 6 larger faces.5 This makes a total of 6 x 9 = 54 lateral
faces composed of these cubies. Each cubie is part of 3 different layers and each of these
layers consists of 9 cubies; this is including the centers of each layers. The mechanism inside of
the cube allows each of the 9 layers to be turned freely in its plane and around its one-colored
face. This makes it relatively easy to recognize the three different types of cubies: 8 corner
cubies with three colored tiles, 12 edge cubies with two colored tiles, and 6 face cubies with one
colored tile. Along these lines, each cubie remains true to its type throughout each of its
permutations, meaning that every time a layer is turned, the cubies maintains the original shape
of a cube but in turn rearranges the cubies positions between each layer.
The next problem in understanding the Rubik’s Cube’s mechanism is by asking how it is
possible to create such an object. For instance, Christoph Bandelow, author of Inside Rubik’s
Cube and Beyond, gave an example, “Why don’t the cubies sitting at the corners simply fall off?
They cannot be attached on any side because of the turning mechanics of the three layers they
belong to.” This can be explained through the elaborate mechanism that enables each cubie in
the puzzle to maintain its location without falling out of place. Closer examination of the inner
5 Rectangles within a right prism
5
workings of the cube can be viewed by turning a single layer 45° and prying an edge cubie
outwards.
The final consideration is the movement of each cubie within its given layers. Noted by
the Rubik’s company diagrams, notations for movement are given by layers (which are also the
6 faces of the cube). Each layer is annotated as either Left, Right, Up, Down, Front, or Back
depending upon how the cube is handled (Middle layer movements can also be noted but are
not as readily used). The abbreviation for these annotations are L, R, U, D, F, or B respectively
for which each of these turns are also stated to be 90°due to its clocklike rotation.
Each turn can also have a certain directional rotation being either clockwise or
counterclockwise (also known as prime). Rotations that are counterclockwise are annotated with
an apostrophe following the layer needing to be turned such as L’, R’, and U’ while clockwise
rotations are not noted with an apostrophe. Consecutive movements in the same layer are
6
interpreted with a 2 in front of the annotation. Only a 2 is used for further movements due to the
clockwise and counterclockwise rotations of each layer. It makes sense that if 1 rotation equal
90° while 2 rotations equal 180°, it is unnecessary to move any further because 3 rotations
being 270° is the same as 1 reversed rotation. The reason behind the use of the apostrophe is
to enable a solver to orient each cubie within the puzzle to specific locations more efficiently
while reducing the number of movements required.
Permutations
Without a proper understanding of what a permutation is, solving for total permutation
would be more obscured from the true total number because factors not relevant to the actual
data would be present within the calculations. The first idea to be noted for permutation is
parity6. In mathematics, when X is a finite set of at least two elements, the permutations of X fall
into two classes of equal size: the even permutations and the odd permutations. Every and all
permutations will be either even or odd thus the same should be true for a Rubik’s Cube. To
determine whether a permutation is even or odd, we can count the number pairs out of their
natural order.
For example7, take two sets of numbers with one set being in its natural order.
12345678
21385674
6 The fact of being even or odd
7 Vaughen, Scott. "Counting the Permutations of the Rubik’s Cube." N.d. Microsoft PowerPoint file.
7
First start with the second row and, for each number x, count how many numbers follow that are
reversed out of the natural order with x.
Number
Numbers out of natural order
2 is followed by 1
1 Reversal
1 is followed by only larger numbers
0 Reversals
3 is followed by only larger numbers
0 Reversals
8 is followed by 5, 6, 7, and 4
4 Reversals
5 is followed by 4
1 Reversal
6 is followed by 4
1 Reversal
7 is followed by 4
1 Reversal
4 is the last number in the sequence
0 Reversals
The total reversals8 of the second row is 8 therefore an even permutation. Given this
fact, it must be true that any other permutation created with this set of numbers will result in an
even number of reversals. This process must be used when figuring out if the cube has an even
or odd permutation in order to find how many permutations are of fair nature.
Initial Approach
In order to fully grasp how many permutations are within a Rubik’s Cube, we must first consider
each piece as an individual entity. By scrambling the cube and finding the total number of
permutations for one type of cubie, you can apply that same total to the rest of the cubies at
similar locations. For example, an edge cubie has 12 locations where they can be relocated and
2 orientations for each location. Therefore the total number of permutations for corners must be
8 A number which follows another that is not in the normal counting sequence
8
212 because each location has 2 orientations for each cubie multiplied by itself 12 times to
account for cubies shifting to different locations. Corner cubies should therefore be 38 for having
3 orientations for 8 locations. The next step is to multiply both quantities to find total
permutations. This is because if we added the total permutations for each type of cubie, the
solution would not yield a factor which considers location for each of these cubies respectively.
Thus, 212 x 38 = 26,873,856 should be the total number of permutations within a Rubik’s Cube.
Although this approach assumes some critical points being location and orientation of
each cubie, a factor for location was still not present within the calculations meaning that the
above stated solution for total permutations for a Rubik’s Cube is incorrect. The problem with
the previous solution was that orientation of a cubie does not matter when finding total
permutations for that cubies location. This can be proven true by analyzing location as a
property of a cubie without considering orientation (these properties then should be a
multiplicative factor for that type of cubie). Given that corner and edge properties form a direct
relationship with total permutations, each orientation and location properties for both must be
multiplied together. Thus the newly formulated equation should be the cubie’s properties
multiplied by each other, then total cubie permutations multiplied by each other in order to get a
total for a Rubik’s Cube’s permutations [i.e. (corner properties) x (edge properties) - or (orientation x location) x (orientation x location) = total permutations].
Group Theory
In mathematics and abstract algebra, group theory studies the algebraic structures
known as groups. The concept of a group is central to abstract algebra: other well-known
algebraic structures, such as rings, fields, and vector spaces can all be seen as groups
9
endowed with additional operations and axioms9. In this study, group theory will be used to
evaluate the amount of evenness or oddness is within each permutation of a Rubik’s Cube.
Hannah Provenza, University of Chicago graduate, explains how there are equal amounts of
even and odd permutations. Her first theorem in explaining this was that,
“The set of operations on the cube that can be reached from the solved state by making the
moves listed above in various combinations is a group under the operation of concatenation.
We will then refer to this group as the cube group.”
This theorem was proven by first explaining that the group has associativity, or that
(XY)Z = X(YZ). These variables represent the axes of which the cube remains within a plane.
(XY)Z means to execute move XY, then move Z, while X(YZ) means to execute move X, then
move Y Z. Provenza states that both of these are equivalent to making move X, followed by Y,
followed by Z. The next issue to state is the cubies that do not move such as the centers.
“The identity element, I, is making no move at all. Since this leaves the cube unchanged,
we obviously have IX = X = XI. The inverse, X-1, for any single face turn is X3, which is
equivalent to the face turn X performed counterclockwise instead of clockwise. The inverse of a
sequence of moves XY is Y-1X-1. Intuitively, this is clear: to undo any sequence of moves is to
undo each move in turn; starting with the first move and working back to the first. This can be
easily proven by multiplying the two together: (XY)(Y-1X-1) = X(YY-1)X-1 = X(I)X-1 = XX-1 = I.”
Knowing that the identity of the cube group makes it so that each move needs to be
reversed in order to reach the solved state, it is justifiable then to assume that half of the cubes
permutations are even and the other half are odd. Having the variable “I” be even means that
the two variables must be both odd or both even. By analyzing one face of a Rubik’s Cube, it is
possible to determine how many types of permutations are within a Rubik’s Cube. By shifting
each cubie while having only the corresponding color with the center on that face, one can see
9 A statement or proposition on which an abstractly defined structure is based.
10
that each corner can be in 4 locations and the same with edges. With this, subtract one from the
locations to get possible permutations for one cubie on a face. The result is 6 meaning that
there is an even number of permutations for each fair permutation. Given that half of all
permutations of a given set of elements are even and the other half odd, all fair movements
should be even due to this process while all other movements are odd.
Knowing that there is exactly half even and odd permutations within a Rubik’s Cube
gives a factor of ½ to the equation to find the total fair permutations. This is because by
orienting any cubie on a cube through fair permutations means that there are equal amounts of
fair and unfair permutations.
Corners
Both forms of cubies have similar principles but have slightly different properties. Corner
properties can be deduced by principles of orientation and location. In order to find orientation, a
similar process as the initial one should be taken. It is correct that if each of the 8 corners can
have 3 orientations in 8 locations, the first value for corner orientations should be 38. Location is
the harder of the principles to solve due to the misconception that location is dependent upon
orientation. A simple way to analyze location is by looking at the Rubik’s Cube on a XYZ
coordinate plane.
11
Corners cubies are what you would expect: cubies placed at each of the 8 corners on
the cube. The coordinate representation helps in explaining the corner’s location property by
assuming that each corner represents a point on a 3 dimensional plane. Thus, to find
permutation, you must count the total number of combinations at which a corner can be located
from 8 locations (given that there are 8 corners in a cube). A Rubik's Cube has corners that can
be located at 8 possible locations for each piece. It is then inaccurate to assume that each
corner has 8 location permutations if solved because the total location permutations must
consider the movements of each corner while one corner remains stationary. If one corner
remains stationary while the other 7 shift throughout the cube, the number of times a corner will
be accounted for will be twice as much making the value larger than what is possible through
fair methods of solving the puzzle. Since each corner would be accounted for twice without
considering multiple countings, each corner after the initial must be one increment less than its
previous to find the total number of combinations for one corner. Total location permutations per
corner would therefore be 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 40,320 or 8!.
These two principles of orientation and location should then be multiplied together to
achieve the total number of permutations for corners. 8! x 38 = 264,539,520 is the total
permutations for corner cubies. The main limiting factor for this solution is the movements of the
cubies themselves. The number stated above symbolizes the total permutations possible
through disassembly of the cube. While it may be true that there can be 38 total orientations for
12
corners, the fair method of solving a rubiks cube only allows 37 ways to orient corners. This is
because 7 of these corners can be oriented independently while the orientation of the eighth
depends on the preceding seven, giving a total of 37 (2,187) possible orientations for corners.
The new equation for total corner permutations without disassembly is therefore 8! x 37 =
88,179,840 permutations.
Edges
Edge cubies within the Rubik’s Cube hold the same type of properties that corner cubies
do. To find the total possible permutations without limitations, the first consideration is to find the
location orientations and then progress to the application of orientations with location. Thus, in
order to find total location permutations, refer back to figure 4 and assume that each edge cubie
is a point on a 3 dimensional plane. Since counting location would mean that each cubie would
be accounted for more than once per location, if one cubie was stationary while the others are
being relocated, factorials should be the method of choice for finding total location permutations.
For a Rubik’s cube, there are 12 locations per edge cubie. Thus, following the same method as
corner cubies, the total permutation for location should be 12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3
x 2 x 1 = 479,001,600 or 12!.
With this in mind, we know as previously discussed that the total number of orientations
that an edge cubie has is 212 given that there is 2 orientations for each of the 12 edge cubies.
And similarly to corner cubies, the Rubik’s cube can have 212 permutations without considering
the fair methods of solving the cube meaning that the power should be n-1 with respect to the
fair movements of each edge permutation. Thus the permutations for and edge cubie for unfair
permutations should be 12! x 212 and 12! x 211 for fair permutations.
13
Total Permutations
Through this method of evaluation to find permutations, the solution for the Rubik’s
Cube’s total unfair permutations is therefore 8! x 38 x 12! x 212 = 5.19 x 1020 permutations while 8!
x 37 x 12! x 211 = 8.65 x 1019 equals the total fair permutations. The solution for total fair
permutations is, however, not complete since the equation does consider the parity error which
is ½ since there is an equal number of even and odd permutations as well as the corner and
edge parity that must be accounted for since one permutation for each form of cubie not
account for the total permutation. Since there is an equal number of even and odd permutations,
the division factor must be 2. The true equation for total fair permutations of a Rubik’s Cube is
therefore ( 8! x 37 x 12! x 211 ) / 2 = 4.32 x 1019.
Further Case Applications
The Rubik’s brand does not only account for a single type of dimensional cube. There
are many variations the original 3 by 3 Rubiks cube that range in a multitude of dimensions
which in turn have varying amounts of permutations. One example of varying permutations
would be a 2x2 Rubik’s Cube. Having only 8 cubies meaning 8 corner cubies only, the total
number of permutations is the same as a 3x3 Rubik’s Cube if you are only considering its
corners as the only form of permutations. This being said, the total number of permutations for a
2x2 is therefore for 8! x 38 unfair movements and 8! x 37 for fair movements. Larger case
scenarios such as a 5x5 Rubik’s Cube need additional considerations when solving for total
permutations.
14
Everything that was true for a 3x3 Rubik’s Cube is true for a 5x5 as well. The new
properties that must be accounted for are the addition of 24 edge cubies and 48 center cubies.
These cubies are added due to the increased amount of space needed to be filled while still
allowing 5 layers of movement. The mechanism as well makes it so that the 24 new edge
cubies cannot be flipped due to the interior shape of those pieces being asymmetrical. The
corresponding outer edges are distinguishable since the pieces are mirror images of each other
and sit a joint to the center edge cubie. Any permutation of the outer edges is possible, including
odd permutations, giving 24! permutations for these additional edge cubies.
With the 48 new center cubies, 24 are corners of the entire center and 24 are the edges
of the entire center. The corner and edge centers cannot be flipped with each other, similarly to
the 24 edges of the cube, due to the internal mechanism. This means that there are 24!2
permutations for both corner centers and edge centers. There is also a division factor that must
be considered when solving for center cubies of a 5x5 Rubik’s Cube. For each of these
additional centers, they can have 4! orientations per face for these edges and centers. Since
there are 6 faces, this factor must be raised to the 6th meaning that there are 4!6 permutations
that are not possible for edges or corner centers. Since each edge and corner center cubies
have 24! / 4!6 permutations, the total permutations for these two combined is 24!2 / 4!12.
The solution for total permutations of a 5x5 Rubik’s Cube, while still considering fair
movements, is therefore ( 8! x 37 x 12! x 211 x 24!3) / ( 2 x 4!12 ) = 2.83 x 1074 permutations.
Bibliography
Bandelow, Christoph. Inside Rubik's Cube and Beyond. Boston: Birkhäuser, 1982.
Print.
Courant, Richard, Herbert Robbins, and Ian Stewart. What Is Mathematics?: An Elementary
Approach to Ideas and Methods. 1941. 2nd ed. New York: Oxford
UP, 1996. Print.
Provenza, Hannah. Group Theory and the Rubik's Cube. Chicago: U of Chicago, n.d.
15
Print.
Stewart, Ian. Symmetry: A Very Short Introduction. N.p.: Oxford UP, 2013. Print.
Vaughen, Scott. "Counting the Permutations of the Rubik’s Cube." N.d.
Microsoft PowerPoint file.
Wolfram Math World. Wolfram Research, n.d. Web. 30 Aug. 2014.
<http://mathworld.wolfram.com/GroupTheory.html>.
16
In my mathematics extended essay, I will attempt to solve the general case of a Rubik’s
Cube through the analysis of mathematical concepts and their constraints. The research
question at hand is, “how many different orientations does a Rubik’s cube have?” In
investigating this question, I first used my own initial approach (which was by examining the
differing orientations of each distinct piece), followed by expressing each orientation of a given
piece in an algebraic expression (i.e. 4! x 4) and finally forming an equation representing the
total “fair” and “unfair” orientations of the cube. In this context, fair represents a single, distinct
movement of four corner and edge pieces around a center noted as a layer while unfair
movements means any other form of movement not permitted in order to solve a Rubik’s cube
through suggested proper methods. The investigation included the usage of mathematical
concepts such as factorials (!), combinations (nCr) and permutations (nPr) to formulate a
working equation along with the use of diagrams provided by the Rubik’s branding company
which details proper movements, directional rotations and a solvers guide. The conclusion
reached by this evaluation is that given a corner has 3 orientations at 8 locations and an edge
piece has 2 orientations at n locations, as higher order cubes hold more edges, the total “unfair”
orientations of a Rubik’s cube equals 8! x 38 x n! x 2n. Conversely, the total “fair” permutations
are expressed as 8! x 37 x (n!/2) x 2(n-1) given that each for each orientation one piece remains
stationary. The centers of these cubes were not considered while forming these conclusions due
to their stationary positions upon the cube as well as their movements not being noticeable. The
conclusions were then validated through the use of several Rubik’s cubes ranging from a 2x2 to
5x5 dimensional cube and the permutation function on a TI-84 Plus graphing calculator enabling
my findings to overlap to find similarities between cube pieces. I finally looked at various other
solutions to solve for total permutations of larger order cubes holding greater number of edges.
Word Count: 350
1
Headings
Page
Introduction
3
Mechanism
4
Permutations
7
Initial Approach
8
Group Theory
9
Corners
11
Edges
12
Total Permutations
13
Further Cases
13
Bibliography
15
2
Introduction
The history of the Rubik’s cube came about by a young Professor of architecture in
Budapest. Born on July 13, 1944, Erno Rubik originally created the Rubik’s Cube, which at the
time was called the Magic Cube. The first working prototype of the Magic Cube was made in
1974 and distributed within Hungary by a toy making company named Politechnika. Due to
Hungary's communist environment, Erno needed a method in introducing his invention to the
world. By garnering the attention of mathematicians and Hungarian entrepreneurs, Erno’s
invention had reached various international conventions including the Nuremberg Toy Fair in
1979 where Tom Kremer, a toy specialist, agreed to sell and distribute Rubik’s Magic Cubes to
the rest of the world.
Kremer’s agreement enabled the Magic Cube to make its international debut in 1980.
Along the way, the Magic Cube became what is currently called the Rubik’s Cube.
Conventionally, the Rubik’s Cube can be challenging, but with proper understanding of the
cube’s fundamentals, it can be easily solved. Cubers, a term signifying someone who partakes
in solving Rubik’s Cubes, have created intuitive and complex algorithms1 that enable quick and
effective patterns that distill these fundamentals. Without these algorithmic patterns, it would be
more daunting to resolve these Rubik’s puzzles.
The primary focus of this Extended Essay will be in covering the total number of permutations2
that a Rubik’s Cube can have. My reason for choosing such a topic is that given how easy it is
1 A set of rules to be followed in calculations or other problem-solving operations
2 A way, especially one of several possible variations, in which a set or number of things can be arranged
depending on order
3
to solve a Rubik’s Cube with current forms of algorithms, many people forget how difficult it is to
be able to solve the puzzle from any given orientation. This being said, I find this topic enjoyable
due to my appreciation for simple and complex puzzles.
Finding a solution to the Rubik’s Cube came about from considering its mechanism,
permutations and factorials in order to create a working solution. Considering both legal and
illegal methods provided by the Rubik’s Company diagrams, I solved both standards of
permutations to enable calculations for total possible permutations and total fair permutations.
With these considerations, I will delve into permutations of each individual part of the Rubik’s
Cube. This analysis will begin with corner and edge permutations. Within this essay, Corners
are defined as cubies3 with three visible faces with differing colors which affects the outcome of
these Cubes due to having certain orientations requiring similar yet different algorithms to solve
while edges are cubies with two visible faces with differing colors. From understanding these
concepts of edges and corners introduces a new notion branded as layers4 which encompasses
both edges and corners. From the analysis of edges, corners and layers, I formulated an
equation to solve for total permutations of a Rubik’s Cube.
Please note that the beginning of this essay (and its solutions) will analyze the 3x3x3
dimensional Rubik’s Cube - however, I will explore further applications and solutions of
permutations in higher order cubes.
Mechanism
3 A cubical shape within the Rubik’s Cube that can be reoriented to form permutations
4 Consisting of 4 corners, 4 edges, and 1 center
4
The difficulty that arises within this investigation comes from inside the Rubik’s Cube. The cube
itself has 9 lateral faces for each of the 6 larger faces.5 This makes a total of 6 x 9 = 54 lateral
faces composed of these cubies. Each cubie is part of 3 different layers and each of these
layers consists of 9 cubies; this is including the centers of each layers. The mechanism inside of
the cube allows each of the 9 layers to be turned freely in its plane and around its one-colored
face. This makes it relatively easy to recognize the three different types of cubies: 8 corner
cubies with three colored tiles, 12 edge cubies with two colored tiles, and 6 face cubies with one
colored tile. Along these lines, each cubie remains true to its type throughout each of its
permutations, meaning that every time a layer is turned, the cubies maintains the original shape
of a cube but in turn rearranges the cubies positions between each layer.
The next problem in understanding the Rubik’s Cube’s mechanism is by asking how it is
possible to create such an object. For instance, Christoph Bandelow, author of Inside Rubik’s
Cube and Beyond, gave an example, “Why don’t the cubies sitting at the corners simply fall off?
They cannot be attached on any side because of the turning mechanics of the three layers they
belong to.” This can be explained through the elaborate mechanism that enables each cubie in
the puzzle to maintain its location without falling out of place. Closer examination of the inner
5 Rectangles within a right prism
5
workings of the cube can be viewed by turning a single layer 45° and prying an edge cubie
outwards.
The final consideration is the movement of each cubie within its given layers. Noted by
the Rubik’s company diagrams, notations for movement are given by layers (which are also the
6 faces of the cube). Each layer is annotated as either Left, Right, Up, Down, Front, or Back
depending upon how the cube is handled (Middle layer movements can also be noted but are
not as readily used). The abbreviation for these annotations are L, R, U, D, F, or B respectively
for which each of these turns are also stated to be 90°due to its clocklike rotation.
Each turn can also have a certain directional rotation being either clockwise or
counterclockwise (also known as prime). Rotations that are counterclockwise are annotated with
an apostrophe following the layer needing to be turned such as L’, R’, and U’ while clockwise
rotations are not noted with an apostrophe. Consecutive movements in the same layer are
6
interpreted with a 2 in front of the annotation. Only a 2 is used for further movements due to the
clockwise and counterclockwise rotations of each layer. It makes sense that if 1 rotation equal
90° while 2 rotations equal 180°, it is unnecessary to move any further because 3 rotations
being 270° is the same as 1 reversed rotation. The reason behind the use of the apostrophe is
to enable a solver to orient each cubie within the puzzle to specific locations more efficiently
while reducing the number of movements required.
Permutations
Without a proper understanding of what a permutation is, solving for total permutation
would be more obscured from the true total number because factors not relevant to the actual
data would be present within the calculations. The first idea to be noted for permutation is
parity6. In mathematics, when X is a finite set of at least two elements, the permutations of X fall
into two classes of equal size: the even permutations and the odd permutations. Every and all
permutations will be either even or odd thus the same should be true for a Rubik’s Cube. To
determine whether a permutation is even or odd, we can count the number pairs out of their
natural order.
For example7, take two sets of numbers with one set being in its natural order.
12345678
21385674
6 The fact of being even or odd
7 Vaughen, Scott. "Counting the Permutations of the Rubik’s Cube." N.d. Microsoft PowerPoint file.
7
First start with the second row and, for each number x, count how many numbers follow that are
reversed out of the natural order with x.
Number
Numbers out of natural order
2 is followed by 1
1 Reversal
1 is followed by only larger numbers
0 Reversals
3 is followed by only larger numbers
0 Reversals
8 is followed by 5, 6, 7, and 4
4 Reversals
5 is followed by 4
1 Reversal
6 is followed by 4
1 Reversal
7 is followed by 4
1 Reversal
4 is the last number in the sequence
0 Reversals
The total reversals8 of the second row is 8 therefore an even permutation. Given this
fact, it must be true that any other permutation created with this set of numbers will result in an
even number of reversals. This process must be used when figuring out if the cube has an even
or odd permutation in order to find how many permutations are of fair nature.
Initial Approach
In order to fully grasp how many permutations are within a Rubik’s Cube, we must first consider
each piece as an individual entity. By scrambling the cube and finding the total number of
permutations for one type of cubie, you can apply that same total to the rest of the cubies at
similar locations. For example, an edge cubie has 12 locations where they can be relocated and
2 orientations for each location. Therefore the total number of permutations for corners must be
8 A number which follows another that is not in the normal counting sequence
8
212 because each location has 2 orientations for each cubie multiplied by itself 12 times to
account for cubies shifting to different locations. Corner cubies should therefore be 38 for having
3 orientations for 8 locations. The next step is to multiply both quantities to find total
permutations. This is because if we added the total permutations for each type of cubie, the
solution would not yield a factor which considers location for each of these cubies respectively.
Thus, 212 x 38 = 26,873,856 should be the total number of permutations within a Rubik’s Cube.
Although this approach assumes some critical points being location and orientation of
each cubie, a factor for location was still not present within the calculations meaning that the
above stated solution for total permutations for a Rubik’s Cube is incorrect. The problem with
the previous solution was that orientation of a cubie does not matter when finding total
permutations for that cubies location. This can be proven true by analyzing location as a
property of a cubie without considering orientation (these properties then should be a
multiplicative factor for that type of cubie). Given that corner and edge properties form a direct
relationship with total permutations, each orientation and location properties for both must be
multiplied together. Thus the newly formulated equation should be the cubie’s properties
multiplied by each other, then total cubie permutations multiplied by each other in order to get a
total for a Rubik’s Cube’s permutations [i.e. (corner properties) x (edge properties) - or (orientation x location) x (orientation x location) = total permutations].
Group Theory
In mathematics and abstract algebra, group theory studies the algebraic structures
known as groups. The concept of a group is central to abstract algebra: other well-known
algebraic structures, such as rings, fields, and vector spaces can all be seen as groups
9
endowed with additional operations and axioms9. In this study, group theory will be used to
evaluate the amount of evenness or oddness is within each permutation of a Rubik’s Cube.
Hannah Provenza, University of Chicago graduate, explains how there are equal amounts of
even and odd permutations. Her first theorem in explaining this was that,
“The set of operations on the cube that can be reached from the solved state by making the
moves listed above in various combinations is a group under the operation of concatenation.
We will then refer to this group as the cube group.”
This theorem was proven by first explaining that the group has associativity, or that
(XY)Z = X(YZ). These variables represent the axes of which the cube remains within a plane.
(XY)Z means to execute move XY, then move Z, while X(YZ) means to execute move X, then
move Y Z. Provenza states that both of these are equivalent to making move X, followed by Y,
followed by Z. The next issue to state is the cubies that do not move such as the centers.
“The identity element, I, is making no move at all. Since this leaves the cube unchanged,
we obviously have IX = X = XI. The inverse, X-1, for any single face turn is X3, which is
equivalent to the face turn X performed counterclockwise instead of clockwise. The inverse of a
sequence of moves XY is Y-1X-1. Intuitively, this is clear: to undo any sequence of moves is to
undo each move in turn; starting with the first move and working back to the first. This can be
easily proven by multiplying the two together: (XY)(Y-1X-1) = X(YY-1)X-1 = X(I)X-1 = XX-1 = I.”
Knowing that the identity of the cube group makes it so that each move needs to be
reversed in order to reach the solved state, it is justifiable then to assume that half of the cubes
permutations are even and the other half are odd. Having the variable “I” be even means that
the two variables must be both odd or both even. By analyzing one face of a Rubik’s Cube, it is
possible to determine how many types of permutations are within a Rubik’s Cube. By shifting
each cubie while having only the corresponding color with the center on that face, one can see
9 A statement or proposition on which an abstractly defined structure is based.
10
that each corner can be in 4 locations and the same with edges. With this, subtract one from the
locations to get possible permutations for one cubie on a face. The result is 6 meaning that
there is an even number of permutations for each fair permutation. Given that half of all
permutations of a given set of elements are even and the other half odd, all fair movements
should be even due to this process while all other movements are odd.
Knowing that there is exactly half even and odd permutations within a Rubik’s Cube
gives a factor of ½ to the equation to find the total fair permutations. This is because by
orienting any cubie on a cube through fair permutations means that there are equal amounts of
fair and unfair permutations.
Corners
Both forms of cubies have similar principles but have slightly different properties. Corner
properties can be deduced by principles of orientation and location. In order to find orientation, a
similar process as the initial one should be taken. It is correct that if each of the 8 corners can
have 3 orientations in 8 locations, the first value for corner orientations should be 38. Location is
the harder of the principles to solve due to the misconception that location is dependent upon
orientation. A simple way to analyze location is by looking at the Rubik’s Cube on a XYZ
coordinate plane.
11
Corners cubies are what you would expect: cubies placed at each of the 8 corners on
the cube. The coordinate representation helps in explaining the corner’s location property by
assuming that each corner represents a point on a 3 dimensional plane. Thus, to find
permutation, you must count the total number of combinations at which a corner can be located
from 8 locations (given that there are 8 corners in a cube). A Rubik's Cube has corners that can
be located at 8 possible locations for each piece. It is then inaccurate to assume that each
corner has 8 location permutations if solved because the total location permutations must
consider the movements of each corner while one corner remains stationary. If one corner
remains stationary while the other 7 shift throughout the cube, the number of times a corner will
be accounted for will be twice as much making the value larger than what is possible through
fair methods of solving the puzzle. Since each corner would be accounted for twice without
considering multiple countings, each corner after the initial must be one increment less than its
previous to find the total number of combinations for one corner. Total location permutations per
corner would therefore be 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 40,320 or 8!.
These two principles of orientation and location should then be multiplied together to
achieve the total number of permutations for corners. 8! x 38 = 264,539,520 is the total
permutations for corner cubies. The main limiting factor for this solution is the movements of the
cubies themselves. The number stated above symbolizes the total permutations possible
through disassembly of the cube. While it may be true that there can be 38 total orientations for
12
corners, the fair method of solving a rubiks cube only allows 37 ways to orient corners. This is
because 7 of these corners can be oriented independently while the orientation of the eighth
depends on the preceding seven, giving a total of 37 (2,187) possible orientations for corners.
The new equation for total corner permutations without disassembly is therefore 8! x 37 =
88,179,840 permutations.
Edges
Edge cubies within the Rubik’s Cube hold the same type of properties that corner cubies
do. To find the total possible permutations without limitations, the first consideration is to find the
location orientations and then progress to the application of orientations with location. Thus, in
order to find total location permutations, refer back to figure 4 and assume that each edge cubie
is a point on a 3 dimensional plane. Since counting location would mean that each cubie would
be accounted for more than once per location, if one cubie was stationary while the others are
being relocated, factorials should be the method of choice for finding total location permutations.
For a Rubik’s cube, there are 12 locations per edge cubie. Thus, following the same method as
corner cubies, the total permutation for location should be 12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3
x 2 x 1 = 479,001,600 or 12!.
With this in mind, we know as previously discussed that the total number of orientations
that an edge cubie has is 212 given that there is 2 orientations for each of the 12 edge cubies.
And similarly to corner cubies, the Rubik’s cube can have 212 permutations without considering
the fair methods of solving the cube meaning that the power should be n-1 with respect to the
fair movements of each edge permutation. Thus the permutations for and edge cubie for unfair
permutations should be 12! x 212 and 12! x 211 for fair permutations.
13
Total Permutations
Through this method of evaluation to find permutations, the solution for the Rubik’s
Cube’s total unfair permutations is therefore 8! x 38 x 12! x 212 = 5.19 x 1020 permutations while 8!
x 37 x 12! x 211 = 8.65 x 1019 equals the total fair permutations. The solution for total fair
permutations is, however, not complete since the equation does consider the parity error which
is ½ since there is an equal number of even and odd permutations as well as the corner and
edge parity that must be accounted for since one permutation for each form of cubie not
account for the total permutation. Since there is an equal number of even and odd permutations,
the division factor must be 2. The true equation for total fair permutations of a Rubik’s Cube is
therefore ( 8! x 37 x 12! x 211 ) / 2 = 4.32 x 1019.
Further Case Applications
The Rubik’s brand does not only account for a single type of dimensional cube. There
are many variations the original 3 by 3 Rubiks cube that range in a multitude of dimensions
which in turn have varying amounts of permutations. One example of varying permutations
would be a 2x2 Rubik’s Cube. Having only 8 cubies meaning 8 corner cubies only, the total
number of permutations is the same as a 3x3 Rubik’s Cube if you are only considering its
corners as the only form of permutations. This being said, the total number of permutations for a
2x2 is therefore for 8! x 38 unfair movements and 8! x 37 for fair movements. Larger case
scenarios such as a 5x5 Rubik’s Cube need additional considerations when solving for total
permutations.
14
Everything that was true for a 3x3 Rubik’s Cube is true for a 5x5 as well. The new
properties that must be accounted for are the addition of 24 edge cubies and 48 center cubies.
These cubies are added due to the increased amount of space needed to be filled while still
allowing 5 layers of movement. The mechanism as well makes it so that the 24 new edge
cubies cannot be flipped due to the interior shape of those pieces being asymmetrical. The
corresponding outer edges are distinguishable since the pieces are mirror images of each other
and sit a joint to the center edge cubie. Any permutation of the outer edges is possible, including
odd permutations, giving 24! permutations for these additional edge cubies.
With the 48 new center cubies, 24 are corners of the entire center and 24 are the edges
of the entire center. The corner and edge centers cannot be flipped with each other, similarly to
the 24 edges of the cube, due to the internal mechanism. This means that there are 24!2
permutations for both corner centers and edge centers. There is also a division factor that must
be considered when solving for center cubies of a 5x5 Rubik’s Cube. For each of these
additional centers, they can have 4! orientations per face for these edges and centers. Since
there are 6 faces, this factor must be raised to the 6th meaning that there are 4!6 permutations
that are not possible for edges or corner centers. Since each edge and corner center cubies
have 24! / 4!6 permutations, the total permutations for these two combined is 24!2 / 4!12.
The solution for total permutations of a 5x5 Rubik’s Cube, while still considering fair
movements, is therefore ( 8! x 37 x 12! x 211 x 24!3) / ( 2 x 4!12 ) = 2.83 x 1074 permutations.
Bibliography
Bandelow, Christoph. Inside Rubik's Cube and Beyond. Boston: Birkhäuser, 1982.
Print.
Courant, Richard, Herbert Robbins, and Ian Stewart. What Is Mathematics?: An Elementary
Approach to Ideas and Methods. 1941. 2nd ed. New York: Oxford
UP, 1996. Print.
Provenza, Hannah. Group Theory and the Rubik's Cube. Chicago: U of Chicago, n.d.
15
Print.
Stewart, Ian. Symmetry: A Very Short Introduction. N.p.: Oxford UP, 2013. Print.
Vaughen, Scott. "Counting the Permutations of the Rubik’s Cube." N.d.
Microsoft PowerPoint file.
Wolfram Math World. Wolfram Research, n.d. Web. 30 Aug. 2014.
<http://mathworld.wolfram.com/GroupTheory.html>.
16
