How to Find the Total Permutations of a Rubiks Cube

7/24/2019 How to Find the Total Permutations of a Rubiks Cube

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Abstract

In my mathematics extended essay, I will attempt to solve the general case of a Rubiks

Cube through the analysis of mathematical concepts and their constraints. The research

uestion at hand is, !how many different orientations does a Rubiks cube have"# In

investigating this uestion, 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. &' x &% and finally forming an euation 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 Rubiks cube

through suggested proper methods. The investigation included the usage of mathematical

concepts such as factorials $'%, combinations $nCr% and permutations $n(r% to formulate a

working euation along with the use of diagrams provided by the Rubiks branding company

which details proper movements, directional rotations and a solvers guide. The conclusion

reached by this evaluation is that given a corner has ) orientations at * locations and an edge

piece has + orientations at nlocations, as higher order cubes hold more edges, the total !unfair#

orientations of a Rubiks cube euals8! x 38x n! x 2n. Conversely, the total !fair# permutations

are expressed as 8! x 37x (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 Rubiks cubes ranging from a +x+ to

x dimensional cube and the permutation function on a TI-*& (lus 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


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Headings Page

Introduction )

/echanism &

(ermutations 0

Initial 1pproach *

2roup Theory 3

Corners

4dges +

Total (ermutations )

5urther Cases )

6ibliography

+


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Introduction

The history of the Rubiks cube came about by a young (rofessor of architecture in

6udapest. 6orn on 7uly ), 3&&, 4rno Rubik originally created the Rubiks Cube, which at the

time was called the /agic Cube. The first working prototype of the /agic Cube was made in

30& and distributed within 8ungary by a toy making company named (olitechnika. 9ue to

8ungary:s communist environment, 4rno needed a method in introducing his invention to the

world. 6y garnering the attention of mathematicians and 8ungarian entrepreneurs, 4rnos

invention had reached various international conventions including the ;uremberg Toy 5air in

303 where Tom


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to solve a Rubiks Cube with current forms of algorithms, many people forget how difficult it is to

be able to solve the pu??le from any given orientation. This being said, I find this topic en@oyable

due to my appreciation for simple and complex pu??les.

5inding a solution to the Rubiks 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 Rubiks Company diagrams, I solved both standards of

permutations to enable calculations for total possible permutations and total fair permutations.

>ith these considerations, I will delve into permutations of each individual part of the Rubiks

Cube. This analysis will begin with corner and edge permutations. >ithin this essay, Corners

are defined as cubies)with three visible faces with differing colors which affects the outcome of

these Cubes due to having certain orientations reuiring similar yet different algorithms to solve

while edges are cubies with two visible faces with differing colors. 5rom understanding these

concepts of edges and corners introduces a new notion branded as layers&which encompasses

both edges and corners. 5rom the analysis of edges, corners and layers, I formulated an

euation to solve for total permutations of a Rubiks Cube.

(lease note that the beginning of this essay $and its solutions% will analy?e the )x)x)

dimensional Rubiks Cube - however, I will explore further applications and solutions of

permutations in higher order cubes.

Mechanism

)1 cubical shape within the Rubiks Cube that can be reoriented to form permutations

&Consisting of & corners, & edges, and center

&


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The difficulty that arises within this investigation comes from inside the Rubiks Cube. The cube

itself has 3 lateral faces for each of the A larger faces.This makes a total of A x 3 B & lateral

faces composed of these cubies. 4ach cubie is part of ) different layers and each of these

layers consists of 3 cubies this is including the centers of each layers. The mechanism inside of

the cube allows each of the 3 layers to be turned freely in its plane and around its one-colored

face. This makes it relatively easy to recogni?e the three different types of cubiesD * corner

cubies with three colored tiles, + edge cubies with two colored tiles, and A face cubies with one

colored tile. 1long 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 Rubiks Cubes mechanism is by asking how it is

possible to create such an ob@ect. 5or instance, Christoph 6andelow, author of Inside Rubiks

Cube and Beyond, gave an example, !>hy dont 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 pu??le to maintain its location without falling out of place. Closer examination of the inner

Rectangles within a right prism


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workings of the cube can be viewed by turning a single layer &E and prying an edge cubie

outwards.

The final consideration is the movement of each cubie within its given layers. ;oted by

the Rubiks company diagrams, notations for movement are given by layers $which are also the

A faces of the cube%. 4ach layer is annotated as either Feft, Right, Gp, 9own, 5ront, or 6ack

depending upon how the cube is handled $/iddle layer movements can also be noted but are

not as readily used%. The abbreviation for these annotations are F, R, G, 9, 5, or 6 respectively

for which each of these turns are also stated to be 3=Edue to its clocklike rotation.

4ach 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 F, R, and G while clockwise

rotations are not noted with an apostrophe. Consecutive movements in the same layer are

A


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interpreted with a + in front of the annotation. Hnly a + is used for further movements due to the

clockwise and counterclockwise rotations of each layer. It makes sense that if rotation eual

3=E while + rotations eual *=E, it is unnecessary to move any further because ) rotations

being +0=E is the same as reversed rotation. The reason behind the use of the apostrophe is

to enable a solver to orient each cubie within the pu??le to specific locations more efficiently

while reducing the number of movements reuired.

Permutations

>ithout 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

parityA. In mathematics, when is a finite set of at least two elements, the permutations of fall

into two classes of eual si?eD the even permutations and the odd permutations. 4very and all

permutations will be either even or odd thus the same should be true for a Rubiks Cube. To

determine whether a permutation is even or odd, we can count the number pairs out of their

natural order.

5or example0, take two sets of numbers with one set being in its natural order.

123456!

213!564

AThe fact of being even or odd

0Jaughen, Kcott. LCounting the (ermutations of the Rubiks Cube.L ;.d. Micoso" #o$e#oin"file.

0


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5irst 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.

"umber "umbers out o# natura$ order

2is followed by Reversal

1is followed by only larger numbers = Reversals

3is followed by only larger numbers = Reversals

!is followed by , A, 0, and & & Reversals

5is followed by & Reversal

6is followed by & Reversal

is followed by & Reversal

4is the last number in the seuence = Reversals

The total reversals*of the second row is * therefore an even permutation. 2iven 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.

Initia$ A%%roach

In order to fully grasp how many permutations are within a Rubiks Cube, we must first consider

each piece as an individual entity. 6y 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. 5or example, an edge cubie has + locations where they can be relocated and

+ orientations for each location. Therefore the total number of permutations for corners must be

*1 number which follows another that is not in the normal counting seuence

*


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++because each location has + orientations for each cubie multiplied by itself + times to

account for cubies shifting to different locations. Corner cubies should therefore be ) *for having

) orientations for * locations. The next step is to multiply both uantities 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, ++x )*B +A,*0),*A should be the total number of permutations within a Rubiks Cube.

1lthough 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 Rubiks 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 analy?ing location as a

property of a cubie without considering orientation $these properties then should be a

multiplicative factor for that type of cubie%. 2iven 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 euation should be the cubies properties

multiplied by each other, then total cubie permutations multiplied by each other in order to get a

total for a Rubiks Cubes permutations Mi.e. $corner properties% x $edge properties% - or -

$orientation x location% x $orientation x location% B total permutationsN.

&rou% 'heor(

In mathematics and abstract algebra, group theory studies the algebraic structures

known as groups. The concept of a group is central to abstract algebraD other well-known

algebraic structures, such as rings, fields, and vector spaces can all be seen as groups

3


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endowed with additional operations and axioms3. In this study, group theory will be used to

evaluate the amount of evenness or oddness is within each permutation of a Rubiks Cube.

8annah (roven?a, Gniversity of Chicago graduate, explains how there are eual amounts of

even and odd permutations. 8er first theorem in explaining this was that,

%&'e se" o oea"ions on "'e cube "'a" can be eac'ed o "'e so*+ed s"a"e by akin, "'e

o+es *is"ed abo+e in +aious cobina"ions is a ,ou unde "'e oea"ion o conca"ena"ion.

e $i** "'en ee "o "'is ,ou as "'e cube ,ou.

This theorem was proven by first explaining that the group has associativity, or that

$O%P B $OP%. These variables represent the axes of which the cube remains within a plane.

$O%P means to execute move O, then move P, while $OP% means to execute move , then

move O P. (roven?a states that both of these are euivalent to making move , followed by O,

followed by P. The next issue to state is the cubies that do not move such as the centers.

%&'e iden"i"y e*een" I is akin, no o+e a" a**. 0ince "'is *ea+es "'e cube unc'an,ed

$e ob+ious*y 'a+e I I. &'e in+ese -1 o any sin,*e ace "un is 3 $'ic' is

eui+a*en" "o "'e ace "un eoed coun"ec*ock$ise ins"ead o c*ock$ise. &'e in+ese o a

seuence o o+es 4 is 4-1-1. In"ui"i+e*y "'is is c*ea5 "o undo any seuence o o+es is "o

undo eac' o+e in "un6 s"a"in, $i"' "'e is" o+e and $okin, back "o "'e is". &'is can be

easi*y o+en by u*"i*yin, "'e "$o "o,e"'e5 (4)(4-1-1) (44-1)-1 (I)-1 -1 I.


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that each corner can be in & locations and the same with edges. >ith this, subtract one from the

locations to get possible permutations for one cubie on a face. The result is A meaning that

there is an even number of permutations for each fair permutation. 2iven 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.


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Corners cubies are what you would expectD cubies placed at each of the * corners on

the cube. The coordinate representation helps in explaining the corners location property by

assuming that each corner represents a point on a ) dimensional plane. Thus, to find

permutation, you must count the total number of combinations at which a corner can be located

from * locations $given that there are * corners in a cube%. 1 Rubik:s Cube has corners that can

be located at * possible locations for each piece. It is then inaccurate to assume that each

corner has * 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 0 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 pu??le. Kince 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 * x 0 x A x x & x ) x + x B &=,)+= or *'.

These two principles of orientation and location should then be multiplied together to

achieve the total number of permutations for corners. *' x )*B +A&,)3,+= 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 symboli?es the total permutations possible

through disassembly of the cube. >hile it may be true that there can be ) *total orientations for

+


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corners, the fair method of solving a rubiks cube only allows )0ways to orient corners. This is

because 0 of these corners can be oriented independently while the orientation of the eighth

depends on the preceding seven, giving a total of )0$+,*0% possible orientations for corners.

The new euation for total corner permutations without disassembly is therefore 8! x 37

881789permutations.

)dges

4dge cubies within the Rubiks 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 & and assume that each edge cubie

is a point on a ) dimensional plane. Kince 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.

5or a Rubiks cube, there are + locations per edge cubie. Thus, following the same method as

corner cubies, the total permutation for location should be 12 x 11 x 19 x x 8 x 7 x : x ; x x 3

x 2 x 1 7991:99or 12!.

>ith 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 + orientations for each of the + edge cubies.

1nd similarly to corner cubies, the Rubiks cube can have 212 permutations without considering

the fair methods of solving the cube meaning that the power should be n- 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.

)


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'ota$ Permutations

Through this method of evaluation to find permutations, the solution for the Rubiks

Cubes total unfair permutations is therefore 8! x 38x 12! x 212 ;.1 x 1929 permutations while 8!

x 37x 12! x 211 8.:; x 191 euals the total fair permutations. The solution for total fair

permutations is, however, not complete since the euation does consider the parity error which

is Q since there is an eual 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. Kince there is an eual number of even and odd permutations,

the division factor must be +. The true euation for total fair permutations of a Rubiks Cube is

therefore ( 8! x 37x 12! x 211) / 2 .32 x 191.

*urther Case A%%$ications

The Rubiks brand does not only account for a single type of dimensional cube. There

are many variations the original ) by ) Rubiks cube that range in a multitude of dimensions

which in turn have varying amounts of permutations. Hne example of varying permutations

would be a +x+ Rubiks Cube. 8aving only * cubies meaning * corner cubies only, the total

number of permutations is the same as a )x) Rubiks Cube if you are only considering its

corners as the only form of permutations. This being said, the total number of permutations for a

+x+ is therefore for 8! x 38 unfair movements and 8! x 37 for fair movements. Farger case

scenarios such as a x Rubiks Cube need additional considerations when solving for total

permutations.

&


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4verything that was true for a )x) Rubiks Cube is true for a x as well. The new

properties that must be accounted for are the addition of +& edge cubies and &* center cubies.

These cubies are added due to the increased amount of space needed to be filled while still

allowing layers of movement. The mechanism as well makes it so that the +& 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 @oint to the center edge cubie. 1ny permutation of the outer edges is possible, including

odd permutations, giving +&' permutations for these additional edge cubies.

>ith the &* new center cubies, +& are corners of the entire center and +& are the edges

of the entire center. The corner and edge centers cannot be flipped with each other, similarly to

the +& edges of the cube, due to the internal mechanism. This means that there are +&' +

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 x Rubiks Cube. 5or each of these

additional centers, they can have &' orientations per face for these edges and centers. Kince

there are A faces, this factor must be raised to the Ath meaning that there are &' A permutations

that are not possible for edges or corner centers. Kince each edge and corner center cubies

have 2! / !:permutations, the total permutations for these two combined is 2!2/ !12.

The solution for total permutations of a x Rubiks Cube, while still considering fair

movements, is therefore ( 8! x 37x 12! x 211x 2!3) / ( 2 x !12) 2.83 x 197permutations.

+ib$iogra%h(

6andelow, Christoph. Inside Rubikoac' "o Ideas and Me"'ods. 3&. +nd ed. ;ew OorkD Hxford

G(, 33A. (rint.

(roven?a, 8annah. @ou &'eoy and "'e Rubik


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(rint.

Ktewart, Ian. 0ye"y5 > Aey 0'o" In"oduc"ion. ;.p.D Hxford G(, +=). (rint.

Jaughen, Kcott. LCounting the (ermutations of the Rubiks Cube.L ;.d.

Micoso" #o$e#oin"file.

o*a Ma"' o*d. >olfram Research, n.d. >eb. )= 1ug. +=&.

ShttpDmathworld.wolfram.com2roupTheory.htmlU.

A

How to Find the Total Permutations of a Rubiks Cube

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