The Mathematics Behind the Rubik’s Cube, Math Problem Example

Rubik’s cube is a puzzle which has three planes which was invented by Ern? Rubik in 1974. Ern? Rubik was respected as a professor of architecture and a sculptor in Hungary. The three plane Rubik’s cube puzzle was patented by Rubik to be sold by Ideal Toy Corp. As of February 2009, over 350 million Rubik’s cubes had been purchased. This quality makes Rubik’s cube the world’s most popular puzzle game.  The point of view of the mathematical nature of the Rubik’s cube will be explored from the works of   Carter (2009), Gianni (1989), Harris (2008), Hernádvölgyi (2001),  Joshi (1997), Joyner (2008), Krebs ( 2011), Mayer (2003) Mayer (2003) MIT (2009) and  Rubik (2007).

In the three by three by three Rubik’s cubes, each of the sides are covered with nine labels. The labels represent each of the six colors, yellow, green, orange, blue, red and white). In the presently sold models, the red is opposite the orange, the green is opposite the blue and the yellow is opposite the white. The blue white and red colors are set in a clockwise order. An internal turning mechanism enables the Rubik’s cube to twist independently. As a result, the colors become disordered. In order to solve the three dimensional Rubik’s cube puzzle, the six faces of the cube must each demonstrate one color. There have been three plane puzzles of the same nature which have been produced since the invention of the Rubik’s cube in the mid-1970s, and not all of them are Rubik’s cubes. . The Rubik’s cube reached the height of popularity in the in the mid-1980s and the Rubik’s cubes continue to be popular.[1]

Permutations

The initial Rubik’s cube which had been developed possessed eight corners and twelve borders. There are 8! (40, 320) methods of composing the corner elements. There are seven elements which can be directed independently. The direction of the eighth cube is reliant on the seven which came before it, which gives 3⁷ potential combinations. There exist 12! / 2 or 2, 389, 500, 800 combinations for the borders of the Rubik’s cube. An even permutation of the borders of the Rubik’s cube implies an even permutation of the cornered elements. There is the potential of turning the edges in an independent manner which provides 2¹¹potenrtial events. The number of potential events which can be formulated by a Rubik’s cube can be demonstrated by the following mathematical formula:

43, 252,003,274,489,856,000 = 8! X 3⁷ X (12! / 2) X 2¹¹

This mathematical formula provides 43 quintillion possibilities for the resolution of the Rubik’s cube. [2] The three plane Rubik’s cube puzzle is produced as having billions of potential positions, as the larger numbers are not familiar to many of the users. In order to place this into the proper perspective, if an individual possessed as many three by three by three Rubik’s cubes as the number of permutations which are in existence, the Earth could be covered over two hundred and seventy five times in Rubik’s cubes. The number which is demonstrated above is limited to the number of permutations which can be achieved by turning the edges of the cube. In the consideration of the potential permutations which can be attained by means of the use of the Rubik’s cube, the number which is shown in the mathematical relationship in equation 1 becomes multiplied by a factor twelve. The mathematical relationship is shown as 8! X 3⁸ X 12! X 2¹² = 519, 024, 039, 293, 86778, 272, 000 which is equivalent to 519 quintillion potential configurations of the Rubik’s cube. However a solution can only be found for 8 1/4% of these estimates.[3] This is due to the nature that the ordering of motions which will switch a solitary pair of cubies or twist solitary corners of the cube edge. There exist twelve potential collections of possible positions which are designated as orbits or universes into which the cube may be placed by deconstructing and reconstructing it.[4]

Central Sides

The originally produced Rubik’s cube possessed none of the directions on the central sides. Consequently, solving the Rubik’s cube does not have the requirement of directing those sides of the central cubie of the white side and it does not have the requisite that a person directs the sides precisely. Markers could be drawn on the central cubies of the Rubik’s cube which is not in a changed and resolved state in a manner where the drawings match each of the central cubie. This nature would identify the cube in a manner which is called a super cube. The drawing on the super cubes raises the level of difficulty due to the nature of increasing the collection of different potential set ups. There are over 200 ways of directing the central parts of the cubes, due to the quality that a corner permutation has the meaning of an even number of rotations of the central sides.[5] The cube can be decoded aside from the orientations of the central faces. There will always be an even number of central squares which need a quarter of a twist. Consequently, the direction of the central sides of the cube increases the potential number of permutations of the three plane Rubik’s cube from 4.3 x 10 ¹⁹ (43,252,003,274,489,856,000) to 8.9 X 10²⁴ (88,580,102,706,155,225,088,000). In the twisting of a Rubik’s cube there is a change in the quantity of permutations which is due to the change in the quantity of central faces. There are basically 6! ways of configuring the central six sides of the cube. Twenty four of these permutations are possible without undoing the cube. In the consideration of the orientations of the centrals aspects, the complete number of potential cube permutations increases from 8.9 x 10²² (88,580,102,706,155,225,088,000 to 2.1 X 10²⁴

(2,125,922,464,947,725,402,112,000).[6]

Formulas

In the inventory which is possessed by the aficionados of the Rubik’s cube, a learned order of motions which has the effect of resolving the Rubik’s cube puzzle is considered to be an algorithm. This word is acquired from the arithmetic application of mathematics in the algorithm which infers a well labeled schedule of instructions which are directed at conducting an activity from the beginning condition by means of well labeled instruction in order to get a final desired condition. Each of the ways of resolving the Rubik’s cube  applies its own set of algorithms, together with the definitions of the effects that the algorithm have  and the order which can be applied in order to resolve the Rubik’s cube. [7]

There are a variety of algorithms which are made in order to change only a tiny section of the cube while avoiding the change of the neighboring sections. These local algorithms can be applied a number of times to the different parts of the cube until the entire puzzle has been completed. There are a variety of famous algorithms which involve twisting three of the corners in the without modifying the other parts of the Rubik’s cube. There are special algorithms which possess a specific effect on the condition of the Rubik’s cube. These algorithms may involve twisting two of the corners. There are specific algorithms which are more basic than others without demonstrating side effects. These algorithms are often used in the beginning parts of solving of the Rubik’s cube puzzle when the side effects are not important. The majority of these algorithms are hard to memorize. In solving the puzzle, the special algorithms are usually applied. [8]

The Implementation and Impotence of Mathematical Group Theories

The Rubik’s cube is made for the use of mathematical group theory. This has worked for the reasoning of specific algorithms. These algorithms have the structures of commutations which are shown by XY/ XY where there are special techniques and the inverses of these techniques. Conjugated formations can also be used which are XY/ X which is normally referenced by speed cubers as beginning moves. Furthermore, the characteristic of having well labeled sub groups in the Rubik’s three plane cube group make the puzzle easier to be mastered and learned by going through a number of difficult levels. One of the difficulty levels could include solving of cube which has been coded by the application of twists which are made of one hundred and eighty degrees.[9]

A significant quality is the scrambling of the Rubik’s cube. This scrambling means that the setting of the Rubik’s cube in a way which would require the maximum amount of twists in order to return the cube to its original condition. This has been shown to be God’s number and has been calculated to be maximum value of 22 in 2009. The lower limit on God’s number is defined. The first twists of the sides of the Rubik’s cube can take place in a dozen distinct ways, (there are six sides which can be twisted in two potential directions and the motion which can affect the twist can take place in an additional eleven set ups, the limits on the maximum number of moves which are different from the original condition can be shown by the mathematical relationship of:

12 * 11^n-1 ? 4.3252 * 10¹⁹. This equation is resolved by n ? 19. [10]

Algorithm Keys

z (rotate) –  the Rubik’s cube should be twisted on the F side.

y (rotate) – the Rubik’s cube should be twisted on the U side.

x (rotate) – the Rubik’s cube should be twisted on the R side.

l (left sided two layers) – the left side of the Rubik’s cube, which includes the middle slice on that side.

r (right two layers) – details the left side of the Rubik’s cube, which includes the middle slice on that side.

d (lower two layers) – details the lower layer of the bottom of the Rubik’s cube, which includes the middle slice on that side.

u (upper two layers) – details the top layer on the upper side of the Rubik’s cube, which includes the middle slice on that side.[11]

b (rear two layers) – details the upper layer on the rear side of the Rubik’s cube, which is opposite the user. The middle slices are included on that side.

f (forward two layers) – details the top layer on the forward side of the Rubik’s cube, which is facing the user. The middle layer of that side is also included.

R (right) – details the ride side of the front of the Rubik’s cube.

L (left) – details the left side of the Rubik’s cube.

D (down) – details the bottom side of the Rubik’s cube.

U (up) – details the top side of the Rubik’s cube.

B (rear) – details the rear side of the Rubik’s cube which is opposite the user.

F (front) – details the front side of the Rubik’s cube.[12]

Groups

A group G is made of a collection of objects and an operator which is binary. The objects must fulfill four conditions. The operation must have a closed nature in order that any of the elements detailed as h and g in group g could be defined by h * g. The operation must also have an associative nature in order to detail that any of the elements of f, g and h (f* g) h = f * (g* h).[13]

There must be an element of identity which means that e ? G in a way that g * e = e * g = g. each of the elements in the group g must possess an inverse 1/ g which is correlated to the operation in order to fulfill the requisite that g * 1/ g = 1/ g * G = e. The commutative characteristic is not a requisite.[14] The motions of the Rubik’s cube can be seen as being three plane group elements. The Rubik’s cube permutations could be represented as elements of a group. The symbol for the permutations of the group can be delegated R.[15]

Rubik’s Group Binary Operator

One of the turns of the sides of the Rubik’s cube will be represented as a binary operator. The symbol which is an asterisk is omitted. This is due to the operation having a closed characteristic. The twists are comprehended to be of an associative nature. There is no distinction in the way that they are grouped, the mathematical operations which are performed inside are considered. When there is no change in the condition of the cube, the identity element is applied.[16]

Inverse Relationships of Group Elements

A group component’s inverse is drawn as g^-1.It had been shown that if h and g are components of a group, they can be described as (hg)^-1 = h^–1 g^-1. The cubie which turns the front side clockwise can be detailed as F. Consequently f which is the inverse of F causes the front side of the cube to turn clockwise. It can be said that FR possesses an inverse of rf. In order to cause the operation to invert, they must be performed in a reverse order. It can be said that the inverse of a components causes its undoing. [17]

Rubik’s Cube Element Permutation

The distinct movement ordering of the cube’s components can be seen as being a rearrangement of permutation of the Rubik’s cube components. The orders which cause the return of the identical cube configurations are seen to be the identical element of the collection of permutations. Each of the movement of the components can be detailed as a permutation. The motion of FFRR is an identical permutation to (ULF URB DRF) (DBR UFR DFL) (BR FR FL) (DR UR) (DF UF).[18] It becomes easier to discuss the permutations with the application of numerical values. A model of a permutation is in drawn in canonical cycle notation which is detailed (1) (234). This means that 1 remains stationary and the components 2, 3, and 4 are placed into cycles. This means that 2 is communicated to 3, 3 is communicated into 4 and 4 is communicated into 2. The phases in describing the mixtures of permutations in the canonical cycle notations are the following:

• Locate the items with the smallest value on the schedule.
• Initiate with the item of the smallest value.
• In this model, one possesses the smallest value.[19]

Conclude the cycle by ordering the motions of the items by means of the permutation. This should be done until the cycle is concluded. In the permutation (124) (35) * (612) (34), the permeation is initiated from the value of one. One communicates to 2 and 2 communicates to 6. This is followed by a motion of 1 moving to 6. In this circumstance, 6 and 1 compose one 2 cycle. [20]

In the event that all of the numbers have been consumed, the job is not complete. There should be a regression to the initial step in order to initiate a cycle with the composed which has not been applied that possess the smallest value. The continuation in this manner provides (1 6) (2 3 5 4). In the event that p is composed of a multitude of cycles of diverse lengths, the order of the permutation in equivalent to the value of n, being that the application of P causes a regression to the initial condition. In the event that P is composed of multiple cycles of diverse length, the order becomes the least frequent multiple of the lengths of the cycles. This is attributed that the quantity of cycles of steps will cause a regression in the chains to their initial conditions.[21]

Parity  

Parity can be applied in order to detail some of the permutations. Any duration of the permutation cycle can be details as being the product of two cycles. This is expressed by the following examples:

(1 2 3 4 5)

(1 2 3 4) = (1 2) (1 3) (1 4)

This format is perpetuated for any duration of cycle. The characteristic of parity of a length which possesses n cycles is provided by the quantity of number 2 cycles w of which it is composed. In the case that n is an odd number; an even number of permutations are needed. In the event that an odd number of cycles are needed, n would be an even number. [22]

It can be detailed that the odd number of Rubik’s cube components transfer odd permutations. There is constantly an even parity in the Rubik’s cube or there are even numbers of components which are transferred from the initial position. As any of the permutations of the Rubik’s cube possess an even parity, there is no motion which will cause the exchange of a solitary pair of components. This means that when two of the Rubik’s cube components are transferred, there must be other components which are exchanged in the process.[23]

This challenge will be avoided in the application of 3 cycles which incorporate the cycling of three components. In the detailing of the cycle formation of the cube motions, the notation which applies ? will facilitate review: The ?_corner details the cycle formation of the corner components. The ?_edge details the cycle formation of the edge components. The characteristics of ?_center are not reviewed as the center of the Rubik’s cube remains stationary.[24]

Subgroups

Assuming a group R, in the circumstance of S C R is a random subset of the group, consequently the subgroup H which is generated by the subset S is the subgroup which possess the minimal value of R which is composed of all of the components of S. F causes the generation of a group which pertains to the subgroup of R. This subgroup of R possesses all of the distinct Rubik’s cube permutations which can be derived by turning the front façade {F, F², F³, F⁴}. The group which is generated by {f, B, u, L, R, D} composes the entire group R. Here are some models of the generators of which the subgroup R consists:

• The two movements {RF}.
• Any pair of side twists which are in opposition {LR}.
• Any solitary side twists {F}.[25]

The order of a component g is defined as the numerical value of m in a way that g^m = e, which is the identity element. The dimension of a subgroup is also the dimension of the order of the component which is created. The motion of order must be applied in order to detail the component motion orders in the contexts of the number of repetitions which are required for a movement prior to the regression of the identity elements. The movement of the F provides a generation which possesses a subgroup of an order of four. This is calculated by rotating the façade for four repetitions in order to return to the initial condition.[26]

The motion FF provides a generation which pertains to a subgroup of an order of two. This is calculated by the two repetitions which are needed in order to return to the initial condition. Any random sequence of motions creates a generator which pertains to a subgroup which possesses a specific finite order. As the cube has the capacity of attaining a finite quantity of arrangements, each of the motions disorders the cubies, it can be said that eventually there would be some arrangement at a minimum which would commence a repeating order. Consequently, it can be proven that if the cube is initiated in its resolved condition, the application of repeating the identical move would eventually return the Rubik’s cube to its initial state after to a specific quantity of twists. This is a theorem of the Rubik’s cube.[27]

Co- Sets

The definition and the characteristics of co- sets are that if J is a group and k is a subgroup which pertains to j then for any of the components of j which pertains to J:

• jK = {jk: k ? K} is a left side co- set of K in J.
• Jk = {jk: k ? K} is a right side co- set of K in J.

Lemma: In K being a finite subgroup of the group J and K possessing n components, consequently any of the right sided co sets of K possesses n components. [28]

Proof: considering any component of J which pertains to J, Jk = {jk? k ? K} provides a definition of the right side co- set. There exists one component in the co- set for each k in K. In consideration, the co-set possesses n components. [29]

Lemma: There are two right side co-sets of a subgroup K in a group J which are similar of disjointed.[30]

Proof. Let it be supposed that Kx and Ky possess a mutual component. It can be said that for a certain k₁ and k_2, k₁x = k₂y. Consequently in the circumstance that x= k^-1k₂y and particular k₃ = k₁^-1k₂ provides x = k₃y. Consequently, each component of Kx can be inscribed as a component of Ky:

Kx = kk₃y[31]

For each of the k in K, this infers that kx and ky possess any mutual components, each of the components of kx is encountered in ky and an identical argument demonstrates the converse aspect.[32] Therefore, if these groups possess any mutual components, they possess each mutual component and are the same. It can now be detailed that the right side co-sets of a group partition which pertain to the group can be separated into disjoint sets. Each of the partitions would possess an identical number of components.

Lagrange’s Theorem

Lagrange’s theorem means the application of the move FFRR on a cube which is in a resolved condition until the starting condition is achieved. The number of repetitions which would be required and the dimensions of the subgroup which is created by FFRR have to be a factor of (8! X 3⁸ X 12! X 2¹²) / 12.[33]

The dimension of any group K C J is required to be a divisor of the dimension of g. consequently, m?K? = ?G? for certain values of m ? 1 ? N ⁺.[34]

Proof: The right co –sets of K in g partition J. let it be supposed that there are m quantities of co –sets of K in J. Every co – set is the dimension of the quantity of components in K or ?K?. J is simply the aggregate of the total of the co-sets: J = k₁J + k₂J + …+ k₂J in order that the dimensions are equivalent to the aggregates of the dimensions of the total of the co-sets: Hence, the relationship can be demonstrated as ?J? = m?K?.[35]

The table demonstrates a schedule of certain group generators and their dimensions which include the factors which are the dimension of R:

Table 1: Group Generators

[36]

The consensus of the moves will generate subgroups which are relatively small. The subgroups which possess smaller dimensions can be applied and they can be observed by the manner which the Rubik’s cube experiences a regression to its initial condition.[37]

Cayley Graphs

An effective manner by which to understand into the formation of subgroups and groups is by means of the application of the Cayley graph. The characteristics which detail a Cayley graph in group J are the following:

• Every j ? J is a vertex.
• Every group generator s ? S is delegated a color which is c_s.[38]

In any of the j ? J, the components which correspond to j and js are connected by a focused edge that is colored c_s. It would be extremely difficult to show a Cayley graph which demonstrates the value of R. The Cayley graph for the values of R would be composed of over 43 thousand trillion vertices. Instead the Cayley graphs for the more minute subgroups that pertain to R will be explored. The Cayley graph which shows the subgroup that is generated by the components F would be demonstrated as I ? F? FF?FFF?I.[39]

Macros

Initially, the characteristic of the cube group components must be defined. The characteristics can be applied in order to produce macros or permutations of moves which would facilitate the particular arrangement which will enable the resolution of the cube.[40]

Commutating Components

The motion sequences on the Rubik’s cube are not overtly communicative. The twisting of the front side can be performed followed by a turning of the right side in order to conduct a move which is defined as FR. This is not identical to turning the right face sequenced by twisting the front side or RF. An effective implement is to detail the relative characteristic of communication of an order of functions is a commutating component. PM/P is detailed as [P, M] in which P and M is dual cube motions. In the circumstance of P and M having a commutative quality, the identity is the commutated. In this motion, the motions can be cancelled since P negates 1/ P. The identical condition is present with the movement of M.[41]

Allow the operator support to be all of the elements which are modified by the motion. Consequently, the dual operations are commutated if they are performed as the identical operation of if they are supportive. This would be detailed as ? = supp (P) ? supp (M). This infers that each of the motions influences a distinct collection of components.[42]

In the event that the identity is not commutated, the relative commutation can be gauged by the number of elements which are modified by the application of the commutated component(s). Reviewing the support interaction of the two functions provides addition understanding into this assessment. The effective motions possess only a minute number of elements which are mutually modified. The macros can be observed to include the commutated elements in this application.[43]

An effective theorem with regards to the commutated elements which will be applied is the following: In the event that supp (J) ? supp (K) is composed of a solitary element it could be expressed that [j, k] have the characteristics of a three cycle. Some of the effective building steps which can be applied for commutated components are:

1. FUDLLUUDDRU which revolves precisely one of the elements on the uppermost facade.
2. rDR which performs the cycling of the three elements on the corners
3. FF which exchanges two edges in a middle slice of the Rubik’s cube.[44]
4. rDRFDf gyrates one of the elements on a facade.

Conjugating

Allow M to be a certain macro which conducts a cube function, a three cycle motion of edge components. It could be perceived that a cube motion which is P, PM/P which is the conjugate of M with regards to P. The conjugation of a group component is a very effective implement which can help to detail and create effective macros. An equivalent relationship is any correlation between two components which have the transitive, symmetric and reflexive characteristic. The correlation of conjugation will be demonstrated. A certain j ? J, x~ y, hence jxj^-1 = x in the event that J= 1, as a result x~ x. If  x~ y, as a result jxj^-1 = y. Performing multiplication on both sides with j on the right side of the equation and j- 1 on the left side of the equation yields  x = j^-1yj. If   x~ y, y~ z, then x~ z.[45]

The equivalence category c (X), x ? J is the collection of all of the y ? J: y ~ x. A partition of J can be achieved into disjointed equivalence categories of conjugate categories. Two permutation components of R are considered to be conjugates in the event of possessing identical cycle formation.[46] In the resolution of the Rubik’s cube, one direct approach is to conduct the resolution layer by layer. As the third layer is achieved, certain edge components may be flipped the wrong way. It is desired to flip the pieces in the correct manner and allow the lower layers to remain intact. This can be achieved by conjugation. The motion which is composed of the commutated j = RUru should be considered. In the application of y to the Rubik’s cube, the effect would be that a cross is achieved on the upper layer.  This demonstrates that seven of the components are influenced which compose the R layer. This can be remedied by conducting a turn prior to applying the macro which will place the upper layer components in the positions which are influenced by the macro. This will allow the upper layer to remain reconfigured.[47]

Subsequently, the conjugate of j by F will be applied in order to derive the motion FRUruf.  In initially conducting F and causing a reversal of this motion with f subsequent to the macro being conducted, it can be assured that only the upper layer is influenced. The permutation notation can be applied in order to detail the function of the macro. Initially j can be perceived prior to its conjugation. The characteristics of ?_corner  possesses the aspect of (12) (34)_, as this function exchanges the rear corner components and the upper right component corners.  The qualities of ?_edge possesses the qualities of (1 2 3) as it performs a three cycle edge components FR, UR and UB.[48]

Singmaster Formula for Solving the Rubik’s cube

This research paper has reviewed the Rubik’s cube from a mathematical approach.  The Rubik’s cube was created in 1974 by Ern? Rubik. In the Rubik’s cube there are 43, 252,003,274,489,856,000 permutations of the manner by which the Rubik’s cube can be manipulated in order to find a solution. This infers that 43, 252,003,274,489,856,999 permutations are erroneous. The objective of this research has been to review the aspects of mathematical group theory and permutations on the resolution of Rubik’s cube. The calculations which were performed in finding the resolution of the cube enabled the discovery of the God’s number which is determined to have a lower limit of nineteen.

Bibliography

Carter, Nathan. Visual Group Theory. USA: Mathematical Association of America, 2009.

Gianni, Patrizia. Symbolic and Algebraic Computation: International Symposium ISSAC ‘ 88 Rome Italy, July 4- 8, 1988 Proceedings. Berlin Heidelberg: Springer 1989.

Harris, Dan. Speedsolving the Cube: Easy to Follow, Step by Step Instructions for Many Popular 3- D puzzles. New York: Sterling Publishing Co., Inc., 2008.

Hernádvölgyi, István T.  “Searching for Macro Operators with automatically generated heuristics” In Advances in Artificial Intelligence: 14^th Biennial Conference of the Canadian Society for Computational Studies of Intelligence, AI 2001, Ottawa, Canada, June 7- 9, 2001 Proceedings. Canada: Canadian Society for Computational Studies of Intelligence, 2001.

Joshi, A. W.  Elements of Group Theory for Physicists. New Delhi: New Age International Publishers, Ltd., 1997.

Joyner, David. Adventures in Group Theory: Rubik’s Cube, Merlin’s Machine and Other Mathematical Toys. Baltimore, MD: John Hopkins University Press, 2008.

Kreb, Mike & Anthony Shaheen. Expander Families and Cayley graphs: A Beginner’s Guide. New York: Oxford University Press, 2011.

Lim. Carter. “Rubik’s Cube Solutions” Rubik’s Cube’s Algorithms, 23 July 2013. Web. 11 May 2014. <http://rubikscubealgorithms.org/rubik-s-cube-solutions>

Mayer, Ulrich, Peter Sanders & Jop Sibelyn. Algorithms for Memory Hierarchies: Advanced Lectures. Berlin Heidelberg: Springer, 2003.

MIT. “The Mathematics of the Rubik’s Cube: Invitation to Group Theory and Permutation Puzzles” MIT, 2009. Web. 10 May 2014<http://web.mit.edu/sp.268/www./rubik.pdf>

Rubik’s.com, “Cube Facts” Rubik’s Brand Ltd., 2014. Web 10 May 2014http://rubiks.com/cubefacts

Rubik, Ern?. Rubik’s Cube Compendium. Oxford, UK: Oxford University Press, 2007.

Robert Snapp. “12. Rubik’s Magic Cube” Department of Computer Science, University of Vermont, 2014. Web. 11 May    2014.<http://www.cems.uvm.edu/~rsnapp/teaching/cs32/lectures/rubik.pdf>

[1] Dan Harris, Speedsolving the Cube: Easy to Follow , Step by Step Instructions for Many Popular 3- D puzzles(New York: Sterling Publishing Co., Inc., 2008), 2

[2]  David Joyner, Adventures in Group Theory: Rubik’s Cube, Merlin’s Machine and Other Mathematical Toys (Baltimore, MD: John Hopkins University Press, 2008), 195.

[3]  Rubik’s.com, “Cube Facts” Rubik’s Brand Ltd., 2014. Web 10 May 2014.<http://rubiks.com/cubefacts>

[4]  Ibid 1.

[5]  Joyner, Adventures in Group Theory: Rubik’s Cube, Merlin’s Machine and Other Mathematical Toys, 6.

[6]  Ibid., 7.

[7]  Ibid., 149.

[8]  Ibid., 150.

[9]   Ibid., 94.

[10]  Ibid., 149.

[11]  Carter Lim., Rubik’s Cube Solutions, Rubik’s Cube’s Algorithms, 23 July 2013. Web. 11 May 2014. <http://rubikscubealgorithms.org/rubik-s-cube-solutions>

[12]  Ibid., 1.

[13]  Nathan Carter, Visual Group Theory (USA: Mathematical Association of America, 2009), 4.

[14]  Ulrich Mayer, Peter Sanders & Jop Sibelyn, Algorithms for Memory Hierarchies:  Advanced Lectures (Berlin Heidelberg: Springer, 2003), 238.

[15] Ibid., 238.

[16] A. W. Joshi, Elements of Group Theory for Physicists (New Delhi: New Age International Publishers, Ltd., 1997), 4.

[17] Ibid., 5.

[18]  Patrizia Gianni, Symbolic and Algebraic Computation: International Symposium ISSAC ‘88 Rome Italy, July 4- 8, 1988 Proceedings (Berlin Heidelberg: Springer 1989), 384.

[19]  Ibid., 377.

[20]  Ibid., 418.

[21]  Ern? Rubik, Rubik’s Cube Compendium (Oxford, UK: Oxford University Press, 2007), 173.

[22]  Ibid., 193.

[23]  Ibid., 34.

[24]  Ibid., 99.

[25]  Carter, Visual Group Theory, 101.

[26]  Ibid., 146.

[27]  Rubik, Rubik’s Cube Compendium, 81.

[28]  Ibid., 102.

[29]  Ibid., 105.

[30]  Ibid., 140.

[31]  Ibid., 103.

[32]  Joshi, Elements of Group Theory for Physicists, 12.

[33]  Carter, Visual Group Theory, 107.

[34]  Carter, Visual Group Theory, 107.

[35]  Joshi, Elements of Group Theory for Physicists, 24.

[36]  Joyner, Adventures in Group Theory: Rubik’s Cube, Merlin’s Machine and Other Mathematical Toys, 206.

[37]  Ibid., 206.

[38]  Mike Kreb & Anthony Shaheen. Expander Families and Cayley graphs: A Beginner’s Guide (New York: Oxford University Press, 2011), 4.

[39]  Ibid., 96.

[40] István T. Hernádvölgyi, “Searching for Macro Operators with automatically generated heuristics” In Advances in  Artificial Intelligence : 14^th Biennial Conference of the Canadian Society for Computational Studies of Intelligence, AI  2001, Ottawa, Canada, June 7- 9, 2001 Proceedings (Canada: Canadian Society for Computational Studies of Intelligence, 2001), 194.

[41]  Ibid., 194.

[42]  Ibid., 194.

[43]  Ibid., 194.

[44]  Ibid., 194.

[45]  MIT, “The Mathematics of the Rubik’s cube: Invitation to Group Theory and Permutation Puzzles” MIT, 2009. Web. 10 May 2014<http://web.mit.edu/sp.268/www./rubik.pdf>

[46]  Ibid., 15.

[47]  Ibid., 15.

[48]  Ibid., 16.

[49]  Robert Snapp. “12.Rubik’s Magic Cube” Department of Computer Science, University of Vermont, 2014. Web. 11 May 2014.<http://www.cems.uvm.edu/~rsnapp/teaching/cs32/lectures/rubik.pdf>