Solving Rubik s Cube with Non-Standard Moves L AURENT VAN E ESBEECK

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1 Solving Rubik s Cube with Non-Standard Moves L AURENT VAN E ESBEECK Master of Science Thesis Stockholm, Sweden 2014

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3 Solving Rubik s Cube with Non-Standard Moves L AURENT VAN E ESBEECK Master s Thesis in Mathematics (30 ECTS credits) Master Programme in Mathematics (120 credits) Royal Institute of Technology year 2014 Supervisors at Université Catholique de Louvain, Belgien, was Olivier Pereira Supervisor at KTH was Tilman Bauer Examiner was Tilman Bauer TRITA-MAT-E 2014:55 ISRN-KTH/MAT/E--14/55--SE Royal Institute of Technology School of Engineering Sciences KTH SCI SE Stockholm, Sweden URL:

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5 Université catholique de Louvain École polytechnique de Louvain Louvain-la-Neuve Royal Institute of Technology Department of Mathematics Stockholm Thesis for the degree of Master in Mathematical Engineering (UCL) Master in Mathematics (KTH) Solving Rubik s Cube with Non-Standard Moves by Laurent Van Eesbeeck Supervisors Olivier Pereira (UCL) Tilman Bauer (KTH) Readers Christophe Petit (UCL) Jean-Pierre Tignol (UCL) Louvain-la-Neuve, August 2014

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7 Abstract Still another thesis on Rubik s cube? Is there still something new to write on that puzzle? In this document, we approach the cube with a rather unusual question: how would you solve the cube if, instead of using the 6 classical rotations, you were restricted to a set of arbitrary moves? To answer that question, we will dive into group theory. Inspired by some previous work on the factorization of the symmetric group, we have developed an algorithm that answers our initial question. However, being able to solve the cube with any set of moves has a trade-off: while some algorithms solve the cube in 20 moves, ours requires several thousands. One could go further than this thesis by: improving our algorithm, providing rigourous bounds on its complexity, or generalizing the algorithm to the n n n cube.

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9 Acknowlegdments I would like to thank my three supervisors: Olivier Pereira for accepting to be my official UCL supervisor; Tilman Bauer for his support and his questions; Christophe petit for his weekly support, for recentering me to the main topic and for having endured the ups and downs of my productivity. And of course, I would like to thank my family for their unconditional support.

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11 Contents Contents 7 Introduction 9 1 Required Notions in Group Theory A Very Short Introduction to Group Theory Quotient Groups and Semi-Direct Products Permutation Groups Conjugated Permutations and Commutators Notations and Conventions Used in this Document Introducing Rubik s Cube Permutations, Positionings and Orientations The Semi-Direct Product Behind G The Structure of G Factorizing the Symmetric Group Finding a Permutation of Degree Less than n/ Finding a 3-Cycle Factorizing any Target from a 3-Cycle Solving Rubik s Cube with Non-Standard Moves Factorizing Small Groups Factorizing A 8 A Solving Rubik s Cube with Non-Standard Moves Conclusion and Further Work 55 Bibliography 57 A Code Snippets 59 A.1 Rubik s Cube A.2 Babai et al s factorization of S n A.3 Small Groups A.4 solving Rubik s Cube with Non-Standard Moves

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13 Introduction Still another thesis on Rubik s cube? Didn t people already do this? Instead of presenting one other effective way to solve the cube or finding an alternative proof to God s number (the minimal number of moves with which one can solve the cube from any configuration), through this thesis we are interested on the following question: how would one solve the cube if, instead of the 6 classical face rotations, he was restricted to use an arbitrary set of moves? If odd, that question perfectly illustrates a far-from-trivial and deeply studied problem in group theory, the factorization problem: given a set S = {s 1,..., s m } generating a group G and an element g in that group G, how does one express g as a product of elements in S? (This problem can be seen as a generalized version of the discrete logarithm problem and has potential applications in cryptography ([PQ11]). For some groups and some generating sets, the problem has well-known solutions. But in general, there is no obvious solution. Solving Rubik s cube is merely the tip of the iceberg of a deeper problem. This thesis is divided into four chapters. Chapter 1 is a warm-up. It introduces the notions in group theory that will be used later on, from the basic definitions to semi-direct products and permutation groups. The main notions are recaped in Section 1.5. Chapter 2 introduces the group structure behind Rubik s cube (it is a semi-direct product) and presents a systematic strategy to solve it: first position the cubits right, then orient them right. As we will see, that solution requires about 1500 moves, far away from the world s record of 20 moves. Chapter 3 dives into the litterature and presents a solution to the factorization problem for the symmetric group. As we will see, the studied algorithm does not work for small symmetric groups such as those encountered in Rubik s cube. Chapter 4 is the central point of this thesis. Building upon the material presented in the Chapters 1 to 3, it presents a solution to Rubik s cube that works for any given set of moves. It also provides a variant of the algorithm of Chapter 3 that works for small symmetric groups. Tests suggest that our algorithm requires about 6000 moves for the standard moves, and about moves for any given set of moves. 9

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15 Chapter 1 Required Notions in Group Theory This chapter provides a quick reminder of the group theory notions needed to understand the next chapters. Section 1.1 defines the very basic notions used in group theory: a group, a group action, a Cayley graph and a Schreier graph, a homomorphism and a subgroup. Section 1.2 presents cosets, Schreier coset graphs, normal subgroups, quotient groups and semi-direct products; the two latter being special groups that we will encounter in Chapter 2 and 4 when studying the group structure of Rubik s cube. Section 1.3 gives a wide overview of permutation groups: definition, cycle representation of a permutation and its properties, parity of a permutation and decomposition of a permutation as a product of 2- and 3-cycles. Permutation groups will be used widely through this document and more specifically, the results of this section will be used in Chapters 3 and 4. Section 1.4 presents some nice properties of conjugated permutations and the commutator of two permutations. These properties will be used mainly in Chapter 3. Section 1.5 summarizes this chapter by listing the notations and conventions used through this document. Readers familiar with all these notions may directly skip to Section 1.5; references to the main results of this chapter will be done when necessary. 1.1 A Very Short Introduction to Group Theory Abstract algebra is a branch of mathematics where one defines mathematical structures by generalizing structures we commonly use and then studying the properties of these general objects. Groups are fundamental objects in abstract algebra, as they are used to define more complex objects such as rings, fields, modules, and so on. Definition 1.1. A group (G, ) is a set G supplied with an operator : G G G satisfying the following axioms: closedness g 1, g 2 G, g 1 g 2 G; 11

16 12 CHAPTER 1. REQUIRED NOTIONS IN GROUP THEORY associativity g 1, g 2, g 3 G, g 1 (g 2 g 3 ) = (g 1 g 2 ) g 3 ; existence of a neutral element G has an identity element Id such that g G, g Id = Id g = g; existence of inverses g G, g 1 G such that gg 1 = g 1 g = Id. If furthermore satisfies commutativity g 1, g 2 G, g 1 g 2 = g 2 g 1 then (G, ) is called a commutative group. Remark. When the group operation is obvious from the context, (G, ) is usually denoted by G and the symbol either is a + or is ommitted: one writes g 1 + g 2 (for so-called additive groups) or g 1 g 2 (for so-called multiplicative groups) instead of g 1 g 2. One also writes g + g + + g }{{} n times = ng and gg g }{{} n times = g n. We are already familiar with different examples of groups: (R 0, ) is a commutative group with identity 1 and as inverse r 1 = 1/r; (Z, +) is a commutative group with identity 0 and as inverse z 1 = z. (Z n, +) is the cyclic group modulo n: Z n = {0,..., n 1} with operation a b = (a + b) mod n. 1 As an example of non-commutative group, consider GL(2, R), the multiplicative group of 2 2 matrices with non-zero determinant. From the axioms of a group follow a couple of elementary properties. The reader interested in a proof of these properties may consult [DF04], 1.1. Proposition 1.2. Let G be a group. Then the following holds: the identity of G is unique; every element g G has a unique inverse g 1 and (g 1 ) 1 = g; the inverse of g 1 g 2 is (g 1 g 2 ) 1 = g 1 2 g 1 1 ; the equations ax = b and xa = b have a unique solution x G for every a, b G; for every g, x, y G, if gx = gy then x = y and if xg = yg then x = y. As we will see when we study permutation groups, some group elements have nice properties. The purpose for the moment is merely to mention their existence. Definition 1.3. Two group elements g 1 and g 2 are said conjugate if there exists an h such that g 1 h = hg 2. The conjugate of g by h, noted g h, is the element h 1 gh. Note that (g h ) 1 = (h 1 gh) 1 = h 1 g 1 h = (g 1 ) h g h 1h 2 = (h 1 h 2 ) 1 gh 1 h 2 = h 1 2 (h 1 1 gh 1)h 2 = (g h 1 ) h 2 (g 1 g 2 ) h = h 1 (g 1 g 2 )h = (h 1 g 1 h)(h 1 g 2 h) = g h 1 g h 2. 1 Observe the difference between the modulo binary operator and the modulo equivalence relation: compare 8 mod 3 = 2 (operator) and 8 11 (mod 3) (equivalence relation). For convenience, through this document the modulo operator has the same precedence as in Magma: (a + b) mod c a + b mod c = a + (b mod c) and a b mod c = (a b) mod c.

17 1.1. A VERY SHORT INTRODUCTION TO GROUP THEORY 13 Definition 1.4. The commutator of two elements g and h, noted [g, h], is defined as [g, h] = g 1 h 1 gh. When g and h commute, [g, h] = Id, so the commutator can be seen as a measure of how much two elements commute. Indeed, gh = hg[g, h]. One also notes that [g, h] 1 = (g 1 h 1 gh) 1 = h 1 g 1 hg = [h, g]. Groups can interact with other structures through a mechanism called group actions. Definition 1.5. A left group action of a group G on a set S is an operation : G S S satisfying the two following actions: for all s S, Id s = s; for all g, h G, s S, (gh) s = g (h s). Similarly, a right group action is an operation : S G S satisfying for all s S, s Id = s; for all g, h G, s S, s (gh) = (s g) h. As an example, the group operation is a (left and right) action of a group G on itself. Another example is the previously defined conjugacy operation: the operation g h = g h = h 1 gh is a right group action of G on itself. We ll see two more examples in the context of permutation groups. Remark. Some authors define the conjugate of g by h as h g = hgh 1 and the commutator of g and h as [g, h] = ghg 1 h 1. With that definition, the operation h g = h g is a left group action. Our choice of using the other convention is made to avoid confusions with Magma (Magma defines g h = h 1 gh and (g, h) = g 1 h 1 gh) and because it is consistent with the multiplication of permutations from left to right. (More about this in Section 1.3.) More generally, to be consistent with this convention, function composition must be done from left to right: (fg)(x) = g(f(x)). That s why through this document, the image of x by a function f, f(x), will be noted x f and (fg)(x) = x fg = (x f ) g. The group actions we will encounter will also, by convention, be right group actions. Definition 1.6. The order of a group G, G, is the number of elements in G. The order of an element g G, g, is the smallest positive integer n such that g n = Id. If such an integer does not exist, then by convention g =. The following result gives some information on the order of an element. Again, the proof can be found in [DF04], 3.2. Proposition 1.7. If G is a finite group, then g divides G. One way to describe a group is to use a generating set S = {s 1, s 2,..., s n }. One then writes G = S to describe the group defined by the set of all finite products of elements of S with respect to the implicit group operation: G = { s i1 s i2 s ik i j {1,..., n} j {1,..., k} }. With a particular generating set S in mind, one can define the Cayley graph of a group. Schreier graphs are similar but are constructed from a group action.

18 14 CHAPTER 1. REQUIRED NOTIONS IN GROUP THEORY Figure 1.1: The Cayley graph of Z 6 generated by {1, 2}, Γ(Z 6, {1, 2}). Definition 1.8. The (colored) Cayley graph of G with respect to a generating set S = {s 1,..., s n }, Γ(G, S), is the oriented graph with vertex set G and edge set { (g, gs i ) g G, i {1,..., n} }. In a colored Cayley graph, the vertex (g, gs i ) is colored in the ith color. Definition 1.9. Given a group G = S that acts on a set X with a left (right) group action : G X X, the left Schreier graph Γ(G, S, X, ) is the oriented graph with vertex set X and edge set {(x, s x) x X, s S}. The right Schreier graph has the same vertex set and {(x, x s) x X, s S} as edge set. Since the group multiplication is a group action, Schreier graphs generalize Cayley graphs. These graphs are regular (i.e., every vertex has the same number of neighbors), so they usually lead to very symmetric and structured drawings (Figure 1.1). A group generated by a set S G can also form a smaller group than G called a subgroup. E.G., 2 = {0, 2, 4} is a subgroup of Z 6. Definition A subgroup H of G is a subset of G which forms a group under the group operation in G. If H is a subgroup of G, one writes H G. Since the associativity of the operation is ensured for every element in G (and hence also in H G), one must only verify the closedness, existence of a neutral element and existence of inverses axioms in H to check that H is a subgroup of G. Note that the identity in H is also the identity in G. An equivalent way to show that H is a subgroup of G is the subgroup criterion. For a proof, see [DF04], 2.1. Proposition 1.11 (The Subgroup Criterion). A subset H of a group G is a subgroup if and only if 1. H 2. x, y H, xy 1 H. Definition For G and H two groups, a map φ : G H is called a homomorphism if, g 1, g 2 G, (g 1 g 2 ) φ = g φ 1 gφ 2. A homomorphism from G to G is called an automorphism. If a homomorphism is bijective, it is called an isomorphism and G and H are said isomorphic, which is noted by G = H. One also has the following properties for homomorphisms (see [DF04], 3.1). Proposition For G and H two groups and for φ : G H,

19 1.2. QUOTIENT GROUPS AND SEMI-DIRECT PRODUCTS 15 Id G φ = Id H ; for every g G, (g 1 ) φ = g (φ 1) ; ker φ = {g G g φ = Id} is a subgroup in G; Im φ = G φ = {g φ g G} is a subgroup in H. Remark. Two isomorphic groups G and H can be seen as the same group up to the difference that its elements have different names from one group to another. The isomorphism linking these groups has the effect of translating an element of G to its equivalent in H. 1.2 Quotient Groups and Semi-Direct Products Let us first review some results related to subgroups. Closely related to a subgroup is the notion of a coset. Definition For H G a subgroup of G and g G, gh = {gh h H} is called a left coset of H and Hg = {hg h H} is a right coset. The elements of a coset are called its representatives. Remark. Needless to say, what we say in this section is written with left cosets but can just as well be written in terms of right cosets. Two cosets of a same subgroup either are disjoint or are identical, and the set of cosets of a particular subgroup H G form a partition of G. Note also for all g gh, gh = g H (see [DF04], 3.1). This implies that all the cosets have the same number of elements. One notes [G : H], the index of H in G, as the number of distinct cosets H has in G. Hence, when G is finite, [G : H] = G / H. Cosets have an equivalent links with graph theory as Cayley graphs and Schreier graphs: Schreier coset graphs. Definition Given a group G = S and a subgroup H G, the Schreier left coset graph Γ(G, S, H) is the oriented graph with vertex set {gh g G} and edge set {(gh, sgh) g G, s S}. The Schreier right coset graph is the graph with vertex set {Hg g G} and edget set {(Hg, Hgs) g G, s S}. For this section we need some particular subgroups called normal subgroups. Definition A subgroup N G is said normal if for all g G, n N, If N G is normal, one notes N G. g 1 ng N. Normal subgroups and kernels of homomorphisms are related. From [DF04], 3.1, Proposition A subgroup N G is normal if and only if it is the kernel of some homomorphism. One main result in group theory is that the set of distinct cosets of a normal subgroup, {gn g G}, forms a group with the group operation (g 1 N) (g 2 N) = g 1 g 2 N.

20 16 CHAPTER 1. REQUIRED NOTIONS IN GROUP THEORY In other words: to multiply two cosets, take any element n 1 g 1 N and any element n 2 g 2 N; the coset (g 1 N)(g 2 N) is the coset to which belongs n 1 n 2. We won t show here that this operation is well-defined, i.e., that the result of this operation is the same independently of the choice of representatives. (The interested reader can consult [DF04].) An equivalent approach of this operation is the following. Suppose that there is a homomorphism φ : G H such that φ is a one-to-one relation between the cosets gn and the elements of H: for every g, for every n gn, n φ = h and h is uniquely and only determined by g. (This implies that N φ = Id, i.e., that N is the kernel of φ.) Then the product of two cosets works as follows: for g 1 N and g 2 N, take the corresponding h 1 and h 2 in H, and compute h 1 h 2. The coset (g 1 N)(g 2 N) is the one corresponding to h 1 h 2. The group of cosets of N G supplied with the previously defined operation is called the quotient group G/N. Its identity is the coset Id N = N and the inverse of gn is (gn) 1 = g 1 N. To clarify this with an example, consider the normal subgroup of Z, 3Z = {..., 3, 0, 3, 6,... }. It has three distinct cosets: 0+3Z, 1+3Z = {..., 2, 1, 4,... }, 2+3Z = {..., 1, 2, 5,... }. 3Z is also the kernel of the homomorphism φ : Z Z 3, x x mod 3. One can then add 1 + 3Z and 2 + 3Z in two different ways: one can choose Z and Z. Their sum, = 12 lies in 0 + 3Z, hence (1 + 3Z) + (2 + 3Z) = 0 + 3Z. φ maps 1 + 3Z to 1 and 2 + 3Z to 2. In the group Z 3, = 0 and 0 is associated to the coset 0 + 3Z. Hence, (1 + 3Z) + (2 + 3Z) = 0 + Z. These two ways to multiply cosets in a quotient group are summarized in what is known as the first isomorphism theorem (see [DF04], 3.3). Theorem If φ : G H is a group homomorphism, then ker φ is a normal subgroup and G/ ker φ = Im φ. We can now move to the last topic: direct and semi-direct products. Definition The direct product between two groups (G 1, ) and (G 2, ), noted G 1 G 2, is the set {(g 1, g 2 ) g 1 G 1, g 2 G 2 } supplied with the operation (g 1, g 2 ) (g 1, g 2) = (g 1 g 1, g 2 g 2). It is not difficult to check that (G 1 G 2, ) is a group. Closedness and associativity are inherited properties from G 1 and G 2, it has (Id G1, Id G2 ) as identity and (g 1, g 2 ) 1 = (g1 1, g 1 2 ). The notion of semi-direct product is a generalization of the direct product. To introduce it, consider a group G with a normal subgroup N G. If g 1 and g 2 G can be written as h 1 n 1 and h 2 n 2 for some n 1, n 2 N, then g 1 g 2 = (h 1 n 1 )(h 2 n 2 ) = h 1 h 2 (h 1 2 n 1h 2 )n 2 But since N is normal, h 1 2 n 1h 2 = n 1 N. Hence, (h 1 n 1 )(h 2 n 2 ) = (h 1 h 2 )(n 1n 2 ) The following definition of the semi-direct product contains a homomorphism φ : H Aut(N). Here, Aut(N) is the group of all the homomorphisms from N to N with as group operation the composition of functions: for f, g Aut(N), f g is the function

21 1.2. QUOTIENT GROUPS AND SEMI-DIRECT PRODUCTS 17 x x fg = (x f ) g. (Observe that with quotient groups we introduced groups of sets, here we introduce a group of functions.) The homomorphism φ : H Aut(N) hence means that for every h H, φ(h) is a homomorphism from N to N: (n 1 n 2 ) φ(h) = n 1 φ(h) n 2 φ(h) ; for h 1, h 2 H, φ(h 1 h 2 ) is the homomorphism n n φ(h 1h 2 ) = (n φ(h 1) ) φ(h 2) ; φ(id) is the identical homomorphism: n φ(id) = n. With this example in mind, we can now state the definition of a semi-direct product. Definition Given two groups H and N and a homomorphism φ : H Aut(N), the semi-direct product H φ N is the set {(h, n) h H, n N} supplied with the operation (h 1, n 1 ) (h 2, n 2 ) = (h 1 h 2, n 1 φ(h 2 ) n 2 ). In the introductary example, n φ(h) = h 1 nh = n h. For a direct product, φ(h) is the identity automorphism: n φ(h) = n. The semi-direct product is a group: Closedness is obvious. Associativity: [ (h1, n 1 )(h 2, n 2 ) ] (h 3, n 3 ) = ( h 1 h 2, n 1 φ(h 2 ) n 2 ) (h3, n 3 ) = ( h 1 h 2 h 3, ( n 1 φ(h 2 ) n 2 ) φ(h3 ) n3 ) = (h 1 h 2 h 3, n 1 φ(h 2 h 3 ) n 2 φ(h 3 ) n 3 ) (h 1, n 1 ) [ (h 2, n 2 )(h 3, n 3 ) ] = (h 1, n 1 )(h 2 h 3, n 2 φ(h 3 ) n 3 ) = (h 1 h 2 h 3, n 1 φ(h 2 h 3 ) n 2 φ(h 3 ) n 3 ). Existence of a neutral element: (Id, Id) is the neutral element of the group. Indeed, (h, n)(id, Id) = (h Id, n φ(id) Id) = (h, n) (Id, Id)(h, n) = (Id h, Id φ(h) n) = (n, h). Existence of inverses: (h, n) 1 = ( h 1, (n 1 ) φ(h 1 ) ). Indeed, (h, n) ( h 1, (n 1 ) φ(h 1 ) ) = ( hh 1, n φ(h 1) (n 1 ) φ(h 1 ) ) = (Id, (nn 1 ) φ(h 1) ) = (Id, Id) ( h 1, (n 1 ) φ(h 1 ) ) (h, n) = (h 1 h, ( (n 1 ) φ(h 1 ) ) φ(h) n) = (Id, (n 1 ) φ(h 1 h) n) = (Id, Id). The discussion behind the introductory example is summarized in the following theorem. Theorem 1.21 ([DF04], 5.5). Suppose G is a group with N, H two subgroups of G such that every g G can be written as a product nh for some n N, h H. If N G and N H = {Id}, then G = H φ N where φ : H Aut(N) maps n n φ(h) = h 1 nh = n h.

22 18 CHAPTER 1. REQUIRED NOTIONS IN GROUP THEORY Remark. When one follows the convention that n h = hnh 1, the reasoning in the introductory example becomes (n 1 h 1 )(n 2 h 2 ) = n 1 (h 1 n 2 h 1 1 )h 1h 2 = (n 1 n 2 h 1 )(h 1 h 2 ) and the definition of the semi-direct product naturally becomes N φ H = { (n, h) N H (n 1, h 1 ) (n 2, h 2 ) = (n 1 n 2 φ(h 1 ), h 1 h 2 ) }. To make the difference between these definitions, one uses either the sign or. 1.3 Permutation Groups Permutation groups are among the most common finite groups and are closely linked to the notion of symmetry. Definition A permutation on a set A is a bijection from A to A. The set of all permutations on A, supplied with the function composition, is a group called S A. If A = { 1,..., n }, S A is called S n. Proposition S A is a group. Proof. One needs to check the axioms of a group: 1. f, g S A, fg S A : true since the composition of two bijections is still a bijection; 2. f, g, h S A, (fg)h = f(gh): true since the function composition is associative; 3. S A has an identity element: the identity function Id : x x is a bijection and f S A, f Id = Id f = f; 4. every f S A has an inverse: true since f is a bijection. Note, however, that S A is not commutative. Through this document, we will only consider permutation groups over the finite set { 1,..., n }. For a given permutation f S n, one notes x f the image of x {1,..., n} by f. This induces the natural right group action : {1,..., n} S n {1,..., n}, x f = x f. It is a group action: x Id = x Id = x, x (fg) = x fg = (x f ) g = (x f) g. Permutations also naturally act on vectors by the right group action : R n S n R n, (x 1,..., x n ) f = (x (1 f ),..., x (n f )). One then naturally notes (x 1 f,..., x n f ) = (x 1,..., x n ) f. The previous action is particular because it also defines a homomorphism φ : S n Aut(R n ) which maps x x φ(f) = x f. (Note that every homomorphism φ : G Aut(H) induces a group action of G on H but that the converse is not true in general.) It is a homomorphism because (x + y) f = (x 1 + y 1,..., x n + y n ) f = (x 1 f + y 1 f,..., x n f + y n f ) = x f + y f. Hence, from the previous section, the group is a semi-direct product. {(x, f) R n S n (x 1, f 1 ) (x 2, f 2 ) = (x 1 + x 2 f 1, f 1 f 2 )}

23 1.3. PERMUTATION GROUPS 19 Definition A permutation group G S n is said k-transitive if for every pair of k-tuples (t 1,..., t k ) and (t 1,..., t k ) with t i t j and t i t j for every i j, there exists a permutation f G such that f(t 1 ) = t 1,..., f(t k) = t k. A 1-transitive permutation group is said transitive. Definition The support of a permutation f, supp(f), is the set of elements in { 1,..., n } that are moved by f. The degree of f, deg f, is the number of elements that it moves. Note that supp(f) = supp(f 1 ) and that deg(f) = deg(f 1 ). Also, i f = i if and only if i supp(f). The most straightforward way to represent a permutation is the line representation: put the elements a A on a first line and put their corresponding values f(a) on a second line. For example, in S 4, the function f mapping 1 to 3, 2 to 4, 3 to 2 and 4 to 1 is represented as ( ) f = Multiplying two functions with this representation is straightforward too. For example, if g maps 1 to 4, 2 to 3, 3 to 2 and 4 to 1, ( ) ( ) fg = ( ) = However, the line representation is quite heavy. A more popular one is the cycle representation: in the previous example, f maps 1 to 3, 3 to 2, 2 to 4 and 4 to 1, so its cycle representation is f = (1, 3, 2, 4). As a second example, g maps 1 to 4 and 4 to 1; 2 to 3 and 3 to 2, which makes 2 distinct cycles: g = (1, 4)(2, 3). As a third example, the product fg is fg = (1, 2)(3)(4); however, since the cycles (3) and (4) do not carry any relevent information, they are omitted from the representation: fg = (1, 2). The cycle representation of a permutation is unique up to a reordering of its cycles, g = (1, 4)(2, 3) = (2, 3)(1, 4), and up to starting each cycle with another element of the cycle: f = (1, 3, 2, 4) = (3, 2, 4, 1) = (2, 4, 1, 3) = (4, 1, 3, 2). We can then count the number of permutations of a particular cycle structure.

24 20 CHAPTER 1. REQUIRED NOTIONS IN GROUP THEORY Proposition Write n = 1n 1 + 2n kn k. There are n! ki=1 i n i ni! different permutations that have n 1 1-cycles, n 2 2-cycles,..., n k k-cycles. Proof. We can represent every permutation as a string containing all the integers 1,..., n exactly once and split these strings according to the fixed cycle structure. (This representation explicitly includes the 1-cycles.) There are n! such strings. But amongst these strings, some are equivalent representations of the same permutations: every k-cycle can be written in k equivalent different ways, which makes a total factor i in i ; in this string representation, we can swap cycles of same length among them: e.g., for n = 9, n 1 = n 2 = 1 and n 3 = 2, the strings and represent the same permutation. This makes the factor i n i!. Multiplying two permutations from their cycle representation is a bit less obvious and is given by the following algorithm. 1. Start a new cycle with the smallest element x in supp(f) supp(g) that hasn t been used yet. 2. Do this for every cycle in the product, starting from the cycle on the left and moving to the right: if x belongs to the cycle, replace x by the element next to it in the cycle. 3. If the new x is equal to the first element of the cycle under construction, close that cycle and go to step 4. Else, add x to the current cycle, add it to the list of elements already used and go back to step If the list of elements that haven t been used yet is empty, remove the 1-cycles; the algorithm is done. Else, go back to step 1. As an example, consider the product f 2 g = ffg with f and g defined as previous: f 2 g = (1, 3, 2, 4)(1, 3, 2, 4)(1, 4)(2, 3). The multiplication algorithm goes as follows. First, start a new cycle with 1. In the first cycle, 1 goes to 3. In the second cycle, 3 goes to 2. 2 does not appear in the third cycle, so it remains 2. In the fourth cycle, 2 goes to 3. Hence, f 2 g = (1, 3... In the first cycle, 3 goes to 2. In the second, 2 goes to 4. In the third, 4 goes to 1. 1 does not appear in the fourth cycle, closing the first cycle of the product: f 2 g = (1, 3)...

25 1.3. PERMUTATION GROUPS 21 We then start a new product with an element that hasn t been used yet: 2. In the first cycle, 2 goes to 4. In the second, 4 goes to 1. In the third, 1 goes to 4. 4 does not appear in the last cycle. Then, f 2 g = (1, 3)(2,... One easily checks that 2 goes to 4, closing so the second cycle: f 2 g = (1, 3)(2, 4). The cycle representation is convenient to extract some characteristics of a permutation. Indeed, the support of a permutation is the set of elements in its cycle representation and its degree is the number of elements in it. It also makes the following result trivial. Proposition Every permutation of S n can be written as a product of disjoint cycles. From the algorithm of multiplication between permutations, we also have that Proposition Disjoint cycles commute. In addition of providing a compact representation of permutations, the cycle representation of a permutation gives some information about its order. Proposition The order of a k-cycle is k. Proof. Observe that (a 0,..., a k 1 ) n is a permutation that sends a i to a (i+n) mod k. Hence n = k is the smallest positive power such that (a 0,..., a k 1 ) n sends a i to a i. Proposition When a permutation is written as a product of disjoint cycles, its order is the LCM of the length of its cycles. Proof. Write f = C 1 C 2 C k where C 1,..., C k are disjoint cycles and write l 1..., l k their lengths. Since they are disjoint, f n = C n 1 Cn k. By the previous result, f n = Id only if n is a multiple of both l 1,..., l k. Hence the order of f is the LCM of l 1,..., l k. But there is still more to say about permutations and cycles. Proposition Every permutation f S n can be expressed as a product of at most n 1 2-cycles. Proof. A k-cycle can be expressed as a product of (k 1) 2-cycles in the following way: (a 1, a 2, a 3,..., a k ) = (a 1, a 2 )(a 1, a 3 ) (a 1, a k ). Applying this formula to every cycle of the cycle representation of f proves the result. A permutation of j distinct cycles of lengths l 1,..., l j can be written as a product of (l 1 1) + + (l j 1) n j 2-cycless. The worst case is then the case of one single n-cycle, which is then expressed as a product of (n 1) 2-cycles. Definition A permutation f S n is said even if it can be expressed as an even product of 2-cycles. Else it is said odd. The signature of a permutation f, sgn(f), is defined as { +1 if f is even sgn(f) = 1 if f is odd The set of even permutations in S n is called A n.

26 22 CHAPTER 1. REQUIRED NOTIONS IN GROUP THEORY Proposition The signature function is a homomorphism. Proposition Even cycles are odd, odd cycles are even. Proof. Direct from the previous result. Proposition The signature function, sgn, is a homomorphism. Proposition Every even permutation f can be expressed as a product of at most n 1 3-cycles. Proof. Write first f = c 1 c 2 c 2k as a product 2k n 1 of 2-cycles and then apply the following relation pairwise to the product: (a, b)(a, b) = Id (a, b)(a, c) = (a, b, c) (a, b)(c, d) = (a, b, c)(a, d, c) Proposition A n is a group. Proof. From Proposition 1.11, since A n S n, one must only check the following: 1. Id A n : trivial since (a, b)(a, b) = Id; 2. f A n, f 1 A n : if f is an even product of 2-cycles, f = c 1 c 2 c 2k, since c i = c 1 i then f 1 = c 2k c 2k 1 c 2 c 1 is also even; 3. f, g A n, fg A n : if f = c 1 c 2 c 2k and g = c 1 c 2 c 2k, then obviously fg = c 1 c 2 c 2k c 1 c 2 c 2k is even. 1.4 Conjugated Permutations and Commutators As announced in Section 1.1, conjugated permutations and commutators have some nice properties. Proposition The conjugate of a k-cycle c = (c 1,..., c k ) is a k-cycle. In particular, c g = g 1 cg = (c 1 g,..., c k g ). Proof. g 1 cg acts on c i g as follows: (c i g ) g 1 cg = c i cg = { ci+1 g if i { 1,..., k 1 } c 1 g if i = k Hence, g 1 cg contains the k-cycle (c 1 g,..., c k g ). To show that g 1 cg is that k-cycle, notice that if i c = c i = j, then (i g ) g 1 cg = (i c ) g = j g. Hence, if i supp(c), then i c = i and (i g ) g 1 cg = i g and i g supp(g 1 cg). Hence, deg(g 1 cg) deg(c) = k. But since g 1 cg contains a k-cycle, deg(g 1 cg) = k and gcg 1 is the k-cycle (c 1 g,..., c k g ).

27 1.4. CONJUGATED PERMUTATIONS AND COMMUTATORS 23 Corollary For any permutation g, f and f g have the same cycle structure. particular, if f = (a 1,..., a k1 )(b 1,..., b k2 ) then f g = (a 1 g,..., a k1 g )(b 1 g,..., b k2 g ) i.e., the cycle decomposition of f g is obtained by replacing each i by i g in the cycle decomposition of f. Proof. Writing f into its disjoint cycle representation, f = C 1 C k, one has f g = g 1 (C 1 C k )g = (g 1 C 1 g)(g 1 C 2 g) (g 1 C k g). Applying the previous proposition to each g 1 C i g proves the result. Corollary For any permutations f and g, supp(f g ) = (supp(f)) g and deg(f) = deg(f g ). The two previous corollaries justify why the notation f g is used for f conjugated by g: conjugating f by g is the same as applying g elementwise to the cycle structure of f. They also justify why the convention f g = g 1 fg is used instead of the convention f g = gfg 1 : if the latter was used, the permutation f g would be such that every element i in the cycle representation of f is replaced i (g 1), which would be confusing to remember. Corollary Two permutations f and g are conjugate in S n if and only if they have the same cycle structure. Proof. We already saw that if f and g are conjugate, then they have the same cycle structure. Suppose that f and g have the same cycle structure when written as a product of disjoint cycles: f = C 1 C 2... C k g = C 1 C 2... C k, C i = (c i,i1,..., c i,im ), C i = (c i,i 1,..., c i,i m ). Since C 1,..., C k have disjoint supports and since C 1,..., C k also have disjoint supports, then there is (at least) one permutation h which sends c i,ij to c i,i j. By the previous corollary, f h = g. This corollary does not apply for permutations in A n : (1, 2, 3) and (1, 3, 2) are not conjugate in A 4. However, one has the following result. Corollary All 3-cycles are conjugate in A n, n 5. Proof. Take f = (f 1, f 2, f 3 ) and g = (g 1, g 2, g 3 ) in A n. Write h = (f 1, g 1 )(f 2, g 2 )(f 3, g 3 ). By Proposition 1.38, f h = g. If h is even, we are done. Suppose h is odd. Since n 5, there exist f 4, f 5 {1,..., n} \ {f 1, f 2, f 3 } such that f 4 f 5. Write h = (f 4, f 5 )h. By construction, f and (f 4, f 5 ) commute, hence f h = h 1 (f 4, f 5 ) 1 (f 1, f 2, f 3 )(f 4, f 5 )h = h 1 (f 4, f 5 ) 1 (f 4, f 5 )(f 1, f 2, f 3 )h = h 1 (f 1, f 2, f 3 )h = f h = g. Since h is even, f and g are conjugate in A n. As the two following propositions show, looking at how the support of two permutations overlap tells us some information on their commutator. 2 Proposition If f, g S n are such that supp(f) supp(g) = { a }, then the commutator [f, g] is the 3-cycle (a, a g, a f ). 2 Special thanks to Ewan Delanoy for his help for the proof of Proposition In

28 24 CHAPTER 1. REQUIRED NOTIONS IN GROUP THEORY Proof. First, observe the action of [f, g] on a. Since a f ±1 supp(g) and a g±1 supp(f), Hence, (a f ±1 ) g±1 = a f ±1 and (a g±1 ) f ±1 = a g±1. a f 1 g 1 fg = ((a f 1 ) g 1 ) fg = (a f 1 ) fg = a g ; (a g ) f 1 g 1 fg = (a g ) g 1 fg = a fg = a f ; (a f ) f 1 g 1 fg = a g 1 fg = a g 1g = a. Hence, [f, g] includes the 3-cycle (a, a g, a f ). Second, observe the action on [f, g] on k {a, f(a), g(a)}. If k supp(f) supp(g), then obviously k f 1 g 1 fg = k. If k supp(g) \ {a, g(a)}, then k g 1 supp(g) \ {a} and k, k g 1 supp(f). Hence, k f 1 = k and (k g 1 ) f = k g 1 and k f 1 g 1 fg = ((k f 1 ) g 1 ) fg = (k g 1 ) fg = k g 1g = k. If k supp(f) \ {a, f(a)}, then k f 1 supp(f) \ {a} and k, k f 1 supp(g). Hence, as in the previous case, (k f 1 ) g 1 = k f 1 and k g = k k f 1 g 1 fg = ((k f 1 ) g 1 ) fg = (k f 1 ) fg = k g = k Hence, supp([f, g]) = {a, a g, a f } and [f, g] = (a, a g, a f ). Proposition If f, g S n such that supp(f) supp(g) = k, then deg([f, g]) 3k. Proof. Write A = supp(f) supp(g). The proof consists of showing that for all x supp([f, g]), x A A f A g and thus, supp([f, g]) 3 A = 3k. For the proof, let us write F = supp(f) and G = supp(g). (Then, A = F G.) Let us also note the following statements: x f 1 = x x F (1.1) (x g 1 ) f = x g 1 x g 1 F (1.2) (x f 1 ) g 1 = x f 1 x f 1 G (1.3) x g = x x G (1.4) It is not hard to see that either when (1.1) and (1.2) or (1.3) or (1.4) are satisfied, x [f,g] = x. Logically written, ( (1.1) and (1.2) ) or ( (1.3) and (1.4) ) x [f,g] = x i.e., x supp([f, g]) or equivalently, by contraposition, x supp([f, g]) ( (1.1) or (1.2) ) and ( (1.3) or (1.4) ) Hence, if x supp([f, g]), there are 4 different cases: 1. (1.1) and (1.3): x F and x f 1 G. Since x F, x f 1 F and hence x f 1 A, hence x A f.

29 1.4. CONJUGATED PERMUTATIONS AND COMMUTATORS (1.1) and (1.4): x F and x G. Then, obviously, x A. 3. (1.2) and (1.3): x g 1 F and x f 1 G. We can t say anything from here, but looking at the 4 subcases we see that x A A f A g : a) (1.1) and (1.2) and (1.3) and (1.4): this subcase never occurs because (1.1) and (1.4) imply that x f = x g 1 = x, hence (x g 1 ) f = x f = x = x g 1. Yet, this contradicts (1.2) which states that (x g 1 ) f x g 1. b) (1.1) and (1.2) and (1.3) and (1.4): this subcase is dealt with in the case 1: x A f. c) (1.1) and (1.2) and (1.3) and (1.4): this subcase is dealt with in case 4: x A g. d) (1.1) and (1.2) and (1.3) and (1.4): this case is dealt with in cases 1, 2 and 4: x A A f A g. 4. (1.2) and (1.4): (x g 1 F and x G. Since x G, x g 1 G and hence x g 1 A, hence x A g. Remark. Proposition 1.44 does not exactly generalize Proposition While the latter gives an equality on the degree of the commutator, the former only provides an upper bound, which: could not be better: indeed, by Proposition 1.43, we know that the bound is reached when supp(f) supp(g) = 1; may be far too pessimistic: if f = g, then obviously supp(f) supp(g) = deg(f) but deg([f, g]) = 0 3 deg(f). Note also that [g, h] is an even permutation. Hence, it can not be a 2-cycle.

30 26 CHAPTER 1. REQUIRED NOTIONS IN GROUP THEORY 1.5 Notations and Conventions Used in this Document Following Magma s convention, permutations are multiplied from left to right. That is, fg is the permutation that sends i to g(f(i)) (also noted i fg = (i f ) g ). Group elements are multiplied from left to right and group actions in this document are all right group actions. x f (also f(x)). The image of x by a function f (a homomorphism, a permutation,... ). If x is a set, x f = {y f y x}. If x is a vector of n elements and f a permutation in S n, x f = (x (1 f ),..., x (n f )). x i. The ith element of a vector x. Graphs. Given a group G generated by a set S, a left (or a right) group action : G X X and a subgroup H G, 1. the Cayley graph Γ(G, S) is the oriented graph with vertex set G and edge set {(g, gs) g G, s S}; 2. the left Schreier graph Γ(G, S, X, ) is the oriented graph with vertex set X and edge set {(x, s x) x X, s S}; the right Schreier graph has the same vertex set and {(x, x s) x X, s S} as edge set; 3. the Schreier left coset graph is the oriented graph with vertex set {gh g G} and edge set {(gh, sgh) g G, s S}; the Schreier right coset graph Γ(G, S, H) has {Hg g G} as vertex set and {(Hg, Hgs) g G, s S} as edge set. Order of an element/a group. The smallest positive integer n such that x n = Id/the number of elements of a group. Parity/Signature. If a permutation g can be written as an even product of 2-cycles, its parity is even/has signature +1. Else, it is odd/has signature 1. Permutation groups. S n is the group of all permutations over { 1,..., n }, A n is the group of all even permutations over { 1,..., n }. Support. The set of elements that are moved by a permutation π (noted supp(π)) Degree. The number of elements that are moved by a permutation π (noted deg(π)) Conjugate. The conjugate of a permutation f by a permutation g is f g = g 1 fg. Commutator. The commutator between two permutations f and g is [f, g] = f 1 g 1 fg. Semi-direct product. Given N, H two groups and φ : H Aut(N) a homomorphism, N φ H = { (n, h) N H (n 1, h 1 ) (n 2, h 2 ) = (n 1 n 2 φ(h 1 ), h 1 h 2 ) } H φ N = { (h, n) H N (h 1, n 1 ) (h 2, n 2 ) = (h 1 h 2, n 1 φ(h 2 ) n 2 ) }

31 Chapter 2 Introducing Rubik s Cube Rubik s cube is a famous 3-D puzzle consisting of 26 colored small cubes forming together one bigger cube, Rubik s cube. Through a ingenious mechanism, each face of the cube can rotate by 90, 180 or 270 degrees (Figure 2.1). The aim of the puzzle is to rotate the faces cleverly in order to put back each small cube at its initial position. In this chapter, we present the mathematical structure behind the cube. As we will see, two different groups are linked to the cube: a group G consisting of all the possible arrangements of the smaller cubes, and a group G of the arrangements that are reacheable by rotating the faces of the cube. As we will see, these two groups are not equal: G is a subgroup of G. This chapter is divided in three sections: Section 2.1 introduces the groups G and G; Section 2.2 shows that G is a semi-direct product; Section 2.3 presents the structure of G by showing which elements in G are in G. In this section we will see that G is a subgroup of order 12 of G and we will also present one way to solve the cube. The content of this chapter is inspired of [Isa12], [Che78], [BHH + 10] and [Ban82]. [Isa12] presents Rubik s cube from an abstract point of view; [Che78] are the notes of a summer camp and include a very accessible introduction to group theory; [BHH + 10] s approach is not so far from ours; and [Ban82] has a fair list of useful moves to solve the cube. Figure 2.1: Through a ingenious mechanism, each face of the cube can independently rotate by 90, 180 or 270 degrees. 27

32 28 CHAPTER 2. INTRODUCING RUBIK S CUBE left down back front up right Figure 2.2: When one fixes the centers of the cube in space, the faces are called up, down, left, right, front and back Figure 2.3: Numbering the facets from 1 to 48 enables us to describe any state of Rubik s cube by a permutation in S Permutations, Positionings and Orientations For clarity, we will call cubits the 26 smaller cubes composing Rubik s cube and cubicles the placeholders where the cubits lie. Without loss of generality, we may fix the center cubits in space and call its faces up, down, left, right, frond and back (Figure 2.2). There are then 20 moveable cubits consisting of 48 colored facets. A given configuration of the cube can hence be described by a permutation in S 48. To ease things for later, we chose to number the corner facets from 1 to 24 and the side facets from 25 to 48. The corner facets are numbered (3k 2), (3k 1), 3k clockwisely and the side facets are numbered (24 + 2k 1) and (24 + 2k) (see Figure 2.3). Observe that there are two numberings in place: one for the cubit facets and one for the cubicle facets. The ith cubicle facet is a mental location in space, while the ith cubit facet is the colored tile of the cube which has the number i written on it. In the solved configuration, the two numberings match each other. Given that context, we represent naturally a configuration c of the cube as the permutation in S 48 which sends i to c i, where c i is the number written on the cubit facet lying in the ith cubicle facet. The clockwise rotations of the 6 faces, U, D, L, R, F and B, correspond to the following permutations (the unmoved elements are removed from the

33 2.1. PERMUTATIONS, POSITIONINGS AND ORIENTATIONS Figure 2.4: The numbering of the cubicles is coherent with the numbering of the facets: the facets 3i 2, 3i 1, 3i lay on the ith corner cubicle and the facets i 1, i lay on the ith side cubicle. line representation): ( ) U = = (1, 10, 7, 4)(2, 11, 8, 5)(3, 12, 9, 6)(25, 31, 29, 27)(26, 32, 30, 28) ( ) D = = (13, 16, 19, 22)(14, 17, 20, 23)(15, 18, 21, 24)(41, 43, 45, 47)(42, 44, 46, 48) ( ) L = = (4, 9, 19, 18)(5, 7, 20, 16)(6, 8, 21, 17)(27, 38, 43, 36)(28, 37, 44, 35) ( ) R = = (1, 14, 22, 11)(2, 15, 23, 12)(3, 13, 24, 10)(31, 34, 47, 40)(32, 33, 48, 39) ( ) F = = (1, 6, 16, 15)(2, 4, 17, 13)(3, 5, 18, 14)(25, 35, 41, 33)(26, 36, 42, 44) ( ) B = = (7, 12, 22, 21)(8, 10, 23, 19)(9, 11, 24, 20)(29, 39, 45, 37)(30, 40, 46, 38) Rotating a face corresponds to apply its permutation to the current configuration of the cube. One defines G as the subgroup of S 48 generated by {U, D, L, R, F, B}, the group of all configurations that can be reached using valid rotations of the cube. Alternatively to the permutation representation, one may describe a configuration of the cube by giving the positioning and the orientation of the cubits in the cubicles. The positioning of the cubits describes which cubit stays in which cubicle independently of its orientation. Since the facets are numbered 3k 2, 3k 1 and 3k for the corners, we define the ith corner cubit and cubicle as the cubit and cubicle including the number 3i. Samewise, we define the ith side cubit and cubicle as the cubit and cubicle with the number i (see Figure 2.4). The corner and side positionings are then permutations in S 8 and S 12 which send i to the cubit lying in the ith cubicle. From the way we chose our facet numbering, the positionings of the corners and sides, π c and π s, are easily derived from the permutation representation of a configuration of the

34 30 CHAPTER 2. INTRODUCING RUBIK S CUBE Figure 2.5: The visual representation of the orientation of a cubit is the following. Choose one reference tile per corner (resp. side) cubicle. In the solved configuration, number 0, 1, 2 (resp. 0, 1) the facets of each corner cubit clockwise starting from the reference facet of the cubicle. The orientation of the cubit in the ith cubicle then corresponds to the number in the reference facet of that cubicle. cube: ( ) π c (c) = 3 c /3 6 c /3 24 c /3 ( ) π s (c) = (26 c 24)/2 (28 c 24)/2 (48 c 24)/2 (Following Section 1.5, i c corresponds to the image of i by the permutation c.) The orientation of a corner cubit is defined as the number of clockwise rotations (1, 2 or 3) it needs to go through to get back to its solved configuration. This definition only works when cubits lie in there initial cubicles. To generalize it, we mark one facet per corner cubit and cubicle as the reference facet of that cubit and cubicle. The definition is then extended as follows: the orientation of the cubit in a cubicle is the number of clockwise rotations it needs to go through to align its reference tile to the reference tile of the cubicle it lies in (see Figure 2.5). We chose these reference tiles as the tiles 3, 6,..., 24. Since the corner cubits all have been numbered clockwisely, their orientations are nicely derived from the permutation representation of a configuration: given a configuration c, the orientations of the cubits in the 8 corner cubicles are described by the vector ρ c (c) = (3 c mod 3, 6 c mod 3,..., 24 c mod 3). The orientation of a side cubit is defined in a similar way: given a reference facet per side cubit and cubicle, the orientation of the cubit lying in the ith cubit is the number of flips (0 or 1) that cubit must go through to align its reference facet with the cubicle s one. For the side cubits and cubicles, we chose the reference facets as the facets 26, 28,..., 48. As for the corners, the side orientations are nicely derived from the permutation representation: given a configuration c, the orientations of the cubits in the 12 side cubicles are described by the vector ρ s (c) = (26 c mod 2, 28 c mod 2,..., 48 c mod 2). The positioning permutations and orientation vectors associated to the 6 clockwise face rotations are as follows: x π c (x) π s (x) ρ c (x) ρ s (x) U (1, 4, 3, 2) (1, 4, 3, 2) (0, 0, 0, 0, 0, 0, 0, 0) (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0) D (5, 6, 7, 8) (9, 10, 11, 12) (0, 0, 0, 0, 0, 0, 0, 0) (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0) L (2, 3, 7, 6) (2, 7, 10, 6) (0, 2, 1, 0, 0, 1, 2, 0) (0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0) R (1, 5, 8, 4) (4, 5, 12, 8) (1, 0, 0, 2, 2, 0, 0, 1) (0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1) F (1, 2, 6, 5) (1, 6, 9, 5) (2, 1, 0, 0, 1, 2, 0, 0) (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0) B (3, 4, 8, 7) (3, 8, 11, 7) (0, 0, 2, 1, 0, 0, 1, 2) (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0) (2.1)