God s Number and the Robotic Turn Metric

Transcription

1 Saint Peter s University Honors Thesis God s Number and the Robotic Turn Metric Author: Nykosi H. Hollingsworth Advisor: Dr. Brian Hopkins A thesis submitted in partial fulfilment of the requirements for a baccalaureate degree in Mathematics in Cursu Honorum Submitted to: The Honors Program, Saint Peter s University April 28, 2015

2 Abstract Since the invention of the Rubik s cube in the 1970s, mathematicians have been captivated with finding out the maximum number of moves needed to solve the Rubik s cube optimally from any of the cube s approximately 43 trillion possible positions. This value, known as God s number, depends on the metric, which is the set of allowed moves. This thesis proposes a new robotic turn metric based on allowing antipodal faces of the cube to be turned simultaneously. Although human solvers cannot easily accomplish such moves, they are used by various robotic cube solvers, thus the name. We explore lower and upper bounds for God s number in this metric and how it compares to God s number in the Face Turn Metric and Quarter Turn Metric. 1

3 Acknowledgements Let me take this opportunity to commend those who have had a crucial role in assisting me in completing this thesis. I would first like to thank my adviser, Dr. Hopkins, for his constructive criticism, lucid guidance and his unending patience. This thesis would not exist without his support, as well as the influence of Thomas Rokicki who proposed the idea for the Robotic-Turn Metric. My family, friends, and fellow students have also played an integral part by showing their support when things seemed to be at a dead end. I cannot begin listing the ways they have helped as it would be endless. I would also like to show my appreciation for Dr. Wifall, the honors program director, who showed passion and genuine interest in all the students who took up the challenge of completing a thesis. Whether you provided a minute or grand contribution, I appreciate your efforts. Thank you. 2

4 Contents 1 The Cube 4 2 Mathematical Concepts Recursively Defined Sequences Cycle Notation Cayley Graphs Permutations of the Cube Existing Metrics Face-Turn Metric Establishing a Lower Bound Establishing an Upper Bound Quarter-Turn Metric Establishing a Lower Bound Establishing an Upper Bound Robotic-Turn Metric Search for a Lower Bound Search for an Upper Bound Appendices: Mathematica Code Evaluation of Moves Reduction of 20-length FTM sequences

5 1 The Cube In the 1974, Ernő Rubik, an architectural professor of Budapest College of Applied Arts, invented a device in which the each face of a cube rotates around its axis [1]. Although it is widely reported that the cube was built as a teaching tool to help his students understand 3D objects, his actual purpose was solving the structural problem of moving the parts independently without the entire mechanism falling apart. He did not realize that he had created a puzzle until the first time he scrambled his new cube and then tried to restore it; which took him about a month. When its popularity exponentially grew in the 1980s due to its international launch, the cube garnered the attention of mathematicians, and those of similar minds, into finding out optimal move sequences that will solve the cube in the shortest amount of moves. The length of this optimal sequence is what we call God s number. God s number, however, changes depending on what moves are allowed, which determines a metric. For instance, consider the most popular metrics: the Face-Turn Metric (FTM) and Quarter-Turn Metric (QTM). FTM counts all face rotations of 90 and 180 as one move. However, only 90 turns in either direction is allowed as one move in QTM. Thus, in contrast to one 180 face rotation in the FTM, it would be considered be two moves in QTM. Instinctively, we would assume that God s number in FTM should be less compared to QTM and it is. As of today, God s number in the FTM and QTM are 20 and 26 respectively [10]. Regardless, each metric determines moves which are required to solve the Rubik s cube from any of its approximately 43 quintillion positions. The Rubik s cube s many positions are a result of each face of each cube being permuted. The cube consists of 27 cubies; however, only 20 are adjustable since the 6 center cubes are fixed and the unseen middle cube is also fixed. The cube s 8 corners that may be permuted in 8! = 40,320 ways, and 12 edges that can be permuted in 12! = 479,001,600 ways. Once seven corner rotations have been made, the orientation of the eighth corner is forced. This leads to 3 7 = 2187 possible orientations by forcing the orientation of the fixed corner. Similarly, once 11 edge orientations have been made, the last edge cubie is forced. This leads to a total number of edge orientations is 2 11 = 2,048 [9]. However, because the Rubik s cube only allows even permutations of its cubies, only half of these permutations are 4

6 allowed since there are equal numbers of even and odd permutations. From these numbers, we can tell that the number of permutations of the Rubik s cube is 8! 12! = 43, 252, 003, 274, 489, 856, Mathematical Concepts 2.1 Recursively Defined Sequences A recursive sequence is a sequence of numbers indexed by an integer n and generated by a recurrence equation. The recurrence equation is a discrete equation which related past indexes of n to future indexes. Recurrence equations are differential equations, and may not always be solvable. When a recurrence equation is solvable, the resulting equation is considered a closedform solution [5]. One of the most famous closed-form recursive sequences is the Fibonacci sequence. The Fibonacci sequence, F (n), is defined by as F (0) = 0, F (1) = 1, and F (n) = F (n 1) + F (n 2) for n 2. With this definition, the first few terms are 0, 1, 1, 2, 3, 5, 8, 13, 21. The asymptotic behavior of some sequences can also be found as the index n approaches infinity. For the Fibonacci sequence, this asymptote is the golden ration, which is approximately The recursive sequence found in the later portion of this paper for the Robotic-Turn Metric is defined for only finitely many terms since it is limited by the total number of positions of the cube. Thus, we do not need to observe the limit as n approaches infinity. 5

7 2.2 Cycle Notation A permutation is a one-to-one and onto function for some set G. Our permutation functions for the cube group are our selected moves that are within G. Permutations may be written in cycle notation. To illustrate this, consider the following permutation on G = {1, 2, 3, 4, 5, 6}, [ ] α = This notation means 1 2 1, , 5 5. A more succinct expression of this is the cycle α = (12)(346)(5). In fact, numbers like 5 that are sent to themselves are excluded by convention, so we write α = (12)(346). Recall that the cube group, however, is a permutation group which only allows even permutations of the smaller cubies. A permutation is considered even or odd depending on whether the permutation can be written as the product of an even or odd number of 2-cycles. (While a permutation can be written as such a product in many ways, the number of 2-cycles is always even or odd, see [7]). For instance, consider α = (12)(346) = (12)(34)(36). Since the number of 2-cycles in α is three, α is an odd permutation. The permutation functions (moves) are even permutations of the cubies. For each move, 9 cubies are moved with one being shifted around its axis, making it 8 cubies that are actually permuted. The permutation of these 8 cubies results in the product of two cycles each of length-4, each an odd permutation. The product of two odd permutations is even (because the sum of two odd numbers is even), thus all moves allowed on the cube are even permutations. It is also proven that exactly 1 2 of all the permutations on a set are even. When considering the permutation of all edges and corners together, the overall permutation must be even. However, when considering only edges or 6

8 corners alone, it is possible for them to be either even or odd. To obey the laws of the cube [8] which require an even permutation, if the permutation of edges is even then the permutation of corners must also be even, and if the edge permutation is odd then the corner permutation must also be odd. 2.3 Cayley Graphs A Cayley graph is used to analyze the structure of finite groups. Given a finite group S and a generating subset G (written S = G ), the Cayley graph of S and G is denoted Γ(S, G). Figure 1 shows the Cayley graph of symmetric group, S 4. The 4! = 24 vertices are all the permutations of 1, 2, 3, 4. The generators are the transpositions (12) (shown as blue edges), (23) (red), and (34) (green). Figure 1: The Cayley graph of S 4 with generators (12), (23), (34). http: //en.wikiversity.org/wiki/symmetric_group_s4 Each vertex is a representation of a possible permutation of the set, and each edge represents a move performed. For Rubik s cube, considering the approximately 43 quintillion possible states, there would be that many vertices, making the Cayley graph for the cube, if graphically represented, considerably large and complex. The diameter of a group is the smallest number of edges needed to move from one vertex to the other without any 7

9 repeated vertices. In Figure 1, the diameter is Permutations of the Cube The cube may be analyzed using permutations, where each possible arrangement of the cube can be found through the product of disjoint cycles. In group theory, God s number is the diameter or maximum distance between the vertices of the Cayley graph associated with the particular metric. Imagine a cube on your desk, untouched. The face directly facing you is the front, then the back, the top, the bottom, the right, and the left. We denote these faces as F, B, U, D, R, and L respectively. Figure 2: Diagram of the numbered mapping of each cube. sfu.ca/~jtmulhol/math302/puzzles-rc.html The disjoint cycles whose product reproduces the permutation space, S (all possible positions), are these moves represented in cycle notation. These cycles are developed by observing what labels are permuted during a move sequence. For example, the first cycle of a 90 clockwise rotation of the corners of the Up face is (1, 3, 8, 6). Using this cycle and Figure 2 to visualize, that clockwise rotation results in the labels moving from 8

10 (a) Front face, denoted F (b) Up face, denoted U (c) Left face, denoted L (d) Back face, denoted B (e) Down face, denoted D (f) Right face, denoted R Figure 3: Illustration of the corresponding side for each allowed move Similarly, the second cycle of a 90 clockwise rotation of the Up face, (2, 5, 7, 4), symbolizes a permutation in edge labels from The other three cycles of this move are derived from observing permutations of the four adjacent faces [1]. The complete notation of U in cycle notation, as well as the other moves, are listed below. (Note that, as a permutation of labels, U is an odd permutation. However, as an action on cubies, it is the product of two cycles of length four, an even permutation, as discussed above). The group, S, is generated by G = {F, B, U, D, R, L}, since all positions of the cube in S can be found from products of the elements of G. U = (1, 3, 8, 6)(2, 5, 7, 4)(9, 33, 25, 17)(10, 34, 26, 18)(11, 35, 27, 19) L = (9, 11, 16, 14)(10, 13, 15, 12)(1, 17, 41, 40)(4, 20, 44, 37)(6, 22, 46, 35) F = (17, 19, 24, 22)(18, 21, 23, 20)(6, 25, 43, 16)(7, 28, 42, 13)(8, 30, 41, 11) R = (25, 27, 32, 30)(26, 29, 31, 28)(3, 38, 43, 19)(5, 36, 45, 21)(8, 33, 48, 24) 9

11 B = (33, 35, 40, 38)(34, 37, 39, 36)(3, 9, 46, 32)(2, 12, 47, 29)(1, 14, 48, 27) D = (41, 43, 48, 46)(42, 45, 47, 44)(14, 22, 30, 38)(15, 23, 31, 39)(16, 24, 32, 40) 3 Existing Metrics 3.1 Face-Turn Metric The Face-Turn Metric (FTM) includes face rotations of ±90 and 180. With each of these rotations applied to each of the six faces of the cube, the set G of the FTM contains 6 3 = 18 elements. These 18 elements are listed below G F T M = {F, F 2, F 3, B, B 2, B 3, U, U 2, U 3, D, D 2, D 3, L, L 2, L 3, R, R 2, R 3 }, where X 2 (X being some face move) is a 180 rotation of that face and X 3 is a 270 clockwise rotation (or 90 counterclockwise rotation). We see here that the set G F T M is larger than that of G QT M, and, as a result, God s number is reduced. In the majority of presentations of the Rubik s cube, the metric of choice was the FTM. Thus, optimal solution analysis occurred primarily in the FTM, and later proceeded into the QTM Establishing a Lower Bound In 1980, the same year as the Rubik s cube s international debut, a lower bound for the FTM was developed through a simple counting method. Since FTM and QTM do not work in antipodal systems like RTM, to solve a cube n moves away from solution, each consecutive move must involve a different face, so there are 5 3 = 15 options at each step after the first. Thus, N F T M = = n 1, Comparing the total number of positions of the cube [10] and the total number of positions reached using a maximum of 16 moves, the latter number was observed to be smaller, and therefore could not be a lower bound. Positions reached with a maximum of 17 moves, however, was larger than 10

12 n N F T M , , , ,668, ,031, ,075,468, ,132,031, ,980,468, ,379,707,031, ,695,605,468, ,335,434,082,031, ,031,511,230,468, ,472,668,457,031, ,882,090,026,855,470, ,231,350,402,832,000,000 Table 1: Simple count of moves needed to encompass all positions of the Rubik s cube in FTM. all possible positions of the Rubik s cube [10]. A refinement of this method gives a lower bound of 18 moves, which was not improved for 14 years as most analysis was concentrated on finding an upper bound. Using Kociemba s Two-Phase algorithm to find optimal solutions in 1995, Micheal Reid raised the lower bound from 18 by proving the minimum number of moves to solve cubes in the superflip position (where all corners are solves, and all edges are flipped in their home position) is 20 [10] Establishing an Upper Bound In 1981, Morwen B. Thistlethwaite developed an algorithm that solves the cube optimally. This method requires computation power beyond that of human capability, but is ideal for computer analysis. Thistlethwaite s procedure is conducted my moving the cube into a position that can be solved 11

13 Figure 4: Development of the lower and upper bounds in the Face-Turn Metric. without using and quarter turns of U and D (U 2 and D 2, however, are still allowed). Those moves are then conducted until the position can be solved, similarly, without F and B quarter turns; then finally without using L and R quarter turns until the cube is in a state that can be solved by only using half-turn rotations [10]. Collectively, the groups contain all possible positions of the cube, and further analysis using many different positions of the cube revealed an upper bound of 52 moves. However, in 1990, Hans Kloosterman, with a slight variation to Thistlethwaite s original algorithm, reduced the upper bound to 42. In 1992, Micheal Reid improved Kloosterman s calculations to reduce the upper bound to 39. Amazingly, the upper bound was reduced once again only a day later to 37 by Dik T. Winter through analysis of Kociemba s two-phase algorithm. In 1995, after re-analyzing Winter s approach, Reid cut the upper bound to 29 moves, and also raised the lower bound to 20 moves moves by specifically analyzing a cube position known as superflip. After this, the cube s upper bound analysis remained at 29 for another 10 years [10]. Using Michael Reid s upper bound proof in 1995, Silviu Radu reduces the upper bound to 28 by proving that 29-length moves can intentionally be avoided in Upon further optimization, Radu improves his analysis and reduces the upper bound once again in 2006 to 27. With a new technique in mathematical group theory in 2007, Dan Kunkle and Gene Cooperman placed positions of the cube in a family of cosets (an internal group structure) and then simulated performing a single move on each coset. This required considerable computation power, but this ap- 12

14 proach showed that the cube can be solved in a maximum of 26 moves. It was from this point on Rubik s cube upper bound analysis began to move quickly. The next three years of upper bound analysis was dominated by Tomas Rokicki and his colleagues. Rokicki lowered the upper bound to 25 in 2008, and again to 23 and 22 later that year when joined by John Welborne using computers that can handle considerable simultaneous computations at Sony Pictures Imageworks. Using Kociemba s two-phase solver, 33GB (gigabytes) of memory and 3.7 million CPU (central processing unit) years were needed to compute all possible cube positions. In a joint effort, Rokicki, Herbert Kociemba, Morley Davidson, and John Dethridge proved that God s number for the cube using FTM is 20 in 2010 [12]. A prominent advantage in Rokicki s approach is his partitioning of the cube s positions into smaller, more easily computed sets. Working with the group generated by {U, F 2, R 2, D, B 2, L 2 }, the set of 43 quintillion positions of the cube is reduced to smaller partitions of sets. These sets are further reduced using symmetry. Partitioning the positions of the cube in this way allowed for faster computation. 3.2 Quarter-Turn Metric The Quarter-Turn Metric (QTM) includes face rotations of ±90. With each of these rotations applied to each of the six faces of the cube, the set G in QTM contains 6 2 = 12 elements, specifically G QT M = {F, F 3, B, B 3, U, U 3, D, D 3, L, L 3, R, R 3 }. Similar to the FTM, a simple counting method was used to determine an appropriate upper bound, N QT M = = n 1. As shown in Table 2, the results of this count established an initial lower bound of

15 n N QT M , , , ,932, ,258, ,846, ,572,306, ,295,372, ,249,095, ,423,740,047, ,661,140,520, ,272,545,727, ,556,998,002,998, ,126,978,032,987, ,396,758,362,866, ,065,364,341,991,530, ,719,007,761,906,800,000 Table 2: Simple count of moves needed to encompass all positions of the Rubik s cube in QTM Establishing a Lower Bound In 1981, Dan Hoey presented the first lower bound for the QTM. In his proof, he defined optimal solutions to be sequences where adjacent moves are ±90 rotations of antipodal faces. Hoey s analysis resulted in a 21 move lower bound [10]. In 1995, Micheal Reid raised this lower bound to 22 moves based on conjecture optimal solutions of superflip positions. Reid raised the lower bound in 1998 after finding that the superflip position composed with four-spot pattern (L, R,F, and B face center pieces are swapped with the antipodal counterpart s centerpiece) is 26. The lower bound for QTM continues to be 26 [10]. 14

16 3.2.2 Establishing an Upper Bound Figure 5: Development of the lower and upper bounds in the Quarter- Turn Metric The first upper bound proposed for the QTM was found through analysis of the worst-case scenario positions of early solution manuals. When Morwen Thistlewaite established a 52 move upper bound for the FTM in 1981, an upper bound of 104 was given for QTM by simply doubling this number. Similarly, in 1992, when Micheal Reid reduces the face-turn upper bound to 39, he also reduced the quarter-turn upper bound to 56, then to 42. The QTM s upper bound contiued to be reduced to 40 by Silviu Radu, then to 38 by Bruce Norskog, then 36, 35, and 35 by Radu after further improvement of his algorithm by 2007 [10]. In the first half of 2009, using coset partitioning, Thomas Rokicki further reduced this upper bound to 32, 31, 30, and 29. It was not until a joint effort by Rokicki and Morely Davidson in 2014 that God s number was proven to be 26 in QTM. This was accomplished with about 29 CPU years provided by the Ohio Supercomputer Center [11]. 4 Robotic-Turn Metric The Robotic-Turn Metric is named after its appropriate user. When holding the cube, it is near impossible for us to perform these moves, and added to the fact that it requires a considerable amount of calculations for optimal solution, this metric is not fit for human use, but robotic. By following the assumption that a larger number of generators would reduce God s number, the Robotic-Turn Metric (RTM) with many more generators should have 15

17 Figure 6: Illustration of a robot capable of solving in RTM. robot-can-solve-any-rubiks-cube-in-under.php a lower God s number. In the RTM, moves are not counted as singular movements, but work as a binary system of antipodal faces. The moves of the generating set G RT M are defined in terms of X i Y j, where X and Y are antipodal faces and i, j {0, 1, 2, 3} which define how many 90 rotations that face undergoes, with 0 = 0, 1 = 90, 2 = 180, and 3 = 270. There are 4 4 = 16 possibilities for i and j, but we remove the move that represents the do-nothing move when i = j = 0, so we have 15 moves for each antipodal face combination. Since we have 3 pairs of antipodal faces, the set G RT M has 45 elements, listed below. This is roughly four times as many as the FTM, and so we expect that God s number for RTM will be less than God s number for the other metrics we have discussed. G RT M ={U 0 D 1, U 0 D 2, U 0 D 3, U 1 D 0, U 1 D 1, U 1 D 2, U 1 D 3, U 2 D 0, U 2 D 1, U 2 D 2, U 2 D 3, U 3 D 0, U 3 D 1, U 3 D 2, U 3 D 3, F 0 B 1,..., F 3 B 3,..., L 0 R 1,..., L 3 R 3 } 16

18 (a) Front and Back face combination, denoted F B (b) Up and Down face combination, denoted UD (c) Left and Right face combination, denoted LR Figure 7: Illustration of the combinations of antipodal faces. 4.1 Search for a Lower Bound To determine an appropriate lower bound for RTM, we used a simple counting method to calculate all the minimum number of moves needed that may solve all positions of the cube, then compared this same counting method to previous metrics where a lower bound had been more carefully established. If the cube was in a state where one move would result in the cube s solution, there would be one of 45 possible moves that can solve this. If the cube state was two moves away from being solved, the two moves would be from different antipodal combinations, and the second move would be one of the 30 from the two families not used in the first move. This was concluded from the fact that any number of successive moves from the same antipodal family could have been performed in one move. In this manner, if we denote n as the number of moves from being solved, then the number of possible positions in this state, a first approximation for N RT M = A(n) is given by A(1) = 45 and A(n) = 30 A(n 1) for n 2. It was found that some sequences, after performing a particular move, does not genuinely progress the sequence to n + 1 moves away from the solution. For instance, consider a cube 2 moves away that may be solved using the sequence U 2 D 2 L 2 R 2 ; however, this sequence may also be solved if these moves were switched to L 2 R 2 U 2 D 2. Since there are three distinct combinations of antipodal systems, then the number of redundancies in a list of n-length sequences is at least 3A(n 1). 17

19 We expect similar redundancies within these lists for greater n-length sequences. To remove these redundancies, calculations were made using Wolfram s M athematica because of its permutation and group analysis capabilities. U 2 D 2 L 2 R 2 L 2 R 2 U 2 D 2 Figure 8: Illustration of redundancy in U 2 D 2, L 2 R 2 sequences. We began with A(0) = 1 from the identity (solved state) and A(1) = 45 from our established 45 elements (moves) in G RT M. The approximation = 1350 overestimates A(2) = 1347 determined by Mathematica. For A(2), these = 3 redundancies can be understood using Figure 8, which shows one such redundancy both possible orders of the U 2 D 2 and L 2 R 2 moves result in the same state. With three antipodal systems (U i D j, F i B j and F 2 B 2, L 2 R 2 being the other two), the three redundancies in A(2) are accounted for. This computation, being more carefully calculated, would have provided an accurate depiction of the lower bound in the RTM. However, due to limited CPU power, the maximum A(n) calculated was for permutations exactly 4 moves away from the identity, A(4). To complete the table, a recursive sequence was developed. Using the three redundancies found above, 18

20 n A(n) , , ,152, ,419, ,730, ,076,156, ,686,232, ,581,550,968, ,418,589,600, ,806,191,488,933, ,004,239,598,314, ,635,245,408,824,725, ,106,870,351,308,403,819 Table 3: Distribution table to identify a lower bound in RTM. The values up to A(4) are exact and subsequent are upper bounds found using the recursive formula. we projected 30 A(2) 3 A(1) = = 40, 275 for A(3), which we know is 39, 631 from Mathematica. We conclude that there are 40, , 631 = 644 redundant length 3 paths. Similarly, we find that there 3, 619 redundant length 4 paths. This gives us the approximation A(n) = 30 A(n 1) 3 A(n 2) 644 A(n 3) 3619 A(n 4), for n 5, shown in Table 3. For values beyond A(4), these are upper bounds since there are almost certainly longer redundant paths. Nonetheless, we can conclude that 13 moves in the RTM are not enough to reach all cube positions, so God s number for this metric is at least Search for an Upper Bound With God s number established to be 20 in the FTM, it serves as a reference and initial proposition for an upper bound in RTM. To analyze the upper 19

21 Figure 9: Tally of permutations vs Number of moves reduced. bound further, a list of 32,625 known 20-move sequences was reduced using the antipodal system of the RTM. The reduction of the sequence occurs when consecutive moves are antipodal faces. A tally of the reductions showed that 19,161 sequences were not reduced; 10,747 were reduced by 1; 2,389 were reduced by 2; 308 reduced by 3; 19 reduced by 4; and 1 reduced by 5. This tally holds that the maximum number of moves needed in the RTM is 20, however, the reduction lacks the ability to optimally solve these 20-move sequences. This shortcoming makes the viability of this tally questionable. Without an optimal solver, a new upper bound was established using the reduction of super flip in the Slice-Turn Metric. Slice-turn metric has more similarity to the robotic-turn metric because it allows permutation of the middle slices of the cube as one move. These slice-moves can be reimagined in RTM to be moves where the outer slices move in unison. An optimal solution for superflip in the slice-turn metric is known to be 16 [13]. If super flip is also a hardest position for RTM, then this suggests 16 as an upper bound for God s number for the RTM. 20

22 5 Appendices: Mathematica Code 5.1 Evaluation of Moves U = Cycles[{{1, 3, 8, 6}, {2, 5, 7, 4}, {9, 33, 25, 17}, {10, 34, 26, 18}, {11, 35, 27, 19}}]; L = Cycles[{{9, 11, 16, 14}, {10, 13, 15, 12}, {1, 17, 41, 40}, {4, 20, 44, 37}, {6, 22, 46, 35}}]; F = Cycles[{{17, 19, 24, 22}, {18, 21, 23, 20}, {6, 25, 43, 16}, {7, 28, 42, 13}, {8, 30, 41, 11}}]; R = Cycles[{{25, 27, 32, 30}, {26, 29, 31, 28}, {3, 38, 43, 19}, {5, 36, 45, 21}, {8, 33, 48, 24}}]; B = Cycles[{{33, 35, 40, 38}, {34, 37, 39, 36}, {3, 9, 46, 32}, {2, 12, 47, 29}, {1, 14, 48, 27}}]; H = Cycles[{{41, 43, 48, 46}, {42, 45, 47, 44}, {14, 22, 30, 38}, {15, 23, 31, 39}, {16, 24, 32, 40}}]; UH = Flatten[Table[PermutationProduct[PermutationPower[U, n], PermutationPower[H, m]], {n, 0, 3}, {m, 0, 3}]]; LR = Flatten[Table[PermutationProduct[PermutationPower[L, n], PermutationPower[R, m]], {n, 0, 3}, {m, 0, 3}]]; FB = Flatten[Table[PermutationProduct[PermutationPower[F, n], PermutationPower[B, m]], {n, 0, 3}, {m, 0, 3}]]; Perm1 = Delete[Union[Flatten[{UH, LR, FB}]], 1]; Step2 = Flatten[Table[PermutationProduct[Extract[Perm1, n], Extract[Perm1, m]], {n, 1, 45}, {m, 1, 45}]]; Length[Step2] 2025 Length[Union[Step2]] 1393 Length[Intersection[Union[Step2], Perm1]] 45 21

23 MemberQ[Union[Step2], Cycles[{}]] True RStep2 = Complement[Union[Step2], Union[Perm1, {Cycles[{}]}]]; Length[RStep2] 1347 Step3 = Flatten[Table[PermutationProduct[Extract[RStep2, n], Extract[Perm1, m]], {n, 1, Length[RStep2]}, {m, 1, 45}]]; RStep3 = Complement[Union[Step3], Union[Perm1, RStep2]]; Length[RStep3] Step4 = Flatten[Table[PermutationProduct[Extract[RStep3, n], Extract[Perm1, m]], {n, 1, Length[RStep3]}, {m, 1, 45}]]; RStep4 = Complement[Union[Step4], Union[RStep3, RStep2]]; Length[RStep4]

24 5.2 Reduction of 20-length FTM sequences Moves = Import[ C:\\Users\\Administrator\\Desktop\\Cube Explorer\\20moves.txt, list ]; Extract[Moves, {1}] U F U F L D F U R D2 F R2 D2 F2 U R2 U R F U2 (20f*) //C2(a){C2h(a)} MoveSequence = Table[Extract[Moves, n], {n, 1, 32625}]; Reduction[sq ]:= StringCount[MoveSequence, D U] + StringCount[MoveSequence, U D]+ StringCount[MoveSequence, F B] + StringCount[MoveSequence, B F]+ StringCount[MoveSequence, L R] + StringCount[MoveSequence, R L] Tally[Reduction[MoveSequence]] {{0, 19161}, {1, 10747}, {2, 2389}, {3, 308}, {4, 19}, {5, 1}} Histogram[Reduction[MoveSequence]] 23

25 References [1] David Joyner. The Man Who Found God s Number. The College Mathematics Journal (2014) [2] David Singmaster. Review: Rubik s Rubrics. American Scientist 91.5 (2003) [3] Ed Pegg Jr., Todd Rowland, and Eric W. Weisstein. Cayley Graph. From MathWorld A Wolfram Web Resource. [4] Edward C. Turner. Rubik s Groups. The American Mathematical Monthly 92.9 (1985) [5] Eric W. Weisstein. Recursive Sequence. From MathWorld A Wolfram Web Resource. [6] Hannah Provenza. Group Theroy and the Rubik s Cube. REUPapers/Provenza.pdf [7] Joseph A. Gallian. Contemporary Abstract Algebra, 8th ed. Boston: Brooks/Cole, [8] Ryan Heise. Rubik s Cube Theory. Parity [9] Rubik Cube Solution. RecPuzzlesArchive. 29 June Web. 27 Jan [10] Tomas Rokicki. God s Number Is [11] Tomas Rokicki. Towards God s Number for Rubik s Cube in the Quarter-Turn Metric. The College Mathematics Journal (2014) [12] Tomas Rokicki, Herbert Kociemba, Morley Davidson, and John Dethridge. The Diameter of the Rubik s Cube Group Is Twenty. SIAM Journal on Discrete Mathematics 27.2 (2013) [13] Superflip. - Speedsolving.com Wiki. Web. 14 Dec