RUBIKS, HEURISITC. Mubbasir. T. Kapadia, D.J Sanghvi College of Engineering. the guidance of: Professor M.V Deshpande and Professor Jayant Umale

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1 DWARKADAS J. SANGHVI COLLEGE OF ENGINEERING, VILE PARLE, MUMBAI. 1 RUBIKS, HEURISITC Mubbasir. T. Kapadia, D.J Sanghvi College of Engineering. guidance of: Professor M.V Deshpande and Professor Jayant Umale Under Abstract This technical paper, titled Rubik s, Heuristic serves as a logical extension to an existing paper and project titled Rubik s, Distributed. Rubik s, Heuristic serves to extend existing project to cater for a user interface representing a three dimensional 3 by 3 cube, a solution to solve Rubik s cube by implementing a heuristics approach and lastly to enhance original engine to distribute solution over multiple processors. Index Terms Rubik s Cube, Distributed Computing, Remote Method Invocation (RMI), Heuristic, A* algorithm, T I. INTRODUCTION HIS document gives a detailed analysis of procedure that was followed in enhancement to an existing application titled Rubik s, Distributed. The original application served to solve a 2 by 2 Rubik s Mini cube by distributing solution over multiple processors. Its logical enhancement, Rubik s, Heuristics serves to extend various features by creating a user interface representing a 3 by 3 cube, designing a solution to solve cube, implementing solution using a heuristic approach and lastly distributing solution over multiple processors. 1) Existing System The existing system had following fully functional features: a. A graphical interface which would display Rubik s 2x2 cube in following modes: a. Two dimensional mode b. Three dimensional mode b. A mechanism to allow user to attempt to solve cube and facilitate easy interaction with cube. c. An algorithm to solve Rubik s 2 by 2 cube. d. Distribution of algorithm over multiple processors. e. A mechanism for evaluation of efficiency of solution The project was carried out as part of our curriculum during third year of engineering at D.J Sanghvi College of Engineering. The project was implemented under able guidance of Professor Rashmi Gugnani. This technical paper, as a supplement to above mentioned project was written under guidance of Professor Jayant Umale. 2) Enhancements to existing system a. A graphical interface which would display Rubik s 3X3 cube in following modes: a. Two dimensional mode b. Three dimensional mode b. A mechanism to allow user to attempt to solve cube and facilitate easy interaction with cube. c. To design a suitable heuristic approach which would accurately measure efficiency of a particular movement on Rubik s Cube. A movement leading away from solution would be given a negative heuristic value whereas a movement

2 2 DWARKADAS. J. SANGHVI COLLEGE OF ENGINEERING, VILE PARLE, MUMBAI. d. e. f. g. leading towards final solved state would be given a high heuristic value. To implement an algorithm to solve Rubik s 3x3 cube. To divide algorithm in stages for ease of distribution of algorithm over multiple processors. To distribute algorithm over multiple processors. To extend algorithm to give it generalized capability to solve an NxN Rubik s Cube. II. RUBIK S 3X3 CUBE 1) Programming Language The choice of programming language for project was JAVA, due to its extensive functionality and ease with which Remote Method Invocation (RMI) could be implemented. NetBeans IDE and Edit Plus were used as aids in coding process while Windows XP was choice of operating system. 2) Graphical User Interface (GUI) Our aim was to create an accurate depiction of Rubik s 2X2 cube and present user with two views: a. Two-dimensional view. b. Three-dimensional view. Fig: 2-D representation of 3x3 cube. From above figure: 1. 1: Top face 2. 2: Bottom face 3. 3: Front face 4. 4: Right face 5. Back face 6. Left face In addition, user would have two options in interacting with cube. a. Buttons. b. Clicking on cube square to rotate it in appropriate direction. 3) Logical Representation The Rubik s cube can be logically represented as an array of integers of 54 elements: int cube[54]; Fig: 3-D Representation of 3x3 Rubik s Cube Fig: Logical Representation of Cube

3 DWARKADAS J. SANGHVI COLLEGE OF ENGINEERING, VILE PARLE, MUMBAI. The various faces of cube can be accessed as follows: i. I. II. III. IV. V. VI. Front face : cube[0] cube[8] Top face : cube[9] cube[17] Back face : cube[18] cube[26] Bottom face : cube[27] cube[35] Left face : cube[36] cube[44] Right face : cube[45] cube[53] j. k. l. 4) Terminology The various faces can be represented using following notation: I. II. III. IV. V. VI. 3 anticlockwise direction Bottom Clockwise: Rotating bottom face in clockwise direction Bottom Anticlockwise: Rotating bottom face in anticlockwise direction Top Clockwise: Rotating top face in clockwise direction Top Anticlockwise: Rotating top face in anticlockwise direction Consider following illustration of Right clockwise rotation: F: Front face T: Top face B: Back face D: Bottom/Down face R: Right face L: Left face Rotation is represented using following terminology: I. II. III. R : Rotate Right face 90 degrees R : Rotate Right face 90 degrees anti R2: Rotate Right face 180 degrees. For example: RU L2 The above notation implies following: a. b. c. Rotate Right face 90 degrees Rotate Top face 90 degrees counter Rotate Left face 180 degrees. The above diagram illustrates working of Right Clockwise Rotation. Equipped with following interface, user can now easily interact with Rubik s cube, thus attempting to solve puzzle. 4) Cube Rotation The virtual cube, an exact simulation of Rubik s cube was given ability of rotating about its central axis in following ways: a. b. c. d. e. f. g. h. Front Clockwise: Rotating front face in clockwise direction Front Anticlockwise: Rotating front face in anticlockwise direction Right Clockwise: Rotating right face in clockwise direction. Right Anticlockwise: Rotating right face in anticlockwise direction. Left Clockwise: Rotating left face in clockwise direction Left Anticlockwise: Rotating left face in anticlockwise direction Back Clockwise: Rotating back face in clockwise direction Back Anticlockwise: Rotating back face in III. THE ALGORITHM Start Initialize Rubik s Cube. Scramble Rubik s Cube. Perform Step 1 Solving first layer a. Solve Cross b. Inserting first layer corners 5. Perform Step 2 Positioning corners 6. Perform Step 3 Orienting corners 7. Perform Step 4 Complete Bottom face 8. Perform Step 5 Position four center edge pieces 9. Perform Step 6 Orient four center edge pieces 10. Display solution 11. Stop The following algorithm is thus divided into three basic

4 DWARKADAS. J. SANGHVI COLLEGE OF ENGINEERING, VILE PARLE, MUMBAI. 4 steps, keeping in mind distribution of intermediate portions of code among multiple processors. 1) Initialization of Rubik s Cube This procedure involves setting up Rubik s cube which is represented as an integer array. It is initially in solved state. a. cube[0] cube[8] = WHITE b. cube[9] cube[17] = GREEN c. cube[18] cube[26] = YELLOW d. cube[27] cube[35] = BLUE e. cube[36] cube[44] = BLACK f. cube[45] cube[53] = BROWN 2) Scrambling Rubik s Cube The cube is n scrambled i.e. using power of random number generation it is made to undergo a random set of moves unknown to user or processor. The result is a scrambled cube, waiting to be solved. 3) Step 1 The First Layer This procedure involves solving first layer of cube. This step can be divided into following stages: I. Solving Cross II. Inserting four first layer corners I. Solving cross: The cross involves orienting all first layer pieces excluding all corner pieces. The cross also includes orienting first layer piece of or faces with its corresponding center piece. The following diagram illustrates orientation of cross of first layer: f. cube[41] = cube[40] // Left face g. cube[48] = cube[49] // Right face h. cube[25] = cube[22] // Back face i. // Bottom face not in use If following conditions hold true, cross of first layer is said to be solved. II. Inserting four first layer corners: Once you have completed cross, completing first layer requires inserting each of 4 corners in separately. The following conditions must hold true for correct insertion of first layer corners: I. First corner i. cube[17] = cube[13] ii. cube[2] = cube[4] iii. cube[51] = cube[49] II. Second corner i. cube[15] = cube[13] ii. cube[44] = cube[40] iii. cube[0] = cube[4] III. Third corner i. cube[9] = cube[13] ii. cube[38] = cube[40] iii. cube[24] = cube[22] IV. Fourth corner i. cube[11] = cube[13] ii. cube[26] = cube[22] iii. cube[45] = cube[49] If following conditions hold true, first layer of cube is said to be solved. 4) Step 2 Positioning Bottom Layer Corners Before proceeding, it is important to note that cube is now repositioned such that solved layer becomes back face. Positioning: A corner is said to be positioned if correct mini cube lies in its corresponding corner position. This, however does not imply that three colours of mini cube have to correspond to middle cube colours Fig: The first layers cross For first layer cross to be solved, following conditions must hold true: a. cube[13]=cube[10] // Top layer b. cube[10]=cube[16] c. cube[12] = cube[14] d. cube[16] = cube[12] e. cube[1] = cube[4] // Front face

5 DWARKADAS J. SANGHVI COLLEGE OF ENGINEERING, VILE PARLE, MUMBAI. 5 Fig: Positioning of corner mini cube Horizontal Positioning: If two positioned corners are horizontal to one anor n it is said to be horizontal positioning.. I. Accept cube with first layer solved after performing Step 1. II. Re-position cube such as solved first layer becomes back face. III. Orient cube in such a manner that atleast two corners are correctly positioned. IV. Check form of positioning: a. Horizontal Positioning i. Perform Move 1 ii. Onto Step 3 b. Diagonal Positioning i. Perform Move 1 ii. Repeat Step 2 V. Stop. 5) Step 3 - Orient four corner pieces of Side 2 Fig: Horizontal Positioning Diagonal Positioning: On or hand, if two corners that are positioned are diagonal to one anor n it is said to be diagonal positioning Orientation: A corner is said to be correctly oriented if correct mini cube lies in its corresponding corner position. Also three colours of mini cube have to correspond to middle cube colours Fig: Diagonal Positioning Move 1: This is a standard algorithm that is utilized in Step 2. It is as follows: T F T L T L T F2 I. Perform Top face rotation: 90 degrees counter II. Perform Front face rotation: 90 degrees III. Perform Top face rotation: 90 degrees IV. Perform Left face rotation: 90 degrees counter V. Perform Top face rotation: 90 degrees VI. Perform Left face rotation: 90 degrees VII. Perform Top face rotation: 90 degrees counter VIII. Perform Front face rotation: 180 degrees. Step 2 Algorithm: Fig: Orientation of corner of cube Move 2: This is a standard algorithm that is utilized in Step 3. It is as follows: T F2 T F T F T F2 I. Perform Top face rotation: 90 degrees counter II. Perform Front face rotation180 degrees. III. Perform Top face rotation: 90 degrees IV. Perform Front face rotation: 90 degrees counter V. Perform Top face rotation: 90 degrees anti VI. Perform Front face rotation: 90 degrees VII. Perform Top face rotation: 90 degrees VIII. Perform Front face rotation: 180 degrees. Step 3 Algorithm:

6 DWARKADAS. J. SANGHVI COLLEGE OF ENGINEERING, VILE PARLE, MUMBAI. 6 I. Accept cube with first layer solved after performing Step 1 and corners correctly positioned after Step2 is performed. II. Check for number of correctly oriented corners: a. Zero Orientation i. Perform Move 2. ii. Repeat Step 3. b. Single Orientation i. Orient cube such that correctly oriented cube is in top right hand corner. ii. Perform Move 2. iii. If all four corners are correctly oriented. 1. Onto Step 4 iv. Else Repeat Step 3 c. Double Orientation i. Orient cube such that two oriented corners are on left hand side. ii. Perform Move 2. iii. Repeat Step 3. d. Triple Orientation i. Orient cube such that two oriented corners are on left hand side. ii. Perform Move 2. iii. Repeat Step 3. III. Stop. Move 3: F T2 B2 F I. Perform Front face rotation: 90 degrees II. Perform Top face rotation: 180 degrees. III. Perform Bottom face rotation: 180 degrees. IV. Perform Front face rotation: 90 degrees counter Move 4: F T B F I. Perform Front face rotation: 90 degrees counter II. Perform Top face rotation: 90 degrees III. Perform Bottom face rotation: 90 degrees IV. Perform Front face rotation: 90 degrees Fig: Cube after Step 3 6) Step 4 Complete Bottom face This procedure involves completing entire bottom face of cube. After Step 4, front layer and bottom layer are solved. Before proceeding, place cube back in its correct position. Various possible cases arise based upon location of colour of bottom face: Case I Fig: Bottom face solved 7) Step 5 Position four center edge pieces

7 DWARKADAS J. SANGHVI COLLEGE OF ENGINEERING, VILE PARLE, MUMBAI. 7 I. Accept cube from Step 4 with first layer and bottom layer solved. II. Estimate number of center edge pieces that are correctly positioned. a. Zero positioning: i. Place top face on right hand side. ii. R L T2 R L T2 iii. Onto b. b. Single positioning: i. While (Quadruple positioning) 1. R L T2 R L T2 ii. Onto d. c. Double positioning: i. R L T R L T R L T2 R L T R L T R L T2 ii. Onto Step 4 d. Quadruple positioning: i. Onto Step 7 III. End. 8) Step 6 Orient four center edge pieces I. Accept cube from Step 5. II. Estimate number of center edge pieces that are correctly oriented a. Parallel mis-orientation: i. Place mis-oriented pieces on top. ii. R L T R L T R L T2 R L T R L T R L T2 iii. Place or two on top iv. R L T R L T R L T2 R L T R L T R L T2 v. Stop. b. Diagonal mis-orientation: i. Convert to parallel misorientation. ii. Onto a. III. Stop. Upon completion of Step 6, 3X3 cube is said to be solved. Heuristic function: IV. RUBIK S: A HEURISTIC APPROACH A heuristic is a function, h(n) defined on nodes of a search tree, which serves as an estimate of cost of cheapest path from that node to goal node. Heuristics are used by informed search algorithms such as Greedy best-first search and A* to choose best node to explore. Greedy best-first search will choose node that has lowest value for heuristic function. A* search will expand nodes that have lowest value for g(n) + h(n), where g(n) is (exact) cost of path from initial state to current node. If h(n) is admissible that is, if h(n) never overestimates costs of reaching goal, n A* will always find an optimal solution. Determining goal state: As can be seen in previous section, algorithm is divided into various intermediate steps each having its own intermediate goal state. For instance, goal states for following steps of algorithm are as follows: 1. Step 1 Solving first layer a. Solve Cross: The goal state would be a cube with cross solved. b. Inserting first layer corners: The goal state would be cube with corners and cross solved of first layer 2. Perform Step 2 Positioning corners: The goal state would be first layer solved and corners correctly positioned. 3. Perform Step 3 Orienting corners: The goal state would be first layer solved and corners correctly oriented. 4. Perform Step 4 Complete Bottom face: The goal state would be first layer solved along with complete bottom face. 5. Perform Step 5 Position four center edge pieces: The goal state would include positioning of four center edge pieces. 6. Perform Step 6 Orient four center edge pieces: The goal state would be completely solved cube. A* Algorithm: A* algorithm is a typical heuristic search algorithm, in which heuristic function is an estimated shortest distance from initial state to closest goal state, and it equals to traveled distance plus predicted distance ahead. That is, f(n) = g(n) + h(n). 1. Create a search graph, G, consisting solely of start node N1. Put N1 in a list called OPEN. 2. Create a list called CLOSED that is initially empty. 3. If OPEN is empty, exit with failure. 4. Select first node on OPEN, remove it from OPEN, and put it on CLOSED. Call this node N.

8 DWARKADAS. J. SANGHVI COLLEGE OF ENGINEERING, VILE PARLE, MUMBAI If N is a goal node, exit successfully with solution obtained by tracing a path along pointers from N to N1 in G. (The pointers define a search tree and are established in step 7.) 6. Expand node N, generating set, M, of its successors that are not already ancestors of N in G. Install se members of M as successors of N in G. 7. Establish a pointer to N from each of those members of M that were not already in G (i.e., not already on eir OPEN or CLOSED). Add se members of M to OPEN. For each member, Mi, of M that was already on OPEN or CLOSED, redirect its pointer to N if best path to Mi found so far is through N. For each member of M already on CLOSED, redirect pointers of each of its descendants in G so that y point backward along best paths found so far to se descendants. 8. Reorder list OPEN in order of increasing f values. (Ties among minimal f values are resolved in favor of deepest node in search tree.) 9. Go to step 3. IDA* Algorithm: IDA* is a depth-first search that looks for increasingly longer solutions in a series of iterations, using a lower-bound heuristic to prune branches once ir estimated length exceeds current iterations bound It works roughly as follows. First problem is divided into a number of sub problems as has already been achieved: 1. Perform Step 1 Solving first layer c. Solve Cross d. Inserting first layer corners 2. Perform Step 2 Positioning corners 3. Perform Step 3 Orienting corners 4. Perform Step 4 Complete Bottom face 5. Perform Step 5 Position four center edge pieces 6. Perform Step 6 Orient four center edge pieces Clearly number of moves required to solve any of se sub problems is a lower bound for number of moves you will need to solve entire cube. Given a random cube C, it is solved as iterative deepening. First all cubes are generated that are result of applying 1 move to m. Next, from this list, all cubes are generated that are result of applying two moves. Then three moves and so on. If at any point a cube is found that needs too many moves based on upper bounds to still be optimal it can be eliminated from list. Determination of Heuristic Function: Consider Step 1a solution of first layer cross. The goal state is as follows: Fig: Step 1a goal state In this case, only first layer is to be taken into consideration. To be more precise, only first layer cross. There are a total of 8 squares that are to be considered, excluding center squares whose colours are fixed. So, if all 8 squares would be perfectly oriented n heuristic would obtain minimum value as distance between current state and goal state would be zero. Thus, intermediate goal state would be reached. Initially Heuristic H is assigned to zero. The following conditions are n tested. a. If cube[13]!= cube[10] b. If cube[10]!= cube[16] c. If cube[12]!= cube[14] d. If cube[16]!= cube[12] e. If cube[1]!= cube[4] f. If cube[41]!= cube[40] g. If cube[48]!= cube[49] h. If cube[25]!= cube[22] Thus, if none of squares would be correctly oriented worst case value of Heuristic would be 8. Incorporating Heuristic Function: A Heuristic function H is designed in a similar manner for all steps of algorithm and each step is implemented using A* algorithm to obtain a faster and more efficient solution of Rubik s Cube. V. SOLVING A NXN RUBIK S CUBE

9 DWARKADAS J. SANGHVI COLLEGE OF ENGINEERING, VILE PARLE, MUMBAI. 9 Solving 4x4 and 5x5 cube are logical extensions to solution of Rubik s 3x3 cube. Thus, solution of an NxN cube can be virtually simulated and is as follows: Step 1: The centers The most popular method of solving centers of a cube of order higher than 3 (that is, larger than 3x3x3) is to solve pairs of opposite centers toger. For example, one might solve centers of cubes with most common color scheme today in following order: white and yellow, n blue and green, and finally red and orange. Step 1a: The first two centers The first two centers are usually solved in rows, often from outside (just inward from edges) to inside. Many people work from outside to inside because it is easier to find pieces that are near outside. To make a row, find pieces that belong in row and maneuver m (without disturbing any previous progress) into a straight line. Then, insert completed row into R or L face using a short, intuitive algorithm such as F2 u' F2 u. On odd cubes (NxNxN where N is odd), "center centers" of white and yellow sides (or first centers being solved) can be kept anywhere, but you should insert first middle row you solve before solving or middle row to avoid disturbing first middle row solved. Then, solve second middle row, but leave a hole in center. When you're done turn F to make row vertical, n insert row with E F' E'. Step 1b: The next two centers Once first two centers are solved, a similar procedure is implemented in solution of next two centers. Step 1 is divided into three intermediate stages in which two centers are solved first, n next two and finally last two centers are solved. Using this procedure, if during step 1b a solution cannot be obtained n it implies that Step 1a is incorrect. In this case Step 1a is repeated until correct solution is obtained. Step 3: The 3x3x3 Now, cube is treated as a 3x3 cube and algorithm for 3x3 cube is followed to reach final solved state. 7. Perform Step 1 Solving first layer e. Solve Cross f. Inserting first layer corners 8. Perform Step 2 Positioning corners 9. Perform Step 3 Orienting corners 10. Perform Step 4 Complete Bottom face 11. Perform Step 5 Position four center edge pieces 12. Perform Step 6 Orient four center edge pieces 13. Display solution VI. DISTRIBUTING THE ALGORITHM. 1) Remote Method Invocation (RMI) If a client has access to an object on a different machine, n using this mechanism, client can invoke remote methods of that object. It is RMI that handles shipping of remote parameters to client machine. 2) Distributed Computing The cube is randomly generated at server and sent to three or more client machines. Using Remote Method Invocation clients attempt to solve cube by invoking procedures step1, step2, step3 of algorithm. At each stage, client machine upon fastest completion of intermediate step would send result to server. The or clients would stop processing upon notification of result having been acquired. The server would n send newly acquired cube to clients which would n perform next step. This procedure would be repeated for all three steps until final solution was obtained. As a result, fastest of three solutions is adopted to solve puzzle, thus proving wrong adage: Too many cooks spoil broth. Step 1c: The last two centers A similar procedure is implemented in solution of last two centers. If, during this procedure, a solution cannot be obtained or or centers are misplaced n Step 1a and Step 1b are repeated. Step 2: The edges This is second step in solution of cube and involves solution of edges of cube. In this stage, edges are solved such that at end of stage an intermediate solution is obtained with centers and edges solved. This is an iterative procedure in which Step 1 is repeated if a solution cannot be obtained. VII. CONCLUSION A heuristic approach for solving Rubik s cube was analyzed and successfully designed. An algorithm for solution of Rubik s 3x3 cube was developed and a strategy for successful deployment of algorithm over multiple processors was devised. It was observed that algorithm for solving 3x3 cube was not much different to that of Rubik s Mini Cube. An interesting observation to note was presence of a central element on each and every face on Rubik s 3x3 cube that remained constant. Thus, colour of faces of 3x3 cube was known in advance which lent a certain degree of simplicity to cube.

10 DWARKADAS. J. SANGHVI COLLEGE OF ENGINEERING, VILE PARLE, MUMBAI. 10 This form of symmetry was not found in Rubiks 2x2 cube which, admittedly easier to solve had its own set of complications in development of an algorithm. It can also be observed that a generalized algorithm for solving Rubik s Cube can be constructed which can n be successfully applied to a 4x4, 5x5 cube with slight extensions. It is thus possible to virtually simulate solution of an NxN cube. VIII. ACKNOWLEDGMENT I would like to take this opportunity to thank Computer Department at D.J Sanghvi College of Engineering for providing us with necessary resources needed for successful implementation of project. I would also like to thank Ms. Rashmi Gugnani in helping me through entire project and Professor Jayant Umale for his guidance through completion of this technical publication. IX. APPENDIX A RUBIK S REVENGE Rubik's Revenge is version of Rubik's Cube. Invented by Péter Sebestény, Rubik's Revenge was nearly called Sebestény Cube until a somewhat last-minute decision changed puzzle's name to attract fans of original Rubik's Cube. Unlike original puzzle, it has no fixed facets: centre facets (four per face) are free to move to different positions. The internal mechanics are rar different: centre cubelets slide in grooves on an internal ball, which cannot be seen unless puzzle is disassembled. The edge and corner cubelets glide on tracks formed by edges of centre cubelets in much same way as in version. A new mechanism was introduced by East Sheen company, where mechanism is an expansion of that of Methods for solving cube work for edges and corners of cube, as long as one has correctly identified relative positions of colours since centre facets can no longer be used for identification. There are 8 corner cubelets, 24 edge cubelets and 24 centre cubelets. Any permutation of corner cubelets is possible, including odd permutations, giving 8! possible arrangements. Seven of corner cubelets can be independently rotated, and eighth cubelet's orientation depends on or seven, giving 3 7 combinations. Assuming 4 centre cubelets of each colour are indistinguishable, re are 24!/(4! 6 ) arrangements, all of which are possible, independently of corner cubelets. (An odd permutation of corner cubelets implies an odd permutation of centre cubelets, and vice versa; however, even and odd permutations are indistinguishable because of identically coloured centre cubelets.) The 24 edge cubelets cannot be flipped. The two edge cubelets in each matching pair are distinguishable, since colours on a cubelet are reversed relative to or. Any permutation of edge cubelets is possible, including odd permutations, giving 24! arrangements, independently of corner or centre cubelets. Assuming cube does not have a fixed orientation in space, and that permutations resulting from rotating cube without twisting cube are considered identical, number of permutations is reduced by a factor of 24. This gives a total number of permutations of The full number is 7,401,196,841,564,901,869,874,093,974,498,574,336,000,00 0,000 possible combinations. X. APPENDIX B - THE PROFESSORS CUBE : 5X5 There are 8 corner cubelets, 36 edge cubelets (two types), and 54 center cubelets (48 movable of two types, 6 fixed). Any permutation of corner cubelets is possible, including odd permutations, giving 8! (40 320) possible arrangements. Seven of corner cubelets can be independently rotated, and eighth cubelet's orientation depends on or seven, giving 3 7 combinations. Assuming 4 center cubelets of each type of each colour are indistinguishable, re are (24!/(4! 6 )) 2 or 24! 2 /4! 12 arrangements, all of which are possible, independently of corner cubelets. (An odd permutation of corner cubelets implies an odd permutation of central edge cubelets, and vice versa) Identically coloured pairs among 24 outer edge cubelets cannot be flipped. The two cubelets in each matching pair are distinguishable, since colours on a cubelet are reversed relative to or. Any permutation of outer edge cubelets is possible, including odd permutations, giving 24! arrangements, independently of corner or center cubelets. The 12 central edge cubelets can be flipped. Eleven can be flipped and arranged independently, giving 12!/ or 12! 2 10 possibilities. Counting outer edge cubelets, re are 24! 12! 2 10 possibilities. The corner, central edge and fixed center cubelets toger form a Rubik's Cube. The remaining cubelets can be arranged independently of it.

11 DWARKADAS J. SANGHVI COLLEGE OF ENGINEERING, VILE PARLE, MUMBAI. 11 This gives a total number of permutations of: XI. REFERENCES [1] Herb Schildt and Patrick Naughton, The Complete Reference JAVA 2, Fourth Edition. [2] JAVA Documentation. [3] David Lee and Winston Miller, Using Best Fast Search Algorithm and Profile Tables to solve Rubik s Cube. [4] Wilkipedia, Distributed Computing [5] Hart, P. E., Nilsson, N. J.; Raphael, B. (1968). "A Formal Basis for Heuristic Determination of Minimum Cost Paths". IEEE Transactions on Systems Science and Cybernetic. [6] Hart, P. E., Nilsson, N. J.; Raphael, B. (1972). "Correction to "A Formal Basis for Heuristic Determination of Minimum Cost Paths"". [7] Nilsson, N. J. (1980). Principles of Artificial Intelligence. Palo Alto, California: Tioga Publishing Company. [8] Russell, S. J.; Norvig, P. (2003). Artificial Intelligence: A Modern Approach,