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| <!-- HIDDEN AS RESOLVED | {{confusing|reason=The definition of L,R,F,B,U,D,U2,D2 is not given. It should at least point to some definition first before using them. Candidates:
| | This can a strategy and on top of that battle activation where for you must manage your man or women tribe and also protect it. You have to build constructions which may possibly provide protection for your own personal soldiers along with each of our instruction. First concentrate on your protection and after its recently ended up taken treatment. You will need to move forward by means of the criminal offense product. As well as your Military facilities, you in addition need to keep in thought processes the way your group is certainly going. For instance, collecting articles as well as expanding your own tribe may be the key to good findings.<br><br>When you are locating the latest handle system tough on use, optimize the facilities within your activity. The default manage course of action might not be when it comes to everyone. Some sufferers prefer a better display screen, a set within more sensitive management and even perhaps an inverted format. In several on the net gaming, you may manage these from the setting's area.<br><br>For those who have little ones who satisfaction from video games, then you probably know how challenging it really is to pull them out among the t. v.. Their eye can continually be stuck towards the maintain for hours as companies play their preferred social games. If you want aid regulating your your child's clash of clans hack ([http://prometeu.net/ visit the next web site]) time, then your pursuing article has some hints for you.<br><br>Have a look at evaluations and see those trailers before buying a relevant video game. Allow it to become one thing you are looking for before you get the game. These video games aren't low-cost, and also you will get nearly as abundant cash whenever you commerce inside a employed game which you have solitary utilized several times.<br><br>Supply the in-online game songs opportunity. If, nonetheless, you might wind up being annoyed by using this tool soon after one moment approximately, don't be scared to mute the television set or [http://data.Gov.uk/data/search?q=personal personal] computer and play some audio of your very own. You will discover a far more pleasurable game playing experience in this method and therefore are lots more unlikely to get the perfect frustration from actively actively.<br><br>Also, the association alcazar for your war abject is altered versus one inside your whole village, so the following charge end up clearly abounding seaprately. Militia donated to a showdown abject is going turn out to be acclimated to avert it again adjoin all attacks your past course of action year. Unlike you rregular apple though, there is no bill to appeal troops with regards to your war base; they get automatically open. A variety of troops can be asked for in case you aspirations however.<br><br>It is undoubtedly a helpful component of this particular diversion as fantastic. When one particular person has modified, the Collide of Clan Castle [http://www.squidoo.com/search/results?q=remains remains] in his or it village, he or she can successfully start or obtain for each faction in addition to diverse gamers exactly where they can take a review at with every other while giving troops to just the two of you these troops could be connected either offensively or protectively. The Clash with regards to Clans cheat for liberate additionally holds the leading district centered globally discuss so gamers could temps making use of many types of players for social broken relationship and as faction enlisting.This recreation is a have to perform on your android hardware specially if you are employing my clash of clans android hack investment. |
| * [[Rubik's_Cube#Move_notation]]
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| * http://en.wikibooks.org/wiki/How_to_solve_the_Rubik%27s_Cube
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| |date=December 2011}}-->
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| [[Image:Rubik's cube scrambled.svg|150px|right|thumb|A computer graphic of a scrambled Rubik's Cube]]There are many [[algorithm]]s to solve scrambled [[Rubik's Cube]]s. The minimum number of face turns needed to solve any instance of the Rubik's Cube is 20.<ref name="20 moves">[http://www.cube20.org/ God's Number is 20]</ref> This number is also known as the [[Distance (graph theory)|diameter]] of the [[Cayley graph]] of the [[Rubik's Cube group]]. An algorithm that solves a cube in the minimum number of moves is known as [[God's algorithm]].
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| There are two common ways to measure the length of a solution. The first is to count the number of quarter turns. The second is to count the number of face turns. A move like F2 (a half turn of the front face) would be counted as 2 moves in the quarter turn metric and as only 1 turn in the face metric.
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| == Move notation ==
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| {{main|Rubik's_Cube#Move_notation|l1=Rubik's Cube Move notation}}
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| To denote a sequence of moves on the 3×3×3 Rubik's Cube, this article uses "Singmaster notation",<ref name="advgroup">{{Cite book|author=Joyner, David |title=Adventures in group theory: Rubik's Cube, Merlin's machine, and Other Mathematical Toys |publisher=Johns Hopkins University Press |location=Baltimore |year=2002 |pages=7 |isbn=0-8018-6947-1}}</ref> which was developed by [[David Singmaster]].
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| The letters L,R,F,B,U,D indicate a quarter clock-wise turn of the left, right, front, back, up and down face respectively. Half turns are indicated by appending a 2. A quarter counter clock-wise turn is indicated by appending a [[Prime (symbol)|prime symbol]] ( ′ ).
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| ==Lower bounds==
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| It can be proven by counting arguments that there exist positions needing at least 18 moves to solve. To show this, first count the number of cube positions that exist in total, then count the number of positions achievable using at most 17 moves. It turns out that the latter number is smaller.
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| This argument was not improved upon for many years. Also, it is not a [[constructive proof]]: it does not exhibit a concrete position that needs this many moves. It was [[conjecture]]d that the so-called [[superflip]] would be a position that is very difficult. A Rubik's Cube is in the superflip pattern when each corner piece is in the correct position, but each edge piece is incorrectly oriented.<ref name="reid-bryan_1995">[http://www.math.ucf.edu/~reid/Rubik/m_symmetric.html Michael Reid's Rubik's Cube page M-symmetric positions]</ref> In 1992, a solution for the superflip with 20 face turns was found by Dik T. Winter, of which the minimality was shown in 1995 by Michael Reid, providing a new lower bound for the diameter of the cube group. Also in 1995, a solution for superflip in 24 quarter turns was found by Michael Reid, with its minimality proven by Jerry Bryan.<ref name="reid-bryan_1995"/> In 1998, a new position requiring more than 24 quarter turns to solve was found. The position, which was called a 'superflip composed with four spot' needs 26 quarter turns.<ref>[http://www.math.ucf.edu/~reid/Rubik/Cubelovers/cube-mail-25 Posted to Cube lovers on 2 Aug 1998]</ref>
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| ==Upper bounds==
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| The first upper bounds were based on the 'human' algorithms. By combining the worst-case scenarios for each part of these algorithms, the typical upper bound was found to be around 100.
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| Perhaps the first concrete value for an upper bound was the 277 moves mentioned by [[David Singmaster]] in early 1979. He simply counted the maximum number of moves required by his cube-solving algorithm.<ref name="the_quest">{{cite web
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| |url = http://digitaleditions.walsworthprintgroup.com/article/The_Quest_For_God%E2%80%99s_Number/532775/50242/article.html
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| |title = The Quest For God’s Number
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| |author = Rik van Grol
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| |date = November 2010
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| |accessdate = 2013-07-26
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| |publisher = Math Horizons
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| }}</ref> Later, Singmaster reported that [[Elwyn Berlekamp]], [[John Horton Conway|John Conway]], and Richard Guy had come up with a different algorithm that took at most 160 moves. Soon after, Conway’s Cambridge Cubists reported that the cube could be restored in at most 94 moves.<ref name="the_quest" /><ref name="singmaster_notes">{{cite book |author = David Singmaster |title = Notes on Rubik's Magic Cube |year = 1981 |publisher = Enslow Publishers |page = 30 }}</ref>
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| ===Thistlethwaite's algorithm===
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| The breakthrough was found by [[Morwen Thistlethwaite]]; details of [[Morwen Thistlethwaite#Thistlethwaite's algorithm|Thistlethwaite's Algorithm]] were published in ''[[Scientific American]]'' in 1981 by [[Douglas Hofstadter]]. The approaches to the cube that lead to algorithms with very few moves are based on [[group (mathematics)|group theory]] and on extensive computer searches. Thistlethwaite's idea was to divide the problem into subproblems. Where algorithms up to that point divided the problem by looking at the parts of the cube that should remain fixed, he divided it by restricting the type of moves you could execute. In particular he divided the [[Rubik's Cube group|cube group]] into the following chain of subgroups:
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| *<math>G_0=\langle L,R,F,B,U,D\rangle</math>
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| *<math>G_1=\langle L,R,F,B,U^2,D^2\rangle</math>
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| *<math>G_2=\langle L,R,F^2,B^2,U^2,D^2\rangle</math>
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| *<math>G_3=\langle L^2,R^2,F^2,B^2,U^2,D^2\rangle</math>
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| *<math>G_4=\{1\}</math>
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| Next he prepared tables for each of the right [[coset]] spaces <math>G_{i+1}\setminus G_i</math>. For each element he found a sequence of moves that took it to the next smaller group. After these preparations he worked as follows. A random cube is in the general cube group <math>G_0</math>. Next he found this element in the right [[coset]] space <math>G_1\setminus G_0</math>. He applied the corresponding process to the cube. This took it to a cube in <math>G_1</math>. Next he looked up a process that takes the cube to <math>G_2</math>, next to <math>G_3</math> and finally to <math>G_4</math>.
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| [[File:Rubik-3-facelet-kociemba.png|thumb|Intermediate state of the Rubik's Cube in Kociemba's algorithm. Any state from G<sub>1</sub> will have the "+" and "–" symbols as shown.<ref name="koci_subgroupH">{{cite web |url=http://kociemba.org/math/20moves/subgroupH.html |title=The Subgroup H and its cosets |author=Herbert Kociemba |accessdate=2013-07-28 }}</ref>|518x518px]]
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| Although the whole cube group <math>G_0</math> is very large (~4.3×10<sup>19</sup>), the right coset spaces <math>G_1\setminus G_0, G_2\setminus G_1, G_3\setminus G_2</math> and <math>G_3</math> are much smaller.
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| The coset space <math>G_2\setminus G_1</math> is the largest and contains only 1082565 elements. The number of moves required by this algorithm is the sum of the largest process in each step.
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| Initially, Thistlethwaite showed that any configuration could be solved in at most 85 moves. In January 1980 he improved his strategy to yield a maximum of 80 moves. Later that same year, he reduced the number to 63, and then again to 52.<ref name="the_quest" /> By exhaustively searching the coset spaces it was later found that the best possible number of moves for each stage was 7, 10, 13, and 15 giving a total of 45 moves at most.<ref>[http://cubeman.org/dotcs.txt Progressive Improvements in Solving Algorithms]</ref>
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| ===Kociemba's Algorithm===
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| Thistlethwaite's algorithm was improved by [[Herbert Kociemba]] in 1992. He reduced the number of intermediate groups to only two:
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| *<math>G_0=\langle U,D,L,R,F,B\rangle</math>
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| *<math>G_1=\langle U,D,L^2,R^2,F^2,B^2\rangle</math>
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| *<math>G_2=\{1\}</math>
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| As with [[Morwen Thistlethwaite#Thistlethwaite's algorithm|Thistlethwaite's Algorithm]], he would search through the right coset space <math>G_1\setminus G_0</math> to take the cube to group <math>G_1</math>. Next he searched the optimal solution for group <math>G_1</math>. The searches in <math>G_1\setminus G_0</math> and <math>G_1</math> were both done with a method equivalent to [[IDA*]]. The search in <math>G_1\setminus G_0</math> needs at most 12 moves and the search in <math>G_1</math> at most 18 moves, as Michael Reid showed in 1995. By generating also suboptimal solutions that take the cube to group <math>G_1</math> and looking for short solutions in <math>G_1</math>, you usually get much shorter overall solutions. Using this algorithm solutions are typically found of fewer than 21 moves, though there is no proof that it will always do so.
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| In 1995 Michael Reid proved that using these two groups every position can be solved in at most 29 face turns, or in 42 quarter turns. This result was improved by Silviu Radu in 2005 to 40.
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| ===Korf's Algorithm===
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| Using these group solutions combined with computer searches will generally quickly give very short solutions. But these solutions do not always come with a guarantee of their minimality. To search specifically for minimal solutions a new approach was needed.
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| In 1997 Richard Korf<ref name="korf_1997">{{cite article
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| | author = Richard Korf
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| | title = Finding Optimal Solutions to Rubik's Cube Using Pattern Databases
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| | url = http://www-compsci.swan.ac.uk/~csphil/CS335/korfrubik.pdf
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| | year = 1997
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| }}</ref> announced an algorithm with which he had optimally solved random instances of the cube. Of the ten random cubes he did, none required more than 18 face turns. The method he used is called [[IDA*]] and is described in his paper "Finding Optimal Solutions to Rubik's Cube Using Pattern Databases". Korf describes this method as follows
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| : 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 a lower bound on their length exceeds the current iterations bound.
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| It works roughly as follows. First he identified a number of subproblems that are small enough to be solved optimally. He used:
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| #The cube restricted to only the corners, not looking at the edges
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| #The cube restricted to only 6 edges, not looking at the corners nor at the other edges.
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| #The cube restricted to the other 6 edges.
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| Clearly the number of moves required to solve any of these subproblems is a lower bound for the number of moves you will need to solve the entire cube.
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| Given a [[random]] cube C, it is solved as [[iterative deepening]]. First all cubes are generated that are the result of applying 1 move to them. That is C * F, C * U, … Next, from this list, all cubes are generated that are the 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 the upper bounds to still be optimal it can be eliminated from the list.
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| Although this [[algorithm]] will always find optimal solutions there is no worst case analysis. It is not known how many moves this algorithm might need. An implementation of this algorithm can be found here.<ref>[http://www.math.ucf.edu/~reid/Rubik/optimal_solver.html Michael Reid's Optimal Solver for Rubik's Cube] (requires a compiler such as gcc)</ref>
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| ===Further improvements===
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| In 2006, Silviu Radu further improved his methods to prove that every position can be solved in at most 27 face turns or 35 quarter turns.<ref name="radu_2007">[http://cubezzz.dyndns.org/drupal/?q=node/view/53 Rubik can be solved in 27f]</ref> Daniel Kunkle and Gene Cooperman in 2007 used a [[supercomputer]] to show that all unsolved cubes can be solved in no more than 26 moves (in face-turn metric). Instead of attempting to solve each of the billions of variations explicitly, the computer was programmed to bring the cube to one of 15,752 states, each of which could be solved within a few extra moves. All were proved solvable in 29 moves, with most solvable in 26. Those that could not initially be solved in 26 moves were then solved explicitly, and shown that they too could be solved in 26 moves.<ref name="kunkle_Cooperman_2007">[http://www.neu.edu/nupr/news/0507/rubik.html Press Release on Proof that 26 Face Turns Suffice]</ref>
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| <ref>
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| {{cite conference
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| | first = D.
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| | last = Kunkle
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| | coauthors = Cooperman, C.
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| | title = Twenty-Six Moves Suffice for Rubik's Cube
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| | booktitle = Proceedings of the International Symposium on Symbolic and Algebraic Computation (ISSAC '07)
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| | year = 2007
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| | publisher = ACM Press
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| | url = http://www.ccs.neu.edu/home/gene/papers/rubik.pdf
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| | format = PDF
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| }}
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| </ref>
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| Tomas Rokicki reported in a 2008 computational proof that all unsolved cubes could be solved in 25 moves or fewer.<ref>
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| {{cite arxiv
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| |eprint=0803.3435
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| |title=Twenty-Five Moves Suffice for Rubik's Cube
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| |author=Tom Rokicki
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| }}</ref> This was later reduced to 23 moves.<ref>[http://cubezzz.dyndns.org/drupal/?q=node/view/117 Twenty-Three Moves Suffice] — Domain of the Cube Forum</ref> In August 2008 Rokicki announced that he had a proof for 22 moves.<ref name="22 moves">[http://cubezzz.dyndns.org/drupal/?q=node/view/121 twenty-two moves suffice]</ref> In 2009, Tomas Rokicki proved that 29 moves in quarter turn metric is enough to solve any scrambled cube.<ref>{{cite web |url=http://cubezzz.dyndns.org/drupal/?q=node/view/143|title=Twenty-Nine QTM Moves Suffice|author = Tom Rokicki|accessdate= 2010-02-19}}</ref> Finally, in 2010, Tomas Rokicki, Herbert Kociemba, Morley Davidson, and John Dethridge gave the final [[computer-assisted proof]] that all cube positions could be solved with a maximum of 20 face turns.<ref name="20 moves">[http://www.cube20.org/]</ref><ref>[http://epubs.siam.org/doi/abs/10.1137/120867366 SIAM J. Discrete Math., 27(2), 1082–1105]</ref>
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| ==References==
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| {{reflist}}
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| ==External links==
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| {{wikibooks|How to solve the Rubik's Cube}}
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| * [[b:How to solve the Rubik's Cube|How to solve the Rubik's Cube]], a Wikibooks article that describes an algorithm that has the advantage of being simple enough to be memorizable by humans, however it will usually not give an ''optimal'' solution which only uses the minimum possible number of moves.
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| * [http://kociemba.org/cube.htm Herbert Kociemba's Two-Phase-Solver and Optimal Solver for Rubik's Cube]
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| * [http://www.ryanheise.com/cube/human_thistlethwaite_algorithm.html Ryan Heise's Human version of the Thistlethwaite algorithm]
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| * [http://arxiv.org/abs/math.CO/0512485 A New Upper Bound on Rubik's Cube Group, Silviu Radu]
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| * [http://rubiksolve.com Online Solver using modified Kociemba's Algorithm to balance optimization vs. compute cycles]
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| {{Rubik's Cube}}
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| [[Category:Rubik's Cube]]
| |
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