Rubik's Cube

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Posted by kaori 04/17/2009 @ 23:09

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Rubik's Cube

A classic Rubik's Cube, solved.

The Rubik's Cube is a 3-D mechanical puzzle invented in 1974 by Hungarian sculptor and professor of architecture Ernő Rubik. Originally called the "Magic Cube", the puzzle was licensed by Rubik to be sold by Ideal Toys in 1980 and won the German Game of the Year special award for Best Puzzle that year. As of January 2009, 350 million cubes have sold worldwide making it the world's top-selling puzzle game. It is widely considered to be the world's best-selling toy.

In a classic Rubik's Cube, each of the six faces is covered by 9 stickers, among six solid colours (traditionally white, red, blue, orange, green, and yellow). A pivot mechanism enables each face to turn independently, thus mixing up the colours. For the puzzle to be solved, each face must be a solid colour.

In March 1970, Larry Nichols invented a 2×2×2 "Puzzle with Pieces Rotatable in Groups" and filed a Canadian patent application for it. Nichols's cube was held together with magnets. Nichols was granted U.S. patent 3,655,201  on April 11, 1972, two years before Rubik invented his improved cube.

On April 9, 1970, Frank Fox applied to patent his "Spherical 3×3×3". He received his UK patent (1344259) on January 16, 1974.

In the mid-1970s, Ernő Rubik worked at the Department of Interior Design at the Academy of Applied Arts and Crafts in Budapest. He sought to find a teaching tool to help his students understand 3D objects. Rubik invented his "Magic Cube" in 1974 and obtained Hungarian patent HU170062 for the Magic Cube in 1975 but did not take out international patents. The first test batches of the product were produced in late 1977 and released to Budapest toy shops. Magic Cube was held together with interlocking plastic pieces that were less expensive to produce than the magnets in Nichols's design. In September 1979, a deal was signed with Ideal Toys to bring the Magic Cube to the Western world, and the puzzle made its international debut at the toy fairs of London, Paris, Nuremberg and New York in January and February 1980.

After its international debut, the progress of the Cube towards the toy shop shelves of the West was briefly halted so that it could be manufactured to Western safety and packaging specifications. A lighter Cube was produced, and Ideal Toys decided to rename it. "The Gordian Knot" and "Inca Gold" were considered, but the company finally decided on "Rubik's Cube", and the first batch was exported from Hungary in May 1980. Taking advantage of an initial shortage of Cubes, many cheap imitations appeared.

Nichols assigned his patent to his employer Moleculon Research Corp., which sued Ideal Toy Company in 1982. In 1984, Ideal lost the patent infringement suit and appealed. In 1986, the appeals court affirmed the judgment that Rubik's 2×2×2 Pocket Cube infringed Nichols's patent, but overturned the judgment on Rubik's 3×3×3 Cube.

Even while Rubik's patent application was being processed, Terutoshi Ishigi, a self-taught engineer and ironworks owner near Tokyo, filed for a Japanese patent for a nearly identical mechanism, which was granted in 1976 (Japanese patent publication JP55-008192). Until 1999, when an amended Japanese patent law was enforced, Japan's patent office granted Japanese patents for non-disclosed technology within Japan without requiring worldwide novelty. Hence, Ishigi's patent is generally accepted as an independent reinvention at that time.

Rubik applied for another Hungarian patent on October 28, 1980, and applied for other patents. In the United States, Rubik was granted U.S. patent 4,378,116  on March 29, 1983, for the Cube.

Greek inventor Panagiotis Verdes patented a method of creating cubes beyond the 5×5×5, up to 11×11×11, in 2003 although he claims he originally thought of the idea around 1985. As of June 19, 2008, the 5x5x5, 6x6x6, and 7x7x7 models are in production.

The Cube celebrated its twenty-fifth anniversary in 2005, when a special edition was released, featuring a sticker in the centre of the reflective face (which replaced the white face) with a "Rubik's Cube 1980-2005" logo, and different packaging.

A standard Rubik's cube measures 5.7 cm (approximately 2¼ inches) on each side. The puzzle consists of the twenty-six unique miniature cubes, also called "cubies" or "cubelets". However, the centre cube of each of the six faces is merely a single square façade; all six are affixed to the core mechanisms. These provide structure for the other pieces to fit into and rotate around. So there are twenty-one pieces: a single core piece consisting of three intersecting axes holding the six centre squares in place but letting them rotate, and twenty smaller plastic pieces which fit into it to form the assembled puzzle. The Cube can be taken apart without much difficulty, typically by turning one side through a 45° angle and prying an edge cube away from a centre cube until it dislodges so it is a very simple process to "solve" a Cube by taking it apart and reassembling it in a solved state.

There are twelve edge pieces which show two coloured sides each, and eight corner pieces which show three colours. Each piece shows a unique colour combination, but not all combinations are present (for example, if red and orange are on opposite sides of the solved Cube, there is no edge piece with both red and orange sides). The location of these cubes relative to one another can be altered by twisting an outer third of the Cube 90°, 180° or 270°, but the location of the coloured sides relative to one another in the completed state of the puzzle cannot be altered: it is fixed by the relative positions of the centre squares and the distribution of colour. However, Cubes with alternative colour arrangements also exist; for example, they might have the yellow face opposite the green, and the blue face opposite the white (with red and orange opposite faces remaining unchanged).

Douglas R. Hofstadter, in the July 1982 issue of Scientific American, pointed out that Cubes could be coloured in such a way as to emphasise the corners or edges, rather than the faces as the standard colouring does; but neither of these alternative colourings has ever become popular.

The original (3×3×3) Rubik's Cube has eight corners and twelve edges. There are 8! (40,320) ways to arrange the corner cubes. Seven can be oriented independently, and the orientation of the eighth depends on the preceding seven, giving 37 (2,187) possibilities. There are 12!/2 (239,500,800) ways to arrange the edges, since an odd permutation of the corners implies an odd permutation of the edges as well. Eleven edges can be flipped independently, with the flip of the twelfth depending on the preceding ones, giving 211 (2,048) possibilities.

There are exactly 43,252,003,274,489,856,000 permutations, which is approximately forty-three quintillion. The puzzle is often advertised as having only "billions" of positions, as the larger numbers could be regarded as incomprehensible to many. To put this into perspective, if every permutation of a 57-millimeter Rubik's Cube were lined up end to end, it would stretch out approximately 261 light years. Alternatively, if laid out on the ground, this is enough to cover the earth with 273 layers of cubes, recognizing the fact that the radius of the earth sphere increases by 57 mm with each layer of cubes.

The full number is 519,024,039,293,878,272,000 or 519 quintillion possible arrangements of the pieces that make up the Cube, but only one in twelve of these are actually solvable. This is because there is no sequence of moves that will swap a single pair of pieces or rotate a single corner or edge cube. Thus there are twelve possible sets of reachable configurations, sometimes called "universes" or "orbits", into which the Cube can be placed by dismantling and reassembling it.

The original Rubik's Cube had no orientation markings on the centre faces, although some carried the words "Rubik's Cube" on the centre square of the white face, and therefore solving it does not require any attention to orienting those faces correctly. However, if one has a marker pen, one could, for example, mark the central squares of an unscrambled Cube with four coloured marks on each edge, each corresponding to the colour of the adjacent face. Some Cubes have also been produced commercially with markings on all of the squares, such as the Lo Shu magic square or playing card suits. Thus one can nominally solve a Cube yet have the markings on the centres rotated; it then becomes an additional test to solve the centers as well.

Marking the Rubik's Cube increases its difficulty because this expands its set of distinguishable possible configurations. When the Cube is unscrambled apart from the orientations of the central squares, there will always be an even number of squares requiring a quarter turn. Thus there are 46/2 = 2,048 possible configurations of the centre squares in the otherwise unscrambled position, increasing the total number of possible Cube permutations from 43,252,003,274,489,856,000 (4.3×1019) to 88,580,102,706,155,225,088,000 (8.9×1022).

In Rubik's cubists' parlance, a memorised sequence of moves that has a desired effect on the cube is called an algorithm. This terminology is derived from the mathematical use of algorithm, meaning a list of well-defined instructions for performing a task from a given initial state, through well-defined successive states, to a desired end-state. Each method of solving the Rubik's cube employs its own set of algorithms, together with descriptions of what the effect of the algorithm is, and when it can be used to bring the cube closer to being solved.

Most algorithms are designed to transform only a small part of the cube without scrambling other parts that have already been solved, so that they can be applied repeatedly to different parts of the cube until the whole is solved. For example, there are well-known algorithms for cycling three corners without changing the rest of the puzzle, or flipping the orientation of a pair of edges while leaving the others intact.

Some algorithms have a certain desired effect on the cube (for example, swapping two corners) but may also have the side-effect of changing other parts of the cube (such as permuting some edges). Such algorithms are often simpler than the ones without side-effects, and are employed early on in the solution when most of the puzzle has not yet been solved and the side-effects are not important. Towards the end of the solution, the more specific (and usually more complicated) algorithms are used instead, to prevent scrambling parts of the puzzle that have already been solved.

Many 3×3×3 Rubik's Cube enthusiasts use a notation developed by David Singmaster to denote a sequence of moves, referred to as "Singmaster notation". Its relative nature allows algorithms to be written in such a way that they can be applied regardless of which side is designated the top or how the colours are organised on a particular cube.

When a prime symbol ( ′ ) follows a letter, it denotes face counter-clockwise, while a letter without a prime symbol denotes a clockwise turn. A letter followed by a 2 (occasionally a superscript ²) denotes two turns, or a 180-degree turn. R is right side clockwise, but R' is right side counter-clockwise. The letters x, y, and z are used to indicate that the entire Cube should be turned about one of its axes. When x, y or z are primed, it is an indication that the cube must be rotated in the opposite direction. When they are squared, the cube must be rotated twice.

For methods using middle-layer turns (particularly corners-first methods) there is a generally accepted "MES" extension to the notation where letters M, E, and S denote middle layer turns. It was used e.g. in Marc Waterman's Algorithm.

The 4×4×4 and larger cubes use an extended notation to refer to the additional middle layers. Generally speaking, uppercase letters (F B U D L R) refer to the outermost portions of the cube (called faces). Lowercase letters (f b u d l r) refer to the inner portions of the cube (called slices). For example: (Rr)' l2 f' means to turn the two rightmost layers counterclockwise, then the left inner layer by a half-turn, and then the inner front layer counterclockwise.

Although there are a significant number of possible permutations for the Rubik's Cube, there have been a number of solutions developed which allow for the cube to be solved in well under 100 moves.

Many general solutions for the Rubik's Cube have been discovered independently. The most popular method was developed by David Singmaster and published in the book Notes on Rubik's "Magic Cube" in 1981. This solution involves solving the Cube layer by layer, in which one layer (designated the top) is solved first, followed by the middle layer, and then the final and bottom layer. After practice, solving the Cube layer by layer can be done in under one minute. Other general solutions include "corners first" methods or combinations of several other methods. In 1982, David Singmaster and Alexander Frey hypothesised that the number of moves needed to solve the Rubik's Cube, given an ideal algorithm, might be in "the low twenties". In 2007, Daniel Kunkle and Gene Cooperman used computer search methods to demonstrate that any 3×3×3 Rubik's Cube configuration can be solved in a maximum of 26 moves. In 2008, Tomas Rokicki lowered the maximum to 22 moves.

Speedcubing (or speedsolving) is the practice of trying to solve a Rubik's Cube in the shortest time possible. There are a number of speedcubing competitions that take place around the world.

The first world championship organised by the Guinness Book of World Records was held in Munich on March 13, 1981. All Cubes were moved 40 times and lubricated with petroleum jelly. The official winner, with a record of 38 seconds, was Jury Froeschl, born in Munich. The first international world championship was held in Budapest on June 5, 1982, and was won by Minh Thai, a Vietnamese student from Los Angeles, with a time of 22.95 seconds.

Since 2003, competitions are decided by the best average (middle three of five attempts); but the single best time of all tries is also recorded. The World Cube Association maintains a history of world records . In 2004, the WCA made it mandatory to use a special timing device called a Stackmat timer.

Of these informal competitions, the World Cube Association only sanctions blindfolded, one-handed, and feet solving as official competition events.

In blindfolded solving, the contestant first studies the scrambled cube (i.e., looking at it normally with no blindfold), and is then blindfolded before beginning to turn the cube's faces. Their recorded time for this event includes both the time spent examining the cube and the time spent manipulating it.

The current world record for single time on a 3×3×3 Rubik's Cube was set by Erik Akkersdijk in 2008, who had a best time of 7.08 seconds at the Czech Open 2008. The world record average solve is currently held by Tomasz Zolnowski; which is 10.63 seconds at the Warsaw Open 2009.

On December 20, 2008, 96 people in Santa Ana, CA broke the Guinness World Record for most people solving a Rubik's cube at once. The previous record was 75 people by a group in Atlanta, GA.

There are different variations of Rubik's Cubes with up to seven layers: the 2×2×2 (Pocket/Mini Cube), the standard 3×3×3 cube, the 4×4×4 (Rubik's Revenge/Master Cube), and the 5×5×5 (Professor's Cube), the 6×6×6 (V-Cube 6), and 7×7×7 (V-Cube 7).

An electronic variation of the 3×3×3 Cube is the Rubik's TouchCube. Sliding a finger across its faces causes its patterns of colored lights to rotate the same way they would on a mechanical cube. The TouchCube was introduced at the American International Toy Fair in New York. on February 15, 2009.

The Cube has inspired an entire category of similar puzzles, commonly referred to as twisty puzzles, which includes the cubes of different sizes mentioned above as well as various other geometric shapes. Some such shapes include the tetrahedron (Pyraminx), the octahedron (Skewb Diamond), the dodecahedron (Megaminx), the icosahedron (Dogic), and even someone cubes that change shape such as the Rubik's Snake and the Square One (puzzle).

In the past, puzzles have been built resembling the Rubik's Cube or based on its inner workings. For example, a cuboid is a puzzle based on the Rubik's Cube, but with different functional dimensions, such as, 2×3×4, 3×3×5, or 2×2×4. Many cuboids are based on 4×4×4 or 5×5×5 mechanisms, via building plastic extensions or by directly modifying the mechanism itself.

Some custom puzzles are not derived from any existing mechanism, such as the Gigaminx v1.5-v2, Bevel Cube, SuperX, Toru, Rua, and 1×2×3. These puzzles usually have a set of masters 3D printed, which then are copied using molding and casting techniques to create the final puzzle.

Other Rubik's Cube modifications include cubes that have been extended or truncated to form a new shape. An example of this is the Trabjer's Octahedron, which can be built by truncating and extending portions of a regular 3×3. Most shape mods can be adapted to higher-order cubes. In the case of Tony Fisher's Rhombic Dodecahedron, there are 3×3, 4×4, 5×5, and 6×6 versions of the puzzle.

Puzzles like the Rubik's Cube can be simulated by computer software, which provide functions such as recording of player metrics, storing scrambled Cube positions, conducting online competitions, analyzing of move sequences, and converting between different move notations. Software can also simulate very large puzzles that are impractical to build, such as 100×100×100 and 1,000×1,000×1,000 cubes, as well as virtual puzzles that cannot be physically built, such as 4- and 5-dimensional analogues of the cube.

Many movies and TV shows have featured characters that solve Rubik's Cubes quickly to establish their high intelligence. The Simpsons twice features a Rubik's Cube as a source of distraction for Homer and the Simpson family. Rubik's cube also regularly feature as motifs in works of art.

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Rubik's Cube in popular culture

Large Rubik's Cube built on the University of Michigan's North Campus

The Rubik's Cube, a mid-1970s invention of Ernő Rubik of Hungary fascinated people around the globe and became one of the most popular games in America at the time. In just seven years worldwide sales surpassed thirty million units with a senior buyer at the New York FAO Schwarz toy emporium noting it had become "the world's most asked-for plaything". Some even argued it could lead to obsessive behavior. Pirated editions turned up in Taiwan, Hong Kong and some American cities. The cube spawned an array of sequels, spinoffs and literary works. As of January 2009 350 million cubes have sold worldwide making it the world's top-selling puzzle game. It earned a place as a permanent exhibit in New York’s Museum of Modern Art and entered the Oxford English Dictionary after just two years. The Cube retains a cult following, with almost 40,000 entries on YouTube featuring tutorials and video clips of quick solutions.

Rubik’s cubes have, in recent, been the subject of several pop art installations. Owing to their popularity as a children’s toy several artists and groups have created large Rubik’s cubes.

Tony Rosenthal's Alamo (“The Astor Cube”) is a spinnable statue of a Cube standing in New York City. Once the cube was covered with colored panels so that it resembled a Rubik's Cube.

Similarly, the University of Michigan students covered Endover creating a large Rubik’s cube on the University of Michigan’s central campus for April fool’s day in 2008. In conjunction with the 2008 April fool’s day cube covering, a student group created a large rotating non-functional Rubik’s cube for the University of Michigan's North Campus. Built out of 600+ lbs. of steel, the cube was an entertaining addition to North Campus. Removed later the same semester, the cube reappeared in the fall of 2009 on the first day of classes. It was later removed, but in response to the cube, the university is planning on a permanent Rubik's Cube art installation on North Campus.

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Optimal solutions for Rubik's Cube

Rubik's cube scrambled.svg

Note: Notation from How to solve the Rubik's Cube is used in this article.

It is not known how many moves is the minimum required to solve any instance of the Rubik's cube. This number is also known as the 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'.

When discussing the length of a solution, there are two common ways to measure this. 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.

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.

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 conjectured that the so-called superflip would be a position that is very difficult. The superflip is a position on the cube where all the cubies are in their correct position, all the corners are correctly oriented but each edge is oriented the wrong way.

One indication that this might be the case is that it is the only element other than the identity that is in the center of the cube group.

In 1992 a solution for the superflip with 20 face turns was found by Dik T. Winter. In 1995, Michael Reid proved its minimality, thereby giving 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, its minimality was proven by Jerry Bryan.

In 1998 Michael Reid found a new position requiring more than 24 quarter turns to solve. The position, named by him as 'superflip composed with four spot' needs 26 quarter turns.

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. The breakthrough was found by Morwen Thistlethwaite; details of Thistlethwaite's Algorithm were published in Scientific American in 1981 by Douglas R. Hofstadter. The approaches to the cube that lead to algorithms with very few moves are based on 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.

Next he prepared tables for each of the right coset spaces G\Gi. 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 G0. Next he found this element in the right coset space G1\G0. He applied the corresponding process to the cube. This took it to a cube in G1. Next he looked up a process that takes the cube to G2, next to G3 and finally to G4.

Although the whole cube group G0 is very large (~4.3×1019), the right coset spaces G1\G0, G2\G1, G3\G2 and G3 are much smaller. The coset space G2\G1 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. In the original version this was 52.

As with Thistlethwaite's Algorithm, he would search through the right coset space G1\G0 to take the cube to group G1. Next he searched the optimal solution for group G1. The searches in G1\G0 and G1 were both done with a method equivalent to IDA*. The search in G1\G0 needs at most 12 moves and the search in G1 at most 18 moves, as Michael Reid showed in 1995. By generating also suboptimal solutions that take the cube to group G1 and looking for short solutions in G1, you usually get much shorter overall solutions. Using this algorithm solutions are typically found of less than 21 moves, though there is no proof that it will always do so.

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.

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.

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.

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.

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 .

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.

In August 2007, Daniel Kunkle and Gene Cooperman 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 15,000 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.

In 2008, Tomas Rokicki was reported to have devised a computational proof that all unsolved cubes could be solved in 25 moves or fewer. This was later reduced to 23 moves. In August 2008 Rokicki announced that he has proof for 22 moves.

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Rubik's Cube group

A CGI Rubik's Cube

The Rubik's Cube provides a tangible representation of a mathematical group. The Rubik's Cube group can be thought of as the set of all cube operations with function composition as the group operation. Any set of operations which returns the cube to the solved state, from the solved state, should be thought of as the identity transformation (the operation that does nothing). Any set of operations which solves the cube from a scrambled state should be thought of as an inverse transformation of the given scrambled state, since it returns the identity transformation.

Formally, the Rubik's Cube group can be defined as a permutation group. A 3×3×3 Rubik's cube consists of 6 faces, each with 9 colored squares called facets for a total of 54 facets. However, the 6 facets in the center of the faces are not moved by any cube operation and may be regarded as fixed in space.

The cube operations consist of rotating the 6 faces and thereby permuting the remaining 48 facets. The cube group G can then be defined as the subgroup of the full symmetric group S48 generated by the 6 face rotations.

By definition, each element of the cube group is a permutation of the 48 movable facets. However, there is a one-to-one correspondence between elements of the cube group and positions of the Rubik's cube. Any element of the cube group is a permutation that when applied to the solved cube results in a (legal) cube position. Conversely, any legal cube position must be the result of some sequence of face rotations applied to the solved cube, and any such sequence is an element of the cube group.

Because of the large size of the cube group it is sometimes useful to analyse the structure with the assistance of a computer algebra system such as GAP.

Let Cube be the group of all legal cube operations. In the following, we assume the notation described in How to solve the Rubik's Cube. Also we assume the orientation of the six centre pieces to be fixed.

Next we can take a closer look at these two groups. Co is an abelian group, it is .

This group can also be described as the subdirect product , in the notation of Griess. It does not make sense to take the possible permutations of the centre pieces into account, as this is simply an artifact of the orientation of the cube in Euclidean 3D-space.

The first factor is accounted for solely by rotations of the centre pieces, the second solely by symmetries of the corners, and the third solely by symmetries of the edges. The latter two factors are examples of wreath products.

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Source : Wikipedia