Stanley Burrell
10-22-2008, 04:33 PM
So can we talk about the cube?
http://www.degraeve.com/wallpaper/rubik4.jpg
The Rubik's Cube is a mechanical puzzle invented in 1974[1] by Hungarian sculptor and professor of architecture Ernő Rubik. Originally called the "Magic Cube" by its inventor, Ernő Rubik, this puzzle was renamed "Rubik's Cube" by Ideal Toys in 1980[1] and won the German Game of the Year special award for Best Puzzle that year. It is said to be the world's best-selling toy, with over 300,000,000 Rubik's Cubes being sold worldwide to date.[2]
In a classic Rubik's Cube, each of the six faces is covered by 9 stickers, among six solid colours (traditionally being white, yellow, orange, red, blue, and green). 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.
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.
There exist four widely available variations of the Cube: the 2×2×2 (Pocket Cube, also Mini Cube, Junior Cube, or Ice Cube), the standard 3×3×3 cube, the 4×4×4 (Rubik's Revenge, or Master Cube), and the 5×5×5 (Professor's Cube). Recently, larger sizes are also on the market (V-Cube 6 and V-Cube 7).
For readability, 3x3x3 is frequently abbreviated 3×3 (and similarly for the other sizes) when there is no ambiguity. Common misspellings include "Rubix cube", "Rubics cube", "Rubick's cube", and "Rubiks cube".
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 such as the tetrahedron (e. g., the Pyraminx), the octahedron (e. g., the Skewb Diamond), the dodecahedron (e. g., the Megaminx), and the icosahedron (e. g., the Dogic). There are even higher-dimensional virtual puzzles, simulated by computer software that lets the user manipulate objects such as a 4-dimensional Rubik's cube, which cannot be physically built.
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.
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.[3]
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 and was granted (Japanese patent publication JP55-008192) in 1976 (55th year of Showa era). Until 1999 amended Japanese patent law was enforced, Japan patent office granted Japanese patent for non-disclosed technology within Japan (not required worldwide novelty until 1999)[4][5], Ishigi's is generally accepted as an independent reinvention at that time.[6][7][8]
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. His designs, which include improved mechanisms for the 3×3×3, 4×4×4, and 5×5×5, are suitable for speedcubing, whereas existing designs for cubes larger than 5×5×5 are prone to break. As of June 19, 2008, 5x5x5, 6x6x6, and 7x7x7 models are available.
A standard 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 face is merely a single square façade; all 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. However, as prying loose a corner cube is a good way to break off a centre cube — thus ruining the Cube — it is far safer to lever a centre cube out using a screwdriver. 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 combinations on edge and corner pieces.
For most recent Cubes, the colours of the stickers are red opposite orange, yellow opposite white, and green opposite blue. 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. Hofstader, in the July 1982 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 been produced commercially.
A normal (3×3×3) Rubik's Cube has eight corners and twelve edges. There are 8! ways to arrange the corner cubies. Seven can be oriented independently, and the orientation of the eighth depends on the preceding seven, giving 37 possibilities. There are 12!/2 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 possibilities.
The full number is 519,024,039,293,878,272,724 or 519 quintillion (on the short scale) 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.
Despite the vast number of positions, all Cubes can be solved in twenty-five or fewer moves (see Optimal solutions for Rubik's Cube). [10] [11] The large number of permutations is often given as a measure of the Rubik's cube's complexity. However, the puzzle's difficulty does not necessarily follow from the large number of permutations; the constraint imposed by the permitted moves is the more significant factor. For example, the number of permutations of the 26 letters of the alphabet (26! = 4.03 × 1026) is larger than that of the Rubik's cube, but a significantly simpler problem is that of sorting a permutation of the 26 letters into alphabetical order when any swap of neighboring letters is permitted.
The original (official) Rubik's Cube has 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 unshuffled 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 scramble and then unscramble the Cube yet have the markings on the centers rotated, and it becomes an additional test to "solve" the centers as well. This is known as "supercubing"[citation needed].
Putting markings on the Rubik's Cube increases the difficulty mainly because it expands the 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 454,988,580,102,706,155,225,088,000 (8.9×1022).
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. Most tutorials teach the layer by layer method, as it gives an easy-to-understand step-by-step guide on how to solve it.
Most 3×3×3 Rubik's Cube solution guides use the same notation, originated by David Singmaster, to communicate sequences of moves. This is generally referred to as "cube notation" or in some literature "Singmaster notation" (or variations thereof), or sometimes (but rarely) it is called "direction inferred notation" or "DIN". 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 organized on a particular cube.
F (Front): the side currently facing you
B (Back): the side opposite the front
U (Up): the side above or on top of the front side
D (Down): the side opposite the top, underneath the Cube
L (Left): the side directly to the left of the front
R (Right): the side directly to the right of the front
f (Front two layers): the side facing you and the corresponding middle layer
b (Back two layers): the side opposite the front and the corresponding middle layer
u (Up two layers) : the top side and the corresponding middle layer
d (Down two layers) : the bottom layer and the corresponding middle layer
l (Left two layers) : the side to the left of the front and the corresponding middle layer
r (Right two layers) : the side to the right of the front and the corresponding middle layer
x (rotate): rotate the Cube up
y (rotate): rotate the Cube to the left
z (rotate): rotate the Cube on its side to the right
When an apostrophe follows a letter, it means to turn the face counter-clockwise a quarter-turn, while a letter without an apostrophe means to turn it a quarter-turn clockwise. Such an apostrophe mark is pronounced prime. A letter followed by a 2 (occasionally a superscript ²) means to turn the face a half-turn (the direction does not matter). So R is right side clockwise, but R' is right side counter-clockwise. When x, y or z are primed, simply rotate the cube in the opposite direction. When they are squared, rotate it twice. For 'z', you should still be viewing the same front face when rotating.
This notation can also be used on the Pocket Cube, and can be extended for the Revenge and Professor cubes (see below).
Less-often used moves include rotating the entire Cube or two-thirds of it. The letters x, y, and z are used to indicate that the entire Cube should be turned about one of its axes. The x-axis is the line that passes through the left and right faces, the y-axis is the line that passes through the up and down faces, and the z-axis is the line that passes through the front and back faces. (This type of move is used infrequently in most solutions, to the extent that some solutions simply say "stop and turn the whole cube upside-down" or something similar at the appropriate point.)
For example, the algorithm F2 U' R' L F2 R L' U' F2, which cycles the top left, top front, and top right "cubies" in a counter-clockwise manner without affecting any other part of the cube, means the following sequence of moves:
Turn the Front face 180 degrees.
Turn the Up face 90 degrees counter-clockwise.
Turn the Right face 90 degrees counter-clockwise.
Turn the Left face 90 degrees clockwise.
Turn the Front face 180 degrees.
Turn the Right face 90 degrees clockwise.
Turn the Left face 90 degrees counter-clockwise.
Turn the Up face 90 degrees counter-clockwise.
Finally, turn the Front face 180 degrees.
For beginning students of the Cube, this notation can be daunting, and many solutions available online therefore incorporate animations that demonstrate the algorithms presented.
[edit] Extension to 4×4×4 and larger cubes
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.
[edit] Alternative notation systems
Some solution guides, including Ideal's official publication, The Ideal Solution, use slightly different conventions. Top and Bottom are used rather than Up and Down for the top and bottom faces, with Back being replaced by Posterior. '+' indicates clockwise rotation and '-' counter-clockwise, with '++' representing a half-turn. For the 4×4×4, the solution is described in terms of layers (vertical slices that rotate about the z-axis), tables (horizontal slices), and books (vertical slices that rotate about the x-axis).
Another less common system of move notation is designed to make it less daunting to new cube solvers. It is called direction displayed notation or DDN for short. Each move is represented by two letters. The first indicates which face to turn, and the second indicates which direction to turn it from the F point of view.
F (Front): The side facing you. A following R means turn it right or clockwise, and a following 'L' means turn it left or counter-clockwise.
U (Up): The top face. A following R means turn it to the right (in other words, clockwise), and an L means turn it to the left (counter-clockwise).
D (Down): The bottom face. A following R means turn it to the right (counter-clockwise when viewed from the bottom), and an L means turn it to the left (clockwise).
R (Right): The right face. Followed by D to mean turning it downward (that is, clockwise when viewed from the right) or U to mean turning it upward (counter-clockwise).
L (Left): The left face. Followed by D to mean turning it downward (counter-clockwise when viewed from the left), or U to mean turning it upward (clockwise).
B (Back): The face on the far side of the cube. Followed by R to mean turning it right (clockwise when viewed from the front), or L to mean turning it to the left (counter-clockwise when viewed from the front).
A half turn is indicated by writing 2 after the first letter. To indicate rotation of the cube as a whole, the same notation is used as in the Singmaster notation (x, y, and z).
The lowercase letters f, b, u, d, l, and r signify to turn the first two layers of the designated face while holding the third layer in place. This is equivalent to rotating the whole cube in that direction, then turning that third face back in the opposite direction; however, this notation is useful for describing certain triggers in speedcubing.
The letters M, E, and S (and respectively their lowercase for larger sized cubes) are used for indicating turns of the center slices of the Cube.
M means to turn the layer that is between L and R downward (clockwise if looking from the left side).
E means to turn the layer between U and D towards the right (counter-clockwise if looking from the top).
S means to turn the layer between F and B clockwise.
These alternative notations failed to catch on, and today the Singmaster scheme is used universally by those interested in the puzzle.
[edit] Algorithms
In Rubik's cubists' parlance, a memorized 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. Different methods of solving the Rubik's cube each employ 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.
[edit] Speedcubing solutions
Speedcubing solutions have been developed for solving the Rubik's Cube as quickly as possible.
The most common speedcubing solution was developed by Jessica Fridrich. It is a very efficient layer-by-layer method that requires a large number of algorithms, especially for orienting and permuting the last layer. The first-layer corners and second layer are done simultaneously, with each corner paired up with a second-layer edge piece.
Another well-known method was developed by Lars Petrus. In this method, a 2×2×2 section is solved first, followed by a 2×2×3, and then the incorrect edges are solved using a three-move algorithm, which eliminates the need for a possible 32-move algorithm later. One of the advantages of this method is that it tends to give solutions in fewer moves. For this reason, the method is also popular for fewest move competitions.
Basic solutions require learning as few as four or five algorithms, but are generally inefficient, needing around 100 twists on average to solve an entire Cube. In comparison, Fridrich's advanced solution requires learning roughly 120 algorithms but allows the Cube to be solved in only 55 moves on average. A different kind of solution developed by Ryan Heise[12] uses no algorithms but rather teaches a set of underlying principles that can be used to solve the Cube in fewer than 40 moves. A number of complete solutions can also be found in any of the books listed in the bibliography, and most can be used to solve any Cube in under five minutes.
[edit] The search for optimal solutions
Main article: Optimal solutions for Rubik's Cube
The manual solution methods described above are intended to be easy to learn, but much effort has gone into finding even faster solutions to the Rubik's Cube.
In 1982, David Singmaster and Alexander Frey hypothesized 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. [10] [13] In 2008, Tomas Rokicki lowered the maximum to 22 moves. [14] [15] [16] Work continues to try to reduce the upper bound on optimal solutions. The arrangement known as the super-flip, where every edge is in its correct position but flipped, requires 20 moves to be solved (Using the notations explained above, these are: U R2 F B R B2 R U2 L B2 R U' D' R2 F R' L B2 U2 F2.). No arrangement of the Rubik's Cube has been discovered so far that requires more than 20 moves to solve.
[edit] Competitions and record times
Many speedcubing competitions have been held to determine who can solve the Rubik's Cube in the shortest time. The number of contests is going up every year; there were 72 official competitions from 2003 to 2006; 33 were in 2006 alone.
The first world championship organized 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 [17]. In 2004, the WCA made it mandatory to use a special timing device called a Stackmat timer.
The current world record for single time is set by Erik Akkersdijk in 2008, he set a best time of 7.08 at the Czech Open 2008. The world record average solve is by Yu Nakajima, when he set a world record average of 11.28 seconds on May 4, 2008.
[edit] Alternative competitions
In addition, informal alternative competitions have been held, inviting participants to solve the Cube under unusual situations. These include:
Blindfolded solving[18]
Solving the Cube with one person blindfolded and the other person saying what moves to do, known as "Team Blindfold"
Solving the Cube underwater in a single breath[19]
Solving the Cube using a single hand[20]
Solving the Cube with one's feet[21]
Of these informal competitions, the World Cube Association only sanctions blindfolded, one-handed, and feet solving as official competition events.[22]
[edit] Custom built puzzles
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 off the Rubik's Cube, but with different functional dimensions, such as, 2x3x4, 3x3x5, or 2x2x4. Many cuboids are based on 4x4x4 or 5x5x5 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 1x2x3. 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 3x3. Most shape mods can be adapted to higher-order cubes. In the case of Tony Fisher's Rhombic Dodecahedron, there are 3x3, 4x4, 5x5, and 6x6 versions of the puzzle.
[edit] Mèffert's Puzzles
Mèffert's Puzzles is a well known distributor of twisty puzzles. Founded by Uwe Mèffert, Mèffert's Puzzles has carried many unique puzzles over the years.
Mèffert's products include the Pyramorphix, which is isomorphic with the Pocket Cube but has the shape of a tetrahedron (four of the Pocket Cube's corners have been turned into tetrahedra, the other four into flat triangles). This makes the Pyramorphix somewhat more difficult to solve than the Pocket Cube, as the isomorphism is not obvious. Mèffert has also been a long time producer of the Pyraminx, another tetrahedral twisty puzzle. More unique puzzles he has produced over the years include the Pyraminx Crystal, Impossiball, Megaminx, Skewb, and variations of the Skewb.
[edit] Rubik's Cube software
Several computer programs have been written to perform various functions, such as, among other things, solving the Cube or animating it. In general, these programs can be considered to fall in one of several categories:
Timers
Solvers
Graphical programs
Animations
Image generators
Analyzers
Some of the software handles not only the 3×3×3 cube, but also other puzzle types. There is even software for virtual puzzles that do not have a real life counterpart. Examples are the four-dimensional cube, the five-dimensional cube and the gliding cube.
In addition these programs may also record player metrics, store and generate scrambled Cube positions or offer either animations or online competition. Solvers are usually given a scramble, after which a solution is generated automatically. Graphical programs can generate a static image or animate the Cube and its motions, e.g. using Java or Flash. Programs may also analyze sequences of moves and transform them to other notations or give player metrics.
[edit] See also
N-dimensional sequential move puzzles
Rubik, the Amazing Cube
[edit] References
Cube Games by Don Taylor & Leanne Rylands
Four-Axis Puzzles by Anthony E. Durham.
Handbook of Cubik Math by Alexander H. Frey, Jr. and David Singmaster
Mathematics of the Rubik's Cube Design by Hana M. Bizek; ISBN 0-8059-3919-9
Mastering Rubik's Cube by Don Taylor
Metamagical Themas by Douglas R. Hofstadter contains two insightful chapters regarding Rubik's Cube and similar puzzles, "Magic Cubology" and "On Crossing the Rubicon", originally published as articles in the March 1981 and July 1982 issues of Scientific American; ISBN 0-465-04566-9
Notes on Rubik's 'Magic Cube' by David Singmaster; ISBN 0-89490-043-9
Rubik's Cube Made Easy by Jack Eidswick, Ph.D.
The Simple Solution To Rubik's Cube by James Nourse
Speedsolving The Cube by Dan Harris
Teach yourself cube-bashing, a layered solution by Colin Cairns and Dave Griffiths, from September 1979, cited by Singmaster's 'Notes on Rubik's Magic Cube'
Unscrambling The Cube by M. Razid Black & Herbert Taylor, Introduction by Professor Solomon W. Golomb
[edit] Notes
^ a b Rubik's Official Online Site
^ Marshall, Ray. Squaring up to the Rubchallenge. icNewcastle. Retrieved August 15, 2005.
^ Moleculon Research Corporation v. CBS, Inc.
^ http://www.wipo.int/clea/en/text_html.jsp?lang=EN&id=2657 Japan: Patents (PCT), Law (Consolidation), 26/04/1978 (22/12/1999), No. 30 (No. 220)
^ http://www.patents.jp/Archive/20030210-02.pdf Major Amendments to the Japanese Patent Law (since 1985)
^ Hofstadter, Douglas R. (1985). Metamagical Themas. Basic Books. Hofstadter gives the name as "Ishige".
^ http://cubeman.org/cchrono.txt
^ The History of Rubik's Cube - Erno Rubik
^ Martin Schönert "Analyzing Rubik's Cube with GAP": the permutation group of Rubik's Cube is examined with GAP computer algebra system
^ a b Kunkle, D.; Cooperman, C. (2007). "Twenty-Six Moves Suffice for Rubik's Cube". Proceedings of the International Symposium on Symbolic and Algebraic Computation (ISSAC '07), ACM Press.
^ KFC (2008). "Rubik’s cube proof cut to 25 moves"..
^ Ryan Heise's method
^ Julie J. Rehmeyer. "Cracking the Cube". MathTrek. Retrieved on 2007-08-09.
^ Tom Rokicki. "Twenty-Five Moves Suffice for Rubik's Cube". Retrieved on 2008-03-24.
^ "Rubik's Cube Algorithm Cut Again, Down to 23 Moves". Slashdot. Retrieved on 2008-06-05.
^ Tom Rokicki. "Twenty-Two Moves Suffice". Retrieved on 2008-08-20.
^ "World Cube Association Official Results". World Cube Association. Retrieved on 2008-02-16.
^ Rubik's 3x3x3 Cube: Blindfolded records
^ Rubik's Cube 3x3x3: Underwater
^ Rubik's 3x3x3 Cube: One-handed
^ Rubik's 3x3x3 Cube: With feet
^ "Competition Regulations, Article 9: Events". World Cube Association (2008-04-09). Retrieved on 2008-04-16.
[edit] External links
Rubik's Cube at the Open Directory Project
Wikimedia Commons has media related to:
Rubik's cubeRubik's official site
World Cube Association (WCA)
Speedcubing.com
Disassembly Photographs of a Rubik's Cube
How to Solve a Rubik’s Cube
[hide]v • d • eRubik's Cube
Inventor Ernő Rubik
Rubik's Cubes Overview • 2×2×2 (Pocket Cube) • 3×3×3 (Rubik's Cube) • 4×4×4 (Rubik's Revenge) • 5×5×5 (Professor's Cube) • 6×6×6 (V-Cube 6) • 7×7×7 (V-Cube 7)
Cubic variations Square 1 • Skewb • Sudoku Cube
Non-cubic variations Megaminx • Impossiball • Alexander's Star • Pyramorphix • Pyraminx • Skewb Diamond • Skewb Ultimate • Dogic • Pyraminx Crystal
Higher-dimensional
virtual variations MagicCube4D • MagicCube5D • Magic 120-cell
Derivatives Rubik's Magic • Rubik's Snake • Missing Link • Rubik's Revolution • Rubik's Clock
World record holders Yu Nakajima • Erik Akkersdijk
Renowned Solvers David Singmaster • Jessica Fridrich • Lars Petrus • Ron van Bruchem • Chris Hardwick • Shotaro "Macky" Makisumi • Tyson Mao • Toby Mao • Leyan Lo
Solutions God's algorithm • Optimal • Speedcubing
Mathematics Rubik's Cube group
Official Organization World Cube Association
Related Articles List of Rubik's Cube software
Retrieved from "http://en.wikipedia.org/wiki/Rubik%27s_Cube"
Categories: Rubik's Cube | Combination puzzles | Mechanical puzzles | Puzzles | Spiel des Jahres winners | Educational toys | 1980s fads | Toys of the 1980s
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