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Rubik's Cube
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factoids
Rubik's Cube is a 3-D combination puzzle invented in 1974BOOK, William Fotheringham, Fotheringham's Sporting Pastimes, Anova Books, 2007, 50, 1-86105-953-1, WEB, de Castella, Tom, The people who are still addicted to the Rubik's Cube,weblink BBC News Magazine, BBC, 28 April 2014, by Hungarian sculptor and professor of architecture Ernő Rubik. Originally called the Magic Cube,'Driven mad' Rubik's nut weeps on solving cube... after 26 years of trying, Daily Mail Reporter, 12 January 2009. the puzzle was licensed by Rubik to be sold by Ideal Toy Corp. in 1980BOOK, Daintith, John, A Biographical Encyclopedia of Scientists, Institute of Physics Pub, Bristol, 1994, 771, 0-7503-0287-9, via businessman Tibor Laczi and Seven Towns founder Tom Kremer,WEB,weblink History of the Cube, Michael Shanks, 8 May 2005, 26 July 2012, Stanford University, yes,weblink" title="web.archive.org/web/20130120065022weblink">weblink 20 January 2013, and won the German Game of the Year special award for Best Puzzle that year. {{As of|January 2009}}, 350 million cubes had been sold worldwideNEWS, William Lee Adams, The Rubik's Cube: A Puzzling Success,weblink Time (magazine), Time, 28 January 2009, 5 February 2009,weblink" title="web.archive.org/web/20090201200141weblink">weblink 1 February 2009, NEWS, Alastair Jamieson, Rubik's Cube inventor is back with Rubik's 360,weblink The Daily Telegraph, 31 January 2009, 5 February 2009, London, making it the world's top-selling puzzle game.NEWS, eGames, Mindscape Put International Twist on Rubik's Cube PC Game,weblink Reuters, 6 February 2008, 6 February 2009, yes,weblink" title="web.archive.org/web/20090212135842weblink">weblink 12 February 2009, Marshall, Ray. Squaring up to the Rubchallenge. icNewcastle. Retrieved 15 August 2005. It is widely considered to be the world's best-selling toy.NEWS, Rubik's Cube 25 years on: crazy toys, crazy times,weblink The Independent, 16 August 2007, 6 February 2009, London, On the original classic Rubik's Cube, each of the six faces was covered by nine stickers, each of one of six solid colours: white, red, blue, orange, green, and yellow. The current version of the cube has been updated to coloured plastic panels instead, which prevents peeling and fadingWEB, Rubik's 3x3x3,weblink 12 April 2018, . In currently sold models, white is opposite yellow, blue is opposite green, and orange is opposite red, and the red, white and blue are arranged in that order in a clockwise arrangement.BOOK, Michael W. Dempsey, Growing up with science: The illustrated encyclopedia of invention, Marshall Cavendish, London, 1988, 1245, 0-87475-841-6, On early cubes, the position of the colours varied from cube to cube.BOOK, Ewing, John, Puzzle It Out: Cubes, Groups and Puzzles, 1982, Press Syndicate of the University of Cambridge, Cambridge, 0 521 28924 6, 4,weblink Czes Kosniowski, 19 May 2014, An internal pivot mechanism enables each face to turn independently, thus mixing up the colours. For the puzzle to be solved, each face must be returned to have only one colour. Similar puzzles have now been produced with various numbers of sides, dimensions, and stickers, not all of them by Rubik.Although the Rubik's Cube reached its height of mainstream popularity in the 1980s, it is still widely known and used. Many speedcubers continue to practice it and similar puzzles: they also compete for the fastest times in various categories. Since 2003, the World Cube Association, the Rubik's Cube's international governing body, has organised competitions worldwide and recognise world records.

Conception and development

Prior attempts

(File:Patent US3655201.png|thumb|Diagram from Nichols' patent showing a cube held together with magnets)In March 1970, Larry D. 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 by magnets. Nichols was granted {{US patent|3655201}} on 11 April 1972, two years before Rubik invented his Cube.On 9 April 1970, Frank Fox applied to patent his "Spherical 3×3×3". He received his UK patent (1344259) on 16 January 1974.WEB, Patent Specification 1344259,weblink 15 June 2012,

Rubik's invention

(File:1980-Rubik's-Cube.jpg|thumb|Packaging of Rubik's Cube, Toy of the year 1980–Ideal Toy Corp., made in Hungary)In the mid-1970s, Ernő Rubik worked at the Department of Interior Design at the Academy of Applied Arts and Crafts in Budapest.BOOK, Kelly Boyer Sagert, The 1970s (American Popular Culture Through History), Greenwood Press, Westport, Conn, 2007, 130, 0-313-33919-8, Although it is widely reported that the Cube was built as a teaching tool to help his students understand 3D objects, his actual purpose was solving the structural problem of moving the parts independently without the entire mechanism falling apart. He did not realise that he had created a puzzle until the first time he scrambled his new Cube and then tried to restore it.WEB,weblink Rubik's Cube, PuzzleSolver, 1 December 2006, 20 June 2012, Rubik obtained Hungarian patent HU170062 for his "(wikt:magic cube|Magic Cube)" in 1975.Rubik's Cube was first called the Magic Cube (Bűvös kocka) in Hungary.The first test batches of the Magic Cube were produced in late 1977 and released in Budapest toy shops. Magic Cube was held together with interlocking plastic pieces that prevented the puzzle being easily pulled apart, unlike the magnets in Nichols's design. With Ernő Rubik's permission, businessman Tibor Laczi took a Cube to Germany's Nuremberg Toy Fair in February 1979 in an attempt to popularise it.BOOK, Holper, Paul, Inventing Millions, 2006, 64–5, Orient, 8122204589, It was noticed by Seven Towns founder Tom Kremer and they signed a deal with Ideal Toys in September 1979 to release the Magic Cube worldwide. Ideal wanted at least a recognisable name to trademark; of course, that arrangement put Rubik in the spotlight because the Magic Cube was renamed after its inventor in 1980. 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 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.

Subsequent history

{{See also|Rubik's Cube in popular culture}}

1980s Cube craze

After the first batches of Rubik's Cubes were released in May 1980, initial sales were modest, but Ideal began a television advertising campaign in the middle of the year which it supplemented with newspaper adverts.NEWS,weblink Dougherty, Philip H., Advertising Ideal Toy's Son of Rubik Cube, 30 July 1981, The New York Times, At the end of 1980 Rubik's Cube won a German Game of the Year special award,BOOK, Carlisle, Rodney P., Encyclopedia of Play in Today's Society, 2009, 612, SAGE, 1452266107, and won similar awards for best toy in the UK, France, and the US.WEB,weblink Interview with ErnÅ‘ Rubik, Europa, 26 October 2016, By 1981 Rubik's Cube had become a craze, and it is estimated that in the period from 1980 to 1983 around 200 million Rubik's Cubes were sold worldwide.BOOK, Singmaster, David, David Singmaster, The Utility of Recreational Mathematics, Guy, Richard K., Woodrow, Robert E., The Lighter Side of Mathematics: Proceedings of the Eugène Strens Memorial Conference on Recreational Mathematics and Its History, 1994, 340, Cambridge University Press, 088385516X, Singmaster estimates the numbers sold were between 100 and 300 million. His estimate is based on sales of 50 to 100 million legitimate cubes and perhaps a higher number of imitations. In March 1981 a speedcubing championship organised by the Guinness Book of World Records was held in Munich, and a Rubik's Cube was depicted on the front cover of Scientific American that same month.BOOK, Batchelor, Bob, Stoddart, Scott, The 1980s, 2007, 97, Greenwood, 031333000X, In June 1981 The Washington Post reported that the Rubik's Cube is "a puzzle that's moving like fast food right now ... this year's Hoola Hoop or Bongo Board",NEWS,weblink Allen, Henry, The Cube, 10 June 1981, The Washington Post, and by September 1981 New Scientist noted that the cube had "captivated the attention of children of ages from 7 to 70 all over the world this summer."NEWS,weblink Herman, Ros, Cubic mastery, 10 September 1981, New Scientist, As most people could only solve one or two sides, numerous books were published including David Singmaster's Notes on Rubik's "Magic Cube" (1980) and Patrick Bossert's You Can Do the Cube (1981). At one stage in 1981 three of the top ten best selling books in the US were books on solving the Rubik's Cube,BOOK, Singmaster, David, David Singmaster, The Utility of Recreational Mathematics, Guy, Richard K., Woodrow, Robert E., The Lighter Side of Mathematics: Proceedings of the Eugène Strens Memorial Conference on Recreational Mathematics and Its History, 1994, 340, Cambridge University Press, 088385516X, and the best-selling book of 1981 was James G. Nourse's The Simple Solution to Rubik's Cube which sold over 6 million copies.NEWS,weblink Hanauer, Joan, The man who wrote the best-selling book of 1981, 5 January 1982, United Press International, In 1981 the Museum of Modern Art in New York exhibited a Rubik's Cube, and at the 1982 World's Fair in Knoxville, Tennessee a six-foot Cube was put on display. ABC Television even developed a cartoon show called Rubik, the Amazing Cube.BOOK, Terrace, Vincent, Encyclopedia of Television Shows, 1925 through 2010, 2008, 915, McFarland, 0786486414, In June 1982 the First Rubik's Cube World Championship took place in Budapest, and would become the only competition recognized as official until the championship was revived in 2003.BOOK, Scheffler, Ian, Cracking the Cube, 2016, 88, Simon and Schuster, 1501121944, In October 1982 The New York Times reported that sales had fallen and that "the craze has died",NEWS,weblink Rubik's Cube: A Craze Ends, 30 October 1982, The New York Times, and by 1983 it was clear that sales had plummeted. However, in some Communist countries, such as China and USSR, the craze had started later and demand was still high because of a shortage of Cubes.NEWS,weblink China facing Rubik Cube shortage, 22 February 1982, United Press International, NEWS,weblink Reed, Steven R., Russians Queue for Rubik's Cube, 8 December 1982, United Press International,

21st-century revival

Rubik's Cubes continued to be marketed and sold throughout the 1980s and 90s, but it was not until the early 2000s that interest in the Cube began increasing again.BOOK, Harris, Dan, Speedsolving the Cube, 2008, 3, Sterling, 1402753136, In the US sales doubled between 2001 and 2003, and The Boston Globe remarked that it was "becoming cool to own a Cube again".NEWS,weblink Noonan, Erica, Let's twist again, 8 November 2003, Boston Globe, The 2003 World Rubik's Games Championship was the first speedcubing tournament since 1982. It was held in Toronto and was attended by 83 participants. The tournament led to the formation of the World Cube Association in 2004. Annual sales of Rubik branded cubes were said to have reached 15 million worldwide in 2008.NEWS,weblink Quenqua, Douglas, Rubik's Cube Twists Back into Limelight, 6 August 2012, The New York Times, Part of the new appeal was ascribed to the advent of Internet video sites, such as YouTube, which allowed fans to share their solving strategies. Following the expiration of Rubik's patent in 2000, other brands of cubes appeared, especially from Chinese companies.NEWS,weblink Hookway, James, One Cube, Many Knockoffs, Quintillions of Possibilities, 14 December 2011, The Wall Street Journal, Many of these Chinese branded cubes have been engineered for speed and are favoured by speedcubers.

Imitations

Taking advantage of an initial shortage of Cubes, many imitations and variations appeared, many of which may have violated one or more patents. Today, the patents have expired and many Chinese companies produce copies of, and in nearly all cases, improvements upon, the Rubik and V-Cube designs.

Patent history

Nichols assigned his patent to his employer Moleculon Research Corp., which sued Ideal 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.WEB,weblink Moleculon Research Corporation v. CBS, Inc, Digital-law-online.info, 20 June 2012, 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.Japan: Patents {{webarchive|url=https://archive.is/20090212000345weblink |date=12 February 2009 }} (PCT), Law (Consolidation), 26 April 1978 (22 December 1999), No. 30 (No. 220)WEB,weblink Major Amendments to the Japanese Patent Law (since 1985), PDF, 20 June 2012, yes,weblink" title="web.archive.org/web/20120216125521weblink">weblink 16 February 2012, Hence, Ishigi's patent is generally accepted as an independent reinvention at that time.BOOK, Metamagical Themas: Questing for the Essence of Mind and Pattern, Hofstadter, Douglas R., Douglas Hofstadter, 1985, Basic Books, New York, 0-465-04566-9, Hofstadter gives the name as 'Ishige'., Rubik's Cube Chronology Researched and maintained by Mark Longridge (c) 1996-2004WEB,weblink The History of Rubik's Cube – Erno Rubik, Inventors.about.com, 9 April 2012, 20 June 2012, Rubik applied for more patents in 1980, including another Hungarian patent on 28 October. In the United States, Rubik was granted {{US patent|4378116}} on 29 March 1983, for the Cube. This patent expired in 2000.Greek inventor Panagiotis Verdes patentedVerdes, PK, Cubic logic game, Greek patent GR1004581, filed 21 May 2003, issued 26 May 2004. a method of creating cubes beyond the 5×5×5, up to 11×11×11, in 2003.WEB,weblink Panagiotis Verdes – Inventor of V-CUBE™ Technology, Verdes Innovations, 22 March 2015, As of 2017, the 5×5×5, 6×6×6, 7×7×7, 8×8×8 and 9×9×9 models are in production in his "V-Cube" line. V-Cube also produces a 2×2×2, 3×3×3 and a 4×4×4.

Trademarks

Rubik's Brand Ltd. also holds the registered trademarks for the word Rubik and Rubik's and for the 2D and 3D visualisations of the puzzle. The trademarks have been upheld by a ruling of the General Court of the European Union on 25 November 2014 in a successful defence against a German toy manufacturer seeking to invalidate them. However, European toy manufacturers are allowed to create differently shaped puzzles that have a similar rotating or twisting functionality of component parts such as for example Skewb, Pyraminx or Impossiball.NEWS, Stephanie Bodoni, Rubik's Cube Wins Trademark Toy Story at EU Court,weblink Bloomberg L.P., Bloomberg, 25 November 2014, 13 December 2014, On 10 November 2016, Rubik's Cube lost a ten-year battle over a key trademark issue. The European Union's highest court, the Court of Justice ruled that the puzzle's shape was not sufficient to grant it trademark protection.NEWS, Rubik's Cube puzzled after losing EU trademark battle,weblink Rebecca Smithers, 10 November 2016, The Guardian, 8 December 2016,

Mechanics

(File:Disassembled-rubix-1.jpg|thumb|Rubik's Cube partially disassembled)(File:Disassembled Rubik's Cube on table, 16 June 2013.png|thumb|Rubik's Cube fully disassembled)(File:Rubiks cube scrambled.jpg|thumb|Rubik's Cube in scrambled state)A standard Rubik's Cube measures {{Convert|5.7|cm|frac=4}} on each side. The puzzle consists of twenty-six unique miniature cubes, also called "cubies" or "cubelets". Each of these includes a concealed inward extension that interlocks with the other cubes while permitting them to move to different locations. However, the centre cube of each of the six faces is merely a single square façade; all six are affixed to the core mechanism. 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.Each of the six centre pieces pivots on a screw (fastener) held by the centre piece, a "3D cross". A spring between each screw head and its corresponding piece tensions the piece inward, so that collectively, the whole assembly remains compact, but can still be easily manipulated. The screw can be tightened or loosened to change the "feel" of the Cube. Newer official Rubik's brand cubes have rivets instead of screws and cannot be adjusted.The Cube can be taken apart without much difficulty, typically by rotating the top layer by 45° and then prying one of its edge cubes away from the other two layers. Consequently, it is a simple process to "solve" a Cube by taking it apart and reassembling it in a solved state.There are six central pieces which show one coloured face, twelve edge pieces which show two coloured faces, and eight corner pieces which show three coloured faces. 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. However, Cubes with alternative colour arrangements also exist; for example, with the yellow face opposite the green, the blue face opposite the white, and red and orange remaining opposite each other.Douglas 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.

Mathematics

Permutations

(File:Rubik's cube colors.svg|thumb|The current colour scheme of a Rubik's Cube)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. Each corner has three possible orientations, although only seven (of eight) can be oriented independently; the orientation of the eighth (final) corner depends on the preceding seven, giving 37 (2,187) possibilities. There are 12!/2 (239,500,800) ways to arrange the edges, restricted from 12! because edges must be in an even permutation exactly when the corners are. (When arrangements of centres are also permitted, as described below, the rule is that the combined arrangement of corners, edges, and centres must be an even permutation.) Eleven edges can be flipped independently, with the flip of the twelfth depending on the preceding ones, giving 211 (2,048) possibilities.Martin Schönert "Analyzing Rubik's Cube with GAP": the permutation group of Rubik's Cube is examined with GAP computer algebra system
{8! times 3^7 times (12!/2) times 2^{11}} = 43,252,003,274,489,856,000
which is approximately 43 quintillion.Counting the Permutations of the Rubik's Cube, Scott Vaughen. Professor of Mathematics. Miami Dade College.The puzzle was originally advertised as having "over 3,000,000,000 (three billion) combinations but only one solution".AV MEDIA,weblink Rubik's Cube Commercial 1981, TELEVISIONARCHIVES, 23 October 2008, 10 October 2017, YouTube, To put this into perspective, if one had as many standard sized Rubik's Cubes as there are permutations, one could cover the Earth's surface 275 times.The preceding figure is limited to permutations that can be reached solely by turning the sides of the cube. If one considers permutations reached through disassembly of the cube, the number becomes twelve times as large:
{8! times 3^8 times 12! times 2^{12}} = 519,024,039,293,878,272,000.
which is approximately 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.

Centre faces

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, with marker pens, 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; a cube marked in this way is referred to as a "supercube". 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. Cubes have also been produced where the nine stickers on a face are used to make a single larger picture, and centre orientation matters on these as well. Thus one can nominally solve a Cube yet have the markings on the centres rotated; it then becomes an additional test to solve the centres as well.Marking the Rubik's Cube's centres increases its difficulty because this expands the set of distinguishable possible configurations. There are 46/2 (2,048) ways to orient the centres since an even permutation of the corners implies an even number of quarter turns of centres as well. In particular, when the Cube is unscrambled apart from the orientations of the central squares, there will always be an even number of centre squares requiring a quarter turn. Thus orientations of centres increases 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).Scientific American, p28, vol 246, 1982 retrieved online 29 January 2009.When turning a cube over is considered to be a change in permutation then we must also count arrangements of the centre faces. Nominally there are 6! ways to arrange the six centre faces of the cube, but only 24 of these are achievable without disassembly of the cube. When the orientations of centres are also counted, as above, this increases the total number of possible Cube permutations from 88,580,102,706,155,225,088,000 (8.9×1022) to 2,125,922,464,947,725,402,112,000 (2.1×1024).

Algorithms

In Rubik's cubers' 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 effect the algorithm has, and when it can be used to bring the cube closer to being solved.Many algorithms are designed to transform only a small part of the cube without interfering with 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 do 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. Most are long and difficult to memorise. Towards the end of the solution, the more specific (and usually more complicated) algorithms are used instead.

Relevance and application of mathematical group theory

Rubik's Cube lends itself to the application of mathematical group theory, which has been helpful for deducing certain algorithms – in particular, those which have a commutator structure, namely XYX−1Y−1 (where X and Y are specific moves or move-sequences and X−1 and Y−1 are their respective inverses), or a conjugate structure, namely XYX−1, often referred to by speedcubers colloquially as a "setup move". In addition, the fact that there are well-defined subgroups within the Rubik's Cube group enables the puzzle to be learned and mastered by moving up through various self-contained "levels of difficulty". For example, one such "level" could involve solving cubes which have been scrambled using only 180-degree turns. These subgroups are the principle underlying the computer cubing methods by Thistlethwaite and Kociemba, which solve the cube by further reducing it to another subgroup.

Solutions

Move notation

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".BOOK, Joyner, David, Adventures in group theory: Rubik's Cube, Merlin's machine, and Other Mathematical Toys, Johns Hopkins University Press, Baltimore, 2002, 7, 0-8018-6947-1, 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.
  • F (Front): the side currently facing the solver
  • 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
  • Æ’ (Front two layers): the side facing the solver 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 entire Cube on R
  • y (rotate): rotate the entire Cube on U
  • z (rotate): rotate the entire Cube on F
When a prime symbol ( â€² ) follows a letter, it denotes an anticlockwise face turn; while a letter without a prime symbol denotes a clockwise turn. These directions are as one is looking at the specified face. 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 anticlockwise. The letters x, y, and z are used to indicate that the entire Cube should be turned about one of its axes, corresponding to R, U, and F turns respectively. 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 180 degrees. The most common deviation from Singmaster notation, and in fact the current official standard, is to use "w", for "wide", instead of lowercase letters to represent moves of two layers; thus, a move of Rw is equivalent to one of r.WEB, World Cube Association Competition Regulations,weblink World Cube Association, 5 May 2012, 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.BOOK, Treep, Anneke, Waterman, Marc, Marc Waterman's Algorithm, Part 2, Cubism For Fun 15, 1987, Nederlandse Kubus Club, 10,
  • M (Middle): the layer between L and R, turn direction as L (top-down)
  • E (Equator): the layer between U and D, turn direction as D (left-right)
  • S (Standing): the layer between F and B, turn direction as F
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). An asterisk (L*), a number in front of it (2L), or two layers in parentheses (Ll), means to turn the two layers at the same time (both the inner and the outer left faces) For example: (Rr)' lf' means to turn the two rightmost layers anticlockwise, then the left inner layer twice, and then the inner front layer anticlockwise. By extension, for cubes of 6×6×6 and larger, moves of three layers are notated by the number 3, for example, 3L.An alternative notation, Wolstenholme notationweblink Wolstenholme notation is designed to make memorising sequences of moves easier for novices. This notation uses the same letters for faces except it replaces U with T (top), so that all are consonants. The key difference is the use of the vowels O, A and I for clockwise, anticlockwise and 180-degree turns, which results in word-like sequences such as LOTA RATO LATA ROTI (equivalent to LU′R′UL′U′RU2 in Singmaster notation). Addition of a C implies rotation of the entire cube, so ROC is the clockwise rotation of the cube around its right face. Middle layer moves are denoted by adding an M to corresponding face move, so RIM means a 180-degree turn of the middle layer adjacent to the R face.Another notation appeared in the 1981 book The Simple Solution to Rubik's Cube. Singmaster notation was not widely known at the time of publication. The faces were named Top (T), Bottom (B), Left (L), Right (R), Front (F) and Posterior (P), with + for clockwise, – for anticlockwise and 2 for 180-degree turns.Another notation appeared in the 1982 "The Ideal Solution" book for Rubik's Revenge. Horizontal planes were noted as tables, with table 1 or T1 starting at the top. Vertical front to back planes were noted as book, with book 1 or B1 starting from the left. Vertical left to right planes were noted as windows, with window 1 or W1 starting at the front. Using the front face as a reference view, table moves were left or right, book moves were up or down, and window moves were clockwise or anticlockwise.

Optimal solutions

File:1982 expedition to Tartu Ülikool 350 (23).jpg|thumb|Mountaineer solving Rubik's Cube during 1982 expedition in Pamir Mountains to peak Tartu Ülikool 350Tartu Ülikool 350Although there are a significant number of possible permutations for the Rubik's Cube, a number of solutions have been developed which allow solving the cube in well under 100 moves.Many general solutions for the Rubik's Cube have been discovered independently. David Singmaster first published his solution in the book Notes on Rubik's "Magic Cube" in 1981.BOOK, Singmaster, David, Notes on Rubik's Magic Cube, 1981, Penguin Books, 0-907395-00-7, Harmondsworth, Eng, 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 sufficient 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".BOOK, Handbook of Cubik Math, Frey, Jr., Alexander H., Singmaster, David, 1982, Enslow Publishers, Hillside, N.J, 0-89490-058-7, 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 26 moves or fewer.CONFERENCE
, D.
, Kunkle
, Cooperman, C.
, Twenty-Six Moves Suffice for Rubik's Cube
, Proceedings of the International Symposium on Symbolic and Algebraic Computation (ISSAC '07)
, 2007
, ACM Press
,weblink
, CONFERENCE
, KFC
, Rubik's cube proof cut to 25 moves
, 2008
,weblink
, WEB,weblink Cracking the Cube, Julie J. Rehmeyer, MathTrek
archiveurl =weblink" title="web.archive.org/web/20071011211147weblink">weblink 11 October 2007, In 2008, Tomas Rokicki lowered that number to 22 moves,ARXIV, 0803.3435, Twenty-Five Moves Suffice for Rubik's Cube, Tom Rokicki, cs.SC, 2008, WEB,weblink Rubik's Cube Algorithm Cut Again, Down to 23 Moves, [Slashdot], 5 June 2008, WEB,weblink Twenty-Two Moves Suffice, Tom Rokicki, 20 August 2008,
and in July 2010, a team of researchers including Rokicki, working with Google, proved the so-called "God's number" to be 20.WEB, Flatley, Joseph F., Rubik's Cube solved in twenty moves, 35 years of CPU time,weblink Engadget, 9 August 2010, 10 August 2010, WEB, Davidson, Morley, Dethridge, John, Kociemba, Herbert, Rokicki, Tomas, God's Number is 20,weblink cube20.org, 10 August 2010, This is optimal, since there exist some starting positions which require a minimum of 20 moves to solve. More generally, it has been shown that an n×n×n Rubik's Cube can be solved optimally in Θ(n2 / log(n)) moves.ARXIV, Demaine, Erik D., Demaine, Martin L., Eisenstat, Sarah, Lubiw, Anna, Anna Lubiw, Winslow, Andrew, 1106.5736v1, Algorithms for Solving Rubik's Cubes, 2011, cs.DS,

Speedcubing methods

A solution commonly used by speedcubers was developed by Jessica Fridrich. This method is called CFOP standing for "cross, F2L, OLL, PLL". It is similar to the layer-by-layer method but employs the use of a large number of algorithms, especially for orienting and permuting the last layer. The cross is done first, followed by first layer corners and second layer edges simultaneously, with each corner paired up with a second-layer edge piece, thus completing the first two layers (F2L). This is then followed by orienting the last layer, then permuting the last layer (OLL and PLL respectively). Fridrich's solution requires learning roughly 120 algorithms but allows the Cube to be solved in only 55 moves on average.A now 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. The principle behind this is that in layer-by-layer you must constantly break and fix the first layer; the 2×2×2 and 2×2×3 sections allow three or two layers to be turned without ruining progress. 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.WEB,weblink Rubik's Cube Solution - Petrus Method, lar5.com, 8 November 2018, The Roux Method, developed by Gilles Roux, is similar to the Petrus method in that it relies on block building rather than layers, but derives from corners-first methods. In Roux, a 3×2×1 block is solved, followed by another 3×2×1 on the opposite side. Next, the corners of the top layer are solved. The cube can then be solved using only moves of the U layer and M slice.WEB,weblink Introduction, Grrroux.free.fr, 20 June 2012,

Beginners' methods

Most beginner solution methods involve solving the cube one layer at a time, using algorithms that preserve what has already been solved. The easiest layer by layer methods require only 3–8 algorithms.WEB, How to solve a Rubik's Cube,weblink how-to-solve-a-rubix-cube.com, 28 June 2016, WEB, Beginner Solution to the Rubik's Cube (available in multiple languages),weblink Jasmine Lee, 17 July 2017, In 1981, thirteen-year-old Patrick Bossert developed a solution for solving the cube, along with a graphical notation, designed to be easily understood by novices.JOURNAL,weblink Restore your cube, Rob, Evans, 24 September 1981, New Scientist, 802, It was subsequently published as You Can Do The Cube and became a best-seller.JOURNAL,weblink Newsweek, 99, 16, Rubik's Cube, In 1997, Denny Dedmore published a solution described using diagrammatic icons representing the moves to be made, instead of the usual notation.WEB,weblink Website with solutions created by Denny Dedmore, Helm.lu, 20 June 2012, Philip Marshall's The Ultimate Solution to Rubik's Cube takes a different approach, averaging only 65 twists yet requiring the memorisation of only two algorithms. The cross is solved first, followed by the remaining edges, then five corners, and finally the last three corners.Philip Marshall (2005), The Ultimate Solution to Rubik's Cube.

Rubik's Cube solver program

The most move optimal online Rubik's Cube solver programs use Herbert Kociemba's Two-Phase Algorithm which can typically determine a solution of 20 moves or less. The user has to set the colour configuration of the scrambled cube and the program returns the steps required to solve it.WEB, Rubik's Cube solver,weblink rubiks-cube-solver.com, 28 June 2016,

Competitions and records

Speedcubing competitions

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.A speedcubing championship organised by the Guinness Book of World Records was held in Munich on 13 March 1981.BOOK, McWhirter, Norris, 1983, Guinness Book of Records, Cubism, 85, Guinness Publishing,weblink The contest used standardised scrambling and fixed inspection times, and the winners were Ronald Brinkmann and Jury Fröschl with times of 38.0 seconds. The first world championship was the 1982 World Rubik's Cube Championship held in Budapest on 5 June 1982, which was won by Minh Thai, a Vietnamese student from Los Angeles, with a time of 22.95 seconds.NEWS,weblink Cube Contest, Joan Hanauer, United Press International, 26 May 1982, Since 2003, the winner of a competition is determined by taking the average time of the middle three of five attempts. However, the single best time of all tries is also recorded.The World Cube Association maintains a history of world records.WEB, World Cube Association Official Results, World Cube Association,weblink 16 February 2008, In 2004, the WCA made it mandatory to use a special timing device called a Stackmat timer.In addition to the main 3x3x3 event, the WCA also holds events where the cube is solved in different ways:WEB,weblink Competition Regulations, Article 9: Events, World Cube Association, 9 April 2008, 16 April 2008,
  • Blindfolded solvingWEB,weblink Rubik's 3x3x3 Cube: Blindfolded records, WorldCubeAssociation.org, 20 June 2012,
  • Multiple blindfolded solving, or "multi-blind", in which the contestant solves any number of cubes blindfolded in a rowWEB,weblink WCA Regulations – World Cube Association, worldcubeassociation.org, 5 April 2018,
  • Solving the Cube using a single handWEB,weblink Rubik's 3x3x3 Cube: One-handed, Worldcubeassociation.org, 20 June 2012,
  • Solving the Cube with one's feetWEB,weblink Rubik's 3x3x3 Cube: With feet, Worldcubeassociation.org, 20 June 2012,
  • Solving the Cube in the fewest possible movesWEB,weblink Fewest Moves Records,
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 memorizing the cube and the time spent manipulating it.In multiple blindfolded, all of the cubes are memorised, and then all of the cubes are solved once blindfolded; thus, the main challenge is memorising many â€“ often ten or more â€“ separate cubes. The event is scored not by time but by the number of solved cubes minus the number of unsolved cubes after one hour has elapsed.In fewest moves solving, the contestant is given one hour to find his or her solution and must write it down.

Records

  • Average time: The world record average of the middle three of five solve times (which excludes the fastest and slowest) is 5.80 seconds, set by Feliks Zemdegs of Australia at the Malaysia Cube Open 2017.WEB,weblink World Cube Association – Official Results, WCA Website, Team, worldcubeassociation.org, 28 March 2018,
  • One-handed solving: The world record fastest one-handed solve is 6.88 seconds, set by Feliks Zemdegs of Australia on 10 May 2015 at Canberra Autumn 2015. The world record fastest average of five one-handed solves is 9.47 seconds, set by Max Park of the United States at WCCT Reno 2018.WEB,weblink 3x3x3 One-handed records,
  • Feet solving: The world record fastest Rubik's Cube solve with one's feet is 16.96 seconds, set by Daniel Rose-Levine of the United States on 9 March 2018 at Heartland Champs 2018. The world record average of five feet solves is 22.22 seconds, also set by Daniel Rose-Levine on 19 July 2018 at WCA Euro 2018.WEB,weblink World Cube Association – Official Results, WCA Website Team, worldcubeassociation.org,
  • Blindfold solving: The world record fastest Rubik's Cube solve blindfolded is 17.33 seconds (including memorization), set by Jeff Park of the United States on 21 July 2018 at DFW Cubing Fiesta July 2018. The world record mean of three blindfold solves is 20.34 seconds, also set by Jeff Park at the same competition.WEB,weblink 3x3x3 blindfolded WCA records, World Cube Association, 21 July 2018,
  • Multiple blindfold solving: The world record for multiple Rubik's Cube solving blindfolded is 48 out of 48 cubes, set by Shivam Bansal of India on 22 July 2018 at Delhi Monsoon Open 2018. Bansal inspected 48 cubes, donned a blindfold, and solved all of them, all under the time limit of one hour.WEB,weblink 3x3x3 multi-blindfolded WCA records, World Cube Association, 21 July 2018,
  • Fewest moves solving: The world record of fewest moves to solve a cube, given one hour to determine one's solution, is 19. This has been achieved by Tim Wong of the United States on 11 October 2015 at Irvine Fall 2015, by Marcel Peters of (Germany) on 9 January 2016 at Cubelonia 2016, by Vladislav Ushakov of Belarus on 27 August 2016 at PSU Open 2016, and by Baiqiang Dong of China on 30 April 2018 at Beijing Open 2018. The world record mean of three fewest moves challenges (with different scrambles) is 24.00, set by Sébastien Auroux of Germany on 28 August 2017 at Only FMC 2017, by Walker Welch of the United States on 15 October 2017 at FMC Americas 2017, and by Reto Bubendorf of Finland on 11–12 May 2018 at Finnish Championship 2018.WEB,weblink World Cube Association – Official Results, WCA Website Team, worldcubeassociation.org, 28 March 2018,
  • Non-human solving: The fastest non-human Rubik's Cube solve was performed by the Rubik's Contraption, a robot made by Ben Katz and Jared Di Carlo. A YouTube video shows a 0.38-second solving time using an Nucleo with the min2phase algorithm.WEB, Rubik's robot solves puzzle in 0.38 seconds, BBC News,weblink 8 March 2018, 8 March 2018,
  • Highest order physical n×n×n cube solving: Douglas Shamlin Jr. solved a 17x17x17 in one hour and six seconds.WEB,weblink 17x17x17 Rubik's cube solve 1:05:43.56 part 1/2 (record broken), Douglas Shamlin Jr., 13 January 2018, YouTube, WEB,weblink 17x17x17 Rubik's cube solve 1:05:43.56 part 2/2, Douglas Shamlin Jr., 13 January 2018, YouTube,

Top 5 solvers by single solve

{| class="wikitable"! Name !! Fastest solve !! Competition
Feliks Zemdegs >| Cube for Cambodia 2018
Max Park >| SacCubing V 2018
| ChicaGhosts 2017
| Pennsylvania 2018
| Ralley in the Valley 2017

Top 5 solvers by average of 5 solves

{| class="wikitable"! Name !! Fastest average !! CompetitionFeliks Zemdegs >| Malaysia Cube Open 2017Max Park >| ABCD 2018| Brooklyn 2018| Cologne Open 2018| China Championship 2017
  • Group solving (12 minutes): The record for most people solving a Rubik's Cube at once in twelve minutes is 134, set on 17 March 2010 by schoolboys from Dr Challoner's Grammar School, Amersham, England, breaking the previous Guinness World Record of 96 people at once.WEB,weblink Pupils break Rubik's Cube Record, BBC News, 17 March 2010, 20 June 2012,
  • Group solving (30 minutes): On 21 November 2012, at the O2 Arena in London, 1414 people, mainly students from schools across London, solved the Rubik's Cube in under 30 minutes, breaking the previous Guinness World Record of 937. The event was hosted by Depaul UK.WEB,weblink Schools Smash World Record, Depaul UK, 21 November 2012, 21 November 2012, yes,weblink" title="web.archive.org/web/20130120065049weblink">weblink 20 January 2013,


On 4 November 2012, 3248 people, mainly students of College of Engineering Pune, successfully solved the Rubik's cube in 30 minutes on college ground. The successful attempt is recorded in the Limca Book of Records. The college will submit the relevant data, witness statements and video of the event to Guinness authorities.WEB,weblink CoEP sets another record, over 3,000 solve Rubik's Cube in 30 minutes, The Indian Express, 5 November 2012, 5 November 2012,

Variations

File:Rubik's Cube variants.jpg|thumb|250px|alt=Rubik's Cube Variants|Variations of Rubik's Cubes. Top row: V-Cube 7, Professor's Cube, V-Cube 6. Bottom row: Rubik's Revenge, original Rubik's Cube, Pocket Cube. Clicking on a cube in the picture will redirect to the respective cube's page. (Note:scrambled states)default none (File:Rubik's Cube variants.jpg)poly 964 370 1082 448 1065 545 970 622 865 545 875 445 Pocket Cubepoly 620 370 844 363 862 536 850 680 628 682 Rubik's Cubepoly 455 280 570 440 580 605 355 705 255 530 220 363 Rubik's Revengepoly 540 75 780 90 780 225 760 360 620 365 605 420 560 420 505 340 500 235 Professor's Cubepoly 890 50 1125 90 1065 420 1040 420 965 365 930 390 850 380 845 365 830 360 840 205 V-Cube 6poly 255 50 320 90 405 225 420 290 210 360 230 460 210 465 150 410 90 320 60 240 45 155 120 100 190 70 V-Cube 7(File:11x11x11.jpg|thumb|An 11×11×11 cube)There are different variations of Rubik's Cubes with up to thirty-three 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) being the most well known. The 17×17×17 "Over The Top" cube (available late 2011) was until December 2017 the largest (and most expensive, costing more than two thousand dollars) commercially sold cube. A working design for a 22×22×22 cube exists and was demonstrated in January 2016,WEB,weblink You Probably Won't Live Long Enough to Solve the World's Largest 22x22 Rubik's Cube, 10 February 2016, and a 33x33 in December 2017.WEB,weblink WORLD RECORD 33x33x33 RUBIK's CUBE !!!!!, Greg's Puzzles, 2 December 2017, 10 February 2018, YouTube, Chinese manufacturer ShengShou has been producing cubes in all sizes from 2×2×2 to 10×10×10 (as of late 2013),WEB,weblink ShengShou puzzles – China Magic Cube, 28 August 2014, and have also come out with an 11x11x11.Non-licensed physical cubes as large as 13×13×13 based on the V-Cube patents are commercially available to the mass-market circa 2015 in China; these represent about the limit of practicality for the purpose of "speed-solving" competitively (as the cubes become increasingly ungainly and solve-times increase quadratically).(File:Rubik's TouchCube.jpg|thumb|left|Rubik's TouchCube)There are many variationsWEB,weblink Rubik's, Variations, 30 December 2012, of the original cube, some of which are made by Rubik. The mechanical products include the Rubik's Magic, 360, and Twist. Also, electronics like the Rubik's Revolution and Slide were also inspired by the original. One of the newest 3×3×3 Cube variants is the Rubik's TouchCube. Sliding a finger across its faces causes its patterns of coloured lights to rotate the same way they would on a mechanical cube. The TouchCube also has buttons for hints and self-solving, and it includes a charging stand. The TouchCube was introduced at the American International Toy Fair in New York on 15 February 2009.NEWS,weblink NY Toy Fair opens with new Rubik's Cube, Lego deals, 23 March 2009, Reuters, 16 February 2009, WEB,weblink Toy Fair Kicks Off at Javits Center, 23 March 2009, yes,weblink" title="web.archive.org/web/20090422104222weblink">weblink 22 April 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). There are also puzzles that change shape such as Rubik's Snake and the Square One.In 2011, Guinness World Records awarded the "largest order Rubiks magic cube" to a 17×17×17 cube, made by Oskar van Deventer.WEB, Largest Order Rubiks / Magic Cube, Guinness World Records,weblink 4 January 2013, WEB, van Deventer, Oskar, Over the Top – 17x17x17,weblink On 2 December 2017, Grégoire Pfennig announced that he had broken this record, with a 33×33×33 cube, and that his claim had been submitted to Guinness for verification. On 8 April 2018, Grégoire Pfennig announced another world record, the 2x2x50 cube.WEB,weblink World Record 2x2x50 Rubik's Cube !!! :D, Greg's Puzzles, 8 April 2018, 13 April 2018, YouTube, Whether this is a replacement for the 33x33x33 record, or an additional record, remains to be seen.(File:Rubiks Platonic Solids.jpg|thumb|All five platonic solid versions of Rubik's cube)Since 2015, with the mass production of the Icosaix, all five platonic solids analogous to Rubik's cube (face-turning with cuts one-third from each face, except the Pyraminx, which also has turnable tips) became available. Besides Rubik's cube, the tetrahedron is available as the Pyraminx, the octahedron as the Face Turning Octahedron, the dodecahedron as the Megaminx, and the icosahedron as the Icosaix.Some puzzles have also been created in the shape of the Kepler-Poinsot polyhedra, such as Alexander's Star (a great dodecahedron).

Custom-built puzzles

(File:Novelty Keychain Rubiks Cube.JPG|thumb|Novelty keychain)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 2×2×4, 2×3×4, and 3×3×5.Martin, W. Eric. "Gamebits: Rubik's Cube... Cubed". Games. Issue 199 (Vol. 28 No. 3). p. 4. April 2004. 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 moulding and casting techniques to create the final puzzle.{{Citation needed|date=March 2009}}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.

Rubik's Cube software

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, analysing 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.WEB, Melinda Green,weblink Magic Cube 4D, Superliminal.com, 25 June 2009, 20 June 2012, WEB,weblink Magic Cube 5D, Gravitation3d.com, 20 June 2012, File:4-cube 4^4.png|Magic Cube 4D, a 4×4×4×4 virtual puzzleFile:5D Rubik's Cube.png|Magic Cube 5D, a 3×3×3×3×3 virtual puzzleFile:4d cube.png|high-dimensional Octagon Magic Cube Puzzle simulated phone applicationFile:4-cube 2^4 highlighted.png|4D virtual 2x2x2x2 sequential move puzzle

Chrome Cube Lab

Google has released the Chrome Cube Lab in association withJOURNAL, Armstrong, Calvin, Goldstine, Susan, September 2014, Review: Beyond Rubik's Cube Exhibit, The College Mathematics Journal, 45, 254–256, 10.4169/college.math.j.45.4.254, 10.4169/college.math.j.45.4.254, Ernő Rubik. The site has various interactive objects based on Rubik's Cube. Customised versions of Rubik's Cube can be created and uploaded.WEB, Chrome Cube Lab,weblink Google, 19 May 2014,

Popular culture

See also

References

{{Reflist|30em}}

Further reading

  • BOOK, Frey, Alexander, Singmaster, David, 1982, Handbook of Cubic Math, Enslow, 0894900587,
  • BOOK, Rubik, ErnÅ‘, Varga, Tamas, Keri, Gerson, Marx, Gyorgy, Vekerdy, Tamas, 1987, Rubik's Cubic Compendium, David, Singmaster, Oxford University Press, 0198532024,
  • BOOK, Bizek, Hana M., Mathematics of the Rubik's Cube Design, 1997, Dorrance Pub. Co, Pittsburgh, Pa, 0805939199,
  • BOOK, Slocum, Jerry, Singmaster, David, Huang, Wei-Hwa, Gebhardt, Dieter, Hellings, Geert, Rubik, ErnÅ‘, 2009, The Cube: The Ultimate Guide to the World's Bestselling Puzzle, Black Dog & Leventhal, 157912805X,

External links

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