Rubik's Cube

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Variations of Rubik's Cubes (from left to right: Rubik's Revenge, Rubik's Cube, Professor's Cube, & Pocket Cube)

Rubik's Cube or (informally) Rubix Cube is a mechanical puzzle invented in 1974 by the Hungarian sculptor and professor of architecture Ernő Rubik. The plastic cube comes in four widely available versions: the 2×2×2 (" Pocket Cube"), the 3×3×3 standard cube, the 4×4×4 (" Rubik's Revenge"), 5×5×5 (" Professor's Cube"). 6×6×6 and 7×7×7 cubes are currently being produced. The 3×3×3 version, which is the version usually meant by the term "Rubik's Cube," has nine square faces on each side, for a total area of fifty-four faces, and occupies the volume of twenty-six unit cubes (not counting the invisible cube in the centre). Typically, the faces of the Cube are covered by stickers in six solid colors, one for each side of the Cube. When the puzzle is solved, each side of the Cube is a solid colour. The original 3×3×3 version celebrated its twenty-fifth anniversary in 2005, when a special edition Cube in a presentation box was released, featuring a sticker in the centre of the white face (which was replaced with a reflective surface) with a "Rubik's Cube 1980-2005" logo.

Originally called the Magic Cube by its inventor, it was renamed Rubik's Cube in 1980 and released worldwide in May of that year, winning a Spiel des Jahres special award for Best Puzzle. It is said to be the world's best-selling toy, with some 300,000,000 Rubik's Cubes and imitations sold worldwide.

History

Conception and development

The Magic Cube was invented in 1974 by Ernő Rubik, a Hungarian sculptor and professor of architecture with an interest in geometry and the study of three-dimensional forms. Ernő 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.

The Cube slowly grew in popularity throughout Hungary as word of mouth spread. Western academics also showed interest in it. In September 1979, a deal was reached with Ideal Toys to release the Magic Cube internationally. It made its international debut at the toy fairs of London, New York, Nuremberg, and Paris in early 1980.

The progress of the Cube towards the toy shop shelves of the West was then 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. In 1984, Ideal lost a patent infringement suit by Larry Nichols for his patent US3655201. Terutoshi Ishigi acquired Japanese patent JP55‒8192 for a nearly identical mechanism while Rubik's patent was being processed, but Ishigi is generally credited with an independent reinvention.

Popularity

Over one hundred million Cubes were sold in the period from 1980 to 1982. It won the BATR Toy of the Year award in 1980 and again in 1981. Many similar puzzles were released shortly after the Rubik's Cube, both from Rubik himself and from other sources, including the Rubik's Revenge, a 4×4×4 version of the Rubik's Cube. There are also 2×2×2 and 5×5×5 Cubes (known as the Pocket Cube and the Rubik's Professor, respectively) and puzzles in other shapes, such as the Pyraminx, a tetrahedron.

In May 2005, the Greek Panagiotis Verdes constructed a 6×6×6 Rubik's Cube. and on May 23 2006, Frank Morris, a world champion Rubik's Cube solver, tested this version. He had previously solved the 3×3×3 in 15 seconds, the 4×4×4 in 1 minute and 10 seconds, and the 5×5×5 in 2 minutes. The 6×6×6 took him 5 minutes and 37 seconds to solve. Morris himself thanked the inventor for making it and purportedly stated that the bigger the Cube is, the greater the pleasure. In July 2006, Mr. Verdes succesfully constructed the 7x7x7 cube, and on October 27 2006, a video of Frank Morris testing the cube was released. Videos of these tests can be viewed at http://www.olympicube.com

In 1981, Patrick Bossert, a twelve-year-old schoolboy from England, published his own solution in a book called You Can Do the Cube ( ISBN 0-14-031483-0). The book sold over 1.5 million copies worldwide in seventeen editions and became the number one book on both The Times and the New York Times bestseller lists for that year.

At the height of the puzzle's popularity, separate sheets of colored stickers were sold so that frustrated or impatient Cube owners could restore their puzzle to its original appearance.

It has been suggested that the international appeal and export achievement of the Cube became one of the contributing factors in the reform and liberalization of the Hungarian economy between 1981 and 1985, which finally led to the move from communism to capitalism.

Financially, the Cube was so successful that Rubik became the first self-made elite in a communist country.

The name "Rubik's Cube" is common in many languages except Hebrew and in Hungarian. In the first, it is known as the "Hungarian Cube", whilst in the second, its name is "Magic Cube" (Bűvös kocka).

Workings

Rubik's Cube partially disassembled

A standard Cube measures approximately 2¼ inches (5.7 cm) on each side. The puzzle consists of the twenty-six unique miniature cubes ("cubies") on the surface. 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 cubie" away from a "centre cubie" until it dislodges (however, prying loose a corner cubie is a good way to break off a centre cubie - thus ruining the cube). It is a simple process to solve a Cube by taking it apart and reassembling it in a solved state; however, this is not the challenge.

There are twelve edge pieces which show two colored sides each, and eight corner pieces which show three colors. Each piece shows a unique color combination, but not all combinations are present (for example, there is no edge piece with both white and yellow sides, if white and yellow are on opposite sides of the solved Cube.). 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 colored 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 colors of the stickers are red opposite orange, yellow opposite white, and green opposite blue. However, there also exist Cubes with alternative colour arrangements. These alternative Cubes have the yellow face opposite the green, and the blue face opposite the white (with red and orange opposite faces remaining unchanged).

Permutations

A Normal (3x3x3) Rubik's Cube can have (8! × 3⁸⁻¹) × (12! × 2¹²⁻¹)/2 = 43,252,003,274,489,856,000 different positions, also referred to by the mathematical term permutations. This number can also be written as (~4.3 × 10¹⁹), about forty-three quintillion ( short scale) or forty-three trillion ( long scale), but the puzzle is advertised as having only " billions" of positions, due to the general incomprehensibility of such a large number to laymen. Despite the vast number of positions, all Cubes can be solved in twenty-nine or fewer moves (see Optimal solutions for Rubik's Cube).

In fact, there are (8! × 3⁸) × (12! × 2¹²) = 519,024,039,293,878,272,000 (about 519 quintillion on the short scale) possible arrangements of the pieces that make up the Cube, but only one in twelve of these is actually reachable. This is because there is no sequence of moves that will swap a single pair or rotate a single corner or edge cubie. 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 and still official Rubik's Cube has no markings on the centre faces. This obscures the fact that the centre faces can rotate independently. If you have a marker pen, you could, for example, mark the central squares of an unshuffled Cube with four colored marks on each edge, each corresponding to the colour of the adjacent square. 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 centres rotated, and it becomes an additional challenge to "solve" the centres as well.

Putting markings on the Rubik's Cube increases the challenge chiefly 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 Cube permutation from 43,252,003,274,489,856,000 to 88,580,102,706,155,225,088,000.

Solutions

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 1980. 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. Other general solutions include "corners first" methods or combinations of several other methods.

Speed cubing solutions have been developed for solving the Rubik's Cube as quickly as possible. The most common speed cubing 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.

Solutions typically consist of a sequence of processes. A process, or algorithm or operator as it is sometimes called, is a series of twists which accomplishes a particular goal. For instance, one process might switch the locations of three corner pieces, while leaving the rest of the pieces in place. These sequences are performed in the appropriate order, as dictated by the current configuration of the puzzle, to solve the Cube. Complete solutions can be found in any of the books listed in the bibliography, and most can be used to solve any Cube in under five minutes. These solutions typically are intended to be easy to learn, but much effort has gone into finding even faster solutions to Rubik's Cube (see Optimal solutions for Rubik's Cube).

Move notation

Rubik's Cube in a scrambled state

Rubik's Cube being solved

Rubik's Cube in solved state

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). 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 colors 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 Up or on bottom
L (Left): the side directly to the left of the front
R (Right): the side directly to the right of the front

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 superscript) means to turn the face a half-turn (the direction does not matter).

(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 - counterclockwise, with ++ representing a half-turn. However, alternative notations failed to catch on, and today the Singmaster scheme is used universally by those interested in the puzzle.)

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.)

Lowercase letters f, b, u, d, l, and r signify to move the first two layers of that face while keeping the remaining layer in place. This is of course equivalent to rotating the whole Cube in that direction, then rotating the opposite face back the same amount in the opposite direction, but is useful notation to describe certain triggers for speedcubing. Furthermore, M, E, and S (and respectively their lowercase for larger sized cubes), are used for inner-slice movements. M signifies turning the layer that is between L and R downward (clockwise if looking from the left side). E signifies turning the layer between U and D towards the right (counter-clockwise if looking from the top). S signifies turning the layer between F and B clockwise.

For example, the algorithm (or operator, or sequence) F2U'R'LF2RL'U'F2, which cycles three edge cubes in the top layer without affecting any other part of the Cube, means:

Turn the Front face 180 degrees
Turn the Up face 90 degrees counterclockwise
Turn the Right face 90 degrees counterclockwise
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 counterclockwise
Turn the Up face 90 degrees counterclockwise
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. For an example, see an animation of the above sequence.

4×4×4 and larger Cubes use slightly different notation to incorporate the middle layers. Generally speaking, upper case letters (FBUDLR) refer to the outermost portions of the cube (called faces). Lower case letters (fbudlr) refer to the inner portions of the cube (called slices). Again Ideal breaks rank by describing their 4×4×4 solution 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).

Competitions

Many speedcubing competitions have been held to determine who can solve the Rubik's Cube in the shortest time.

Ideal Toys held the first UK competition in 1981 less than a year after the cube was launched (sponsored by the Daily Mirror). The following times were recorded in the regional championships;

Edinburgh heat - Alex McNair from Edinburgh in 48.85 secs, Manchester heat - Edgar Whitley from Colwyn Bay in 39.98 secs, York heat - Brian Storey from Sunderland in 41.76 secs, Midlands heat Nicolas Hammond from Nottingham in 35.38 secs, Bristol heat - Julian Bush from Bristol in 52.09 secs, Great Yarmouth heat - Julian Chilvers from Great Yarmouth in 38.67 secs, Southampton heat - Christopher Lennon from Portsmouth in 55.35 secs, London heat - Ben Jones from St Nicholas-at-Wade in 46.12 secs.

The first 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.

Many individuals have recorded shorter times, but these records were not recognized due to lack of compliance with agreed-upon standards for timing and competing. Only records set during official World Cube Association (WCA)-sanctioned tournaments are acknowledged.

In 2004, the WCA established a new set of standards, with a special timing device called a Stackmat timer.

Toby Mao set the current world record of 10.48 seconds at the 2006 US Nationals competition on August 6, 2006. The official world record based on an average of the middle three out of five Cubes is 13.22 seconds, set on March 20, 2006 in Norrköping, Sweden by Anssi Vanhala, a Finn. This record is recognized by the World Cube Association, the official governing body which regulates events and records.

Alternative Competitions

In addition, informal alternative competitions have been held, challenging participants to solve the cube under unusual situations. These include:

Blindfolded solving
Solving a cube in a room with colored lights intended to confuse participants in the colors of the tiles
Solving the cube underwater in a single breath
Solving the cube using a single hand

Rubik's Cube in mathematics and science

The Rubik's Cube is of interest to many mathematicians, partly because it is a tangible representation of a mathematical group. Additionally, a parallel between Rubik's Cube and particle physics was noted by mathematician Solomon W. Golomb and then extended and modified by Anthony E. Durham. Essentially, clockwise and counter-clockwise "twists" of corner cubies may be compared to the electric charges of quarks (+⅔ and −⅓) and antiquarks (−⅔ and +⅓). Feasible combinations of corner twists are paralleled by allowable combinations of quarks and antiquarks—both corner twist and the quark/antiquark charge must total to an integer. Combinations of two or three twisted corners may be compared to various hadrons, though this analogy is not always workable.

Rubik's Cube in popular culture

From 1983 to 1984, a Ruby-Spears produced Saturday morning cartoon based upon the toy Rubik, the Amazing Cube aired on ABC as part of a package program, "The Pac-Man/Rubik, The Amazing Cube Hour."
Saturday Night Live has had two commercial parodies for Rubik's cube-esque products: Rubik's Teeth (a pair of dentures that are multicolored like a Rubik's cube) and Rubik's Grenade (a live hand grenade with a Rubik's cube puzzle on the side that explodes if the puzzle isn't solved correctly)
The Rubik's Cube makes several appearances in The Simpsons, most notably when Homer is distracted by a Rubik's Cube when learning the power plant controls in " Homer Defined", when Marge attempts to solve the Cube while the rest of the family shouts hints at her in " Hurricane Neddy", and when Homer speedcubes a basket full of Cubes after becoming a person with average intelligence (with the IQ of 105 points) in " HOMЯ".
A Rubik's Cube serves as the MacGuffin in Dude, Where's My Car?
It won a Spiel des Jahres Best Puzzle prize in 1980.
The Australian sketch comedy show The Ronnie Johns Half Hour features a fictional Rubik's cube world champion named Sergei Haminov.
Rubik's Cubes are used as a NASA testing method for Harry Stamper's team in the movie Armageddon
In a recent commercial for the Playstation 3, demonstrating the game console's "advanced intelligence," it is presented in a colorless cubic room with the puzzle in front of it. It then telekinetically grasps the cube, levitating it to the centre of the room and speedsolving it in mid-air. The cube then violently explodes and fills each corresponding wall of the room with the respective colour of the Rubik's Cube sides.
It is seen in the Family Guy episode "Saving Private Brian" when Brian solves a Rubik's Cube in order to pass basic training.
In an episode of The Fresh Prince of Bel Air, The Alma Matter, Will Smith solves a rubik's cube in a few seconds during an interview for Princeton and is then recommended to attend the school.
Speedcubing champions Tyson Mao and Toby Mao were hired as consultants in the 2006 Sony Picture's movie The Pursuit of Happyness, to teach Will Smith's character how to solve a Rubik's Cube. There is a scene in the movie where a man tells him how impossible it is to solve the Rubik's cube, yet Smith solves it in mere seconds.

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