Rubik's Cube

Rubik's Cube is a mechanical puzzle invented in 1974 by the Hungarian sculptor and professor of architecture Ernő Rubik. The plastic cube comes in four different versions: the 2×2×2 ("Pocket Cube"), the 3×3×3, the 4×4×4 ("Rubik's Revenge"), and the 5×5×5 ("Professor's Cube"). 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-seven unit cubes. Typically, the faces of the Cube are covered by stickers in six solid colours, 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 center of the white face 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 biggest selling toy, with some three hundred million Rubik's Cubes and imitations sold worldwide. Marshall, Ray. Squaring up to the Rubik challenge. icNewcastle. Retrieved August 15, 2005.

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. [http://inventors.about.com/library/weekly/aa040497.htm

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.

On 23 May 2006, the Greek Panagiotis Verdes constructed a 6×6×6 Rubik's Cube. On the same day, Frank Morris, a world champion Rubik's Cube solver, tested this version. He had previously solved the 3×3×3 in less than 15 seconds, the 4×4×4 in less than 1 minute and 10 seconds, and the 5×5×5 in less than 2 minutes. The 6×6×6 took him less than 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. The inventor has since promised a 7×7×7 Cube to come soon. Note that Rubik himself considered it impossible to create a 6×6×6 Cube.

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 0140314830). 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 coloured 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. ^*, although some sociologists disagree.

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

Workings

A standard Cube measures approximately 2 1/4 inches (5.715 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 facade; 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. 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 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, 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 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, 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 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 coloured marks on each edge, each corresponding to the colour of the adjacent square. Some Cubes have also commercially produced 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 is solved, followed by the middle row, and finally the last 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 their places. 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

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." Its relative nature allows algorithms to be written in such a way that they can be applied regardless of the colors.

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 (can't be B because B means Back)
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 counterclockwise a quarter-turn, while a letter without an apostrophe means to turn it a quarter-turn clockwise. A letter followed by a 2 means to turn the face a half-turn (the direction doesn't matter).

(Some solution guides, including Ideal's official publication, The Ideal Solution, use slightly different conventions. Top and Posterior are used rather than Up and Down for the top and bottom faces, and + indicates clockwise rotation and - counterclockwise, with ++ representing a half-turn.)

Lesser-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 saves a move and is thus favored by "speed Cubers."

For example, the algorithm (or operator) F2U'R'LF2RL'U'F2, which moves three edge cubes in the topmost 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 ninety 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 speed cubing competitions have been held to determine who can solve the Rubik's Cube in the shortest time. 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 sanctioned tournaments are acknowledged.

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

On January 14, 2006, Leyan Lo, a twenty-year-old California Institute of Technology student, set a new world record of 11.13 seconds for solving the Cube. Though he was lucky in that his Cube happened to be scrambled in a way that allowed him to skip a step of his solution, this still counts as the official world record. Lo belongs to Caltech's Rubik's Cube Club, which hosted a competition at the Exploratorium museum in San Francisco (Associated Press). 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.

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 (+2/3 and −1/3) and antiquarks (−2/3 and +1/3). 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."
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 solves a basket full of Cubes after becoming a genius in "HOMЯ".

References

Handbook of Cubik Math by Alexander H. Frey, Jr. and David Singmaster
Notes on Rubik's 'Magic Cube' ISBN 0-89490-043-9 by David Singmaster
Metamagical Themas by Douglas R. Hofstadter contains two insightful chapters regarding Rubik's Cube and similar puzzles, originally published as articles in the March 1981 and July 1982 issues of Scientific American.
Four-Axis Puzzles by Anthony E. Durham.
Mathematics of the Rubik's Cube Design ISBN 0-80593-919-9 by Hana M. Bizek

External links

Rubiks Cube speed solving main web site
online community
WikiCube A Wiki devoted to speed solving and permutation puzzles

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