# Rubik's Cube

A classic Rubik's Cube, solved.

A classic Rubik's Cube, scrambled.

The Rubik's Cube is a 3-D mechanical puzzle invented in 1974[1] by Hungarian sculptor and professor of architecture Ern? Rubik. Originally called the "Magic Cube",[2] the puzzle was licensed by Rubik to be sold by Ideal Toys in 1980[3] and won the German Game of the Year special award for Best Puzzle that year. As of January 2009, 350 million cubes have sold worldwide[4][5] making it the world's top-selling puzzle game.[6][7] It is widely considered to be the world's best-selling toy.[8]

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

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## Conception and development

In the mid-1970s, Ern? Rubik worked at the Department of Interior Design at the Academy of Applied Arts and Crafts in Budapest.[10] He sought to find a teaching tool to help his students understand 3D objects. Rubik invented his "Magic Cube" in 1974 and obtained Hungarian patent HU170062 for the Magic Cube in 1975 but did not take out international patents. The first test batches of the product were produced in late 1977 and released to Budapest toy shops. Magic Cube was held together with interlocking plastic pieces that were less expensive to produce than the magnets in Nichols's design. In September 1979, a deal was signed with Ideal Toys to bring the Magic Cube to the Western world, and the puzzle made its international debut at the toy fairs of London, Paris, Nuremberg and New York in January and February 1980.

Packaging of Rubik's Cube, Toy of the year 1980- Ideal Toy Corp 1980, Made in Hungary.

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

Nichols assigned his patent to his employer Moleculon Research Corp., which sued Ideal Toy Company in 1982. In 1984, Ideal lost the patent infringement suit and appealed. In 1986, the appeals court affirmed the judgment that Rubik's 2×2×2 Pocket Cube infringed Nichols's patent, but overturned the judgment on Rubik's 3×3×3 Cube.[11]

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[12][13]. Hence, Ishigi's patent is generally accepted as an independent reinvention at that time.[14][15][16]

Rubik applied for another Hungarian patent on , and applied for other patents. In the United States, Rubik was granted U.S. Patent 4,378,116  on , for the Cube.

Greek inventor Panagiotis Verdes patented[17] a method of creating cubes beyond the 5×5×5, up to 11×11×11, in 2003 although he claims he originally thought of the idea around 1985.[18] As of June 19, 2008, the 5x5x5, 6x6x6, and 7x7x7 models are in production.

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

## Mechanics

Rubik's Cube partially disassembled.

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

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

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

## Mathematics

### Permutations

The original (3×3×3) Rubik's Cube has eight corners and twelve edges. There are 8! (40,320) ways to arrange the corner cubes. Seven can be oriented independently, and the orientation of the eighth depends on the preceding seven, giving 37 (2,187) possibilities. There are 12!/2 (239,500,800) ways to arrange the edges, since an odd permutation of the corners implies an odd permutation of the edges as well. Eleven edges can be flipped independently, with the flip of the twelfth depending on the preceding ones, giving 211 (2,048) possibilities.[19]

${8! \times 3^7 \times 12! \times 2^{10}} \approx 4.33 \times 10^{19}$

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

The 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:

${8! \times 3^8 \times 12! \times 2^{12}} \approx 5.19 \times 10^{20}.$

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

### 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, if one has a marker pen, one could, for example, mark the central squares of an unscrambled Cube with four coloured marks on each edge, each corresponding to the colour of the adjacent face. Some Cubes have also been produced commercially with markings on all of the squares, such as the Lo Shu magic square or playing card suits. Thus one can nominally solve a Cube yet have the markings on the centres rotated; it then becomes an additional test to solve the centers as well.

Marking the Rubik's Cube increases its difficulty because this expands its set of distinguishable possible configurations. When the Cube is unscrambled apart from the orientations of the central squares, there will always be an even number of squares requiring a quarter turn. Thus there are 46/2 = 2,048 possible configurations of the centre squares in the otherwise unscrambled position, increasing the total number of possible Cube permutations from 43,252,003,274,489,856,000 (4.3×1019) to 88,580,102,706,155,225,088,000 (8.9×1022).[20]

### Algorithms

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

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

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

Last Updated ( Tuesday, 22 June 2010 17:59 )