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Rubik's Cube

Rubik's Cube is a mechanical puzzle invented by the Hungarian sculptor and professor of architecture Erno Rubik in 1974. It has been estimated that over 100,000,000 Rubik's Cubes or imitations have been sold worldwide.


The Rubik's Cube reached its height of popularity during the early 1980s, and it is still a popular toy nowadays. 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 Professor's Cube, respectively), and puzzles in other shapes, such as the Pyraminx, a tetrahedron

"Rubik's Cube" is a trademark of Seven Towns Limited. Erno Rubik holds Hungarian patent HU170062 for the mechanism, but did not take out international patents. Ideal Toys was somewhat reluctant to produce the toy for that reason, and indeed clones appeared almost immediately. Ideal Toys later lost, in 1984, 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.


The standard Rubik's Cube.A Rubik's Cube is a plastic cube with its surface subdivided so that each face consists of nine squares. Each face can be rotated, giving the appearance of an entire slice of the block rotating upon itself. This gives the impression that the cube is made up of 27 smaller cubes (3 × 3 × 3). In its original state each side of the Rubik's Cube is a different color, but the rotation of each face allows the smaller cubes to be rearranged in many different ways.

The challenge is to return the Cube from any state to its original state, in which each face consists of nine squares of a single colour.


A standard cube measures approximately 2 1/8 inches (5.4 cm) on each side. The puzzle consists of the 26 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 21 pieces: a single core, of three intersecting axes holding the six centre squares in place but letting them rotate, and 20 smaller plastic pieces which fit into it to form a cube. The cube can be taken apart without much difficulty, typically by prying an "edge cubie" away from a "center cubie" until it dislodges. It is a simple process to "solve" a cube in this manner, by reassembling the cube in a solved state; however, this is not the challenge.

There are 12 edge pieces which show two colored sides each, and 8 corner pieces which show three colours. Each piece shows a unique colour combination, but not all combinations are realized (For example, there is no edge piece showing both white and yellow, 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 degrees, 180 degrees or 270 degrees; 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 colors of the stickers are red opposite orange, yellow opposite white, and green opposite blue. However, there also exist Cubes with alternative color 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).


Countless general solutions for the Rubik's Cube have been discovered independently (see How to solve the Rubik's Cube for one such solution). A popular method is the "layer by layer", in which one face is solved, following the underlying row, the middle row, and next the last and bottom face. Solutions typically consist of a sequence of processes. A process is a series of cube twists which accomplishes a well-defined 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 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. Also a lot of research has been done on the topic of Optimal solutions for Rubik's Cube.

Patrick Bossert, a 12 year old schoolboy from Britain, 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 17 editions and became the number one book on both The Times and the New York Times bestseller lists for 1981.

A Rubik's Cube can have (8! × 38-1) × (12! × 212-1)/2 = 43,252,003,274,489,856,000 different positions (~4.3 × 1019), about 43 quintillion, but it is advertised only as having "billions" of positions, due to the general incomprehensibility of that number. Despite the vast number of positions, all cubes can be solved in 29 moves or fewer.


Many speedcubing competitions have been held to determine who can solve the Rubik's Cube in the shortest amount of time. The first world championship was held on 5 June 1982 in Budapest and was won by Minh Thai, a Vietnamese student from Los Angeles with a time of 22.95 seconds. The official world record of 20.00 seconds (average of 5 cubes) was set on August 24th 2003 in Toronto by Dan Knights, a San Francisco software developer. This record is recognized by the trademark holders of "Rubik's Cube" as well as by the Guinness Book of Records.

However, virtually all "speedcubers" recognize that Shotaro "Macky" Makisumi is the legitimate world record holder with a time of 12.11 seconds, solved during the 2004 Caltech Spring Tournament in competition. It is relatively common in recent tournaments to see times much faster than Dan Knights'. Makisumi was an 8th grade student at the time at the age of 14.

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. Therefore only records set during official world championships were acknowledged. In 2003 a new set of standards have been agreed-upon, with a special timing device called a Stackmat timer.

Rubik's Cube As A Mathematical Group

Many mathematicians are interested in the Rubik's Cube partly because it is a tangible representation of a mathematical group. Specifically, the cube group is the set of all legal cube operations with composition as the group operation.

The order of the cube group is equal to the number of possible positions obtainable by the cube. This is or 43,252,003,274,489,856,000. This factorizes as 2273145372111. Because of the large size of the cube group it is sometimes useful to analyze the structure with the assistance of a computer algebra system such as GAP .

We consider two subgroups of Cube: First the group of cube orientations, Co, which leaves every block fixed, but can change its orientation. This group is a normal subgroup of the Cube group. It can be represented as the normal closure of some operations that flip a few edges or twist a few corners. For example the normal closure of the following two operation is Co BR'D2RB'U2BR'D2RB'U2, (twist two corners) RUDB2U2B'UBUB2D'R'U'. (flip two edges)

For the second group we take Cube permutations, Cp, which can move the blocks around, but leaves the orientation fixed. For this subgroup there are more choices, depending on the precise way you fix the orientation. One choice is the following group, given by generators: (The last generator is a 3 cycle on the edges).

Cp = [U2, D2, F, B, L2, R2, R2U'FB'R2F'BU'R2 ]

Since Co is a normal subgroup, the intersection of Cube orientation and Cube permutation is the identity, and their product is the whole cube group, it follows that the cube group is the semidirect product of these two groups. That is Cube = Co Cp Next we can take a closer look at these two groups. Co is an abelian group, it is .

Cube permutations, Cp, is little more complicated. It has the following two normal subgroups, the group of even permutations on the corners A8 and the group of even permutations on the edges A12. Complementary to these two groups we can take a permutation that swaps two corners and swaps two edges. We obtain that Putting all the pieces together we get that the cube group is isomorphic to.

Parallel With Particle Physics

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 counterclockwise "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 cubie twists are paralleled by allowable combinations of quarks and antiquarks—both cubie twist and the quark/antiquark charge must total to an integer. Combinations of two or three twisted corners may be compared to various hadrons. This, however, is not always feasible.

A Greater Challenge

Rubik's Cube on a diagonal tiltMost Rubik's Cubes are sold without any markings on the center faces. This obscures the fact that the center 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 color of the adjacent square. Some cubes have also been commercially produced with markings on all of the squares, such as the Lo Shu magic square or playing card suits. You might be surprised to find you could scramble and then unscramble the cube but still leave the markings rotated.

Putting markings on the Rubik's cube increases the challenge of solving the cube, chiefly because it expands the set of distinct possible configurations. It can be shown that, 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.

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