**Rubik's Cube**™ is a mechanical puzzle invented by the Hungarian sculptor and professor of architecture Ernő 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. 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 x 4 x 4 version of the Rubik's Cube. There are also 2 x 2 x 2 and 5 x 5 x 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. Ernő Rubik holds patents related to the cube's mechanism.

Table of contents |

2 Workings 3 Solutions 4 Competitions 5 Rubik's Cube as a mathematical group 6 Parallel with particle physics 7 A greater challenge 8 References 9 External links |

The challenge is to be able to return the Cube to its original state from any position.

Countless general solutions for the Rubik's Cube have been discovered independently (see How to solve the Rubik's Cube for one such solution). 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. 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*. 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! × 3^{8-1}) × (12! × 2^{12-1})/2 = 43,252,003,274,489,856,000 different positions (~4.3 × 10^{19}), 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, see Optimal solutions for Rubik's Cube.

Many 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 20.02 seconds 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.

Many individuals have recorded shorter times, but these records are not recognized due to lack of compliance with agreed-upon standards for timing and competing.

Many mathematicians are interested in the Rubik's Cube partly because it is a tangible representation of a mathematical group.

We analyse the group structure of the cube group. We assume the notation described in How to solve the Rubik's Cube. Also we assume the orientation of the six centre pieces to be fixed. Computations regarding the cube's group structure can be carried out by a computer, for example using GAP computer algebra system

Let Cube be the group of all cube positions. We take two subgroups:
First Cube orientation, *C*_{o}, which leaves every block fixed, but can change it 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 *C*_{o}
BR'D^{2}RB'U^{2}BR'D^{2}RB'U^{2}, (twist two corners)
RUDB^{2}U^{2}B'UBUB^{2}D'R'U'. (flip two edges)

For the second group we take Cube permutations, *C*_{p}, 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).

*C*_{p} = [U^{2}, D^{2}, F, B, L^{2}, R^{2}, R^{2}U'FB'R^{2}F'BU'R^{2} ]

Since *C*_{o} 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 = *C*_{o} *C*_{p}

Next we can take a closer look at these two groups. *C*_{o} is
an abelian group, it is

Cube permutations, *C*_{p}, is little more complicated. It has the following two normal subgroups, the group of even permutations on the corners
and the group of even permutations on the edges .
Complementary to these two groups we can take a permutation that swaps two corners and swaps two edges. We obtain that
*C*_{p} =

Putting all the pieces together we get that the cube group is isomorphic to (

This group can also be described as the quotient group
[(**Z**_{3}^{7}_{8})×(**Z**_{2}^{11} S_{12})]/**Z**_{2}. When one wants to take
the possible permutations of the centre pieces into account, an other direct component arises, which describes the 24 rotations of cube as a whole
, if call this group T, we obtain: T×[(**Z**_{3}^{7} S_{8})×(**Z**_{2}^{11} S_{12})]/**Z**_{2}.

The simple groups that occur as quotients in the composition series of R' are .

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.

*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.

- Rubik's Cube and its variants:
- Rubik's Cube Art:
- Solutions for the puzzle:
- Official world records:
- Online versions of Rubik's Cube:
- Rubik's Cube Java Applet: http://www.schubart.net/rc/
- Another Cube applet: http://user.tninet.se/~ecf599g/aardasnails/java/PuzzleApplet/webpages
- Yet Another Cube applet (with a more 3D feel): http://www.andkon.com/arcade/puzzle/rubikscube/
- An ActiveX version of this puzzle (play it and learn how to solve it): http://www.carobit.com/rubik/rubik.html