Rubik's Cube

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 and it 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 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 Hungarian patent #170062 for the mechanism, but did not take out international patents. (Ideal was somewhat reluctant to produce the toy for that reason, and indeed clones appeared almost immediately.) Terutoshi Ishigi acquired Japanese patent #55-8192 for a nearly identical mechanism while Rubik's patent was being processed, but Ishigi is generally credited with an independent reinvention.[1][2][3]

Description

A Rubik's Cube is a cubic block 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 x 3 x 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 be able to return the Cube to its original state, being a side with all the same colours from any position.

Workings

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 six are affixed to the core mechanisms. These provide structure for the other pieces to fit into and turn 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 prising 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 cubies 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.

Solutions

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 (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! × 3^8-1) × (12! × 2^12-1)/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, see Optimal solutions for Rubik's Cube.

Competitions

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.

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 Stacktimer.

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.

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 (see [4]).

Let Cube be the group of all legal cube operations (for example, swapping two positions cannot be achieved without disassembling the cube, so therefore this permutation of of the cube is not an element of Cube). We consider two subgroups of Cube: First the group of cube orientations, C_o, 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 C_o BR'D²RB'U²BR'D²RB'U², (twist two corners) RUDB²U²B'UBUB²D'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 = [U², D², F, B, L², R², R²U'FB'R²F'BU'R² ]

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 X C_p

Next we can take a closer look at these two groups. C_o is an abelian group, it is $Z_{3}^{7}\times Z_{2}^{11}$

Cube permutations, C_p, is little more complicated. It has the following two normal subgroups, the group of even permutations on the corners $A_{8}$ and the group of even permutations on the edges $A_{12}$ . Complementary to these two groups we can take a permutation that swaps two corners and swaps two edges. We obtain that C_p = $(A_{8}\times A_{12})$ X $Z_{2}$

Putting all the pieces together we get that the cube group is isomorphic to (Z₃⁷ × Z₂¹¹) X( ( A_8 x A_12 )X Z_2 )

This group can also be described as the quotient group [(Z₃⁷XS₈)×(Z₂¹¹X S₁₂)]/Z₂. 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×[(Z₃⁷X S₈)×(Z₂¹¹X S₁₂)]/Z₂.

The simple groups that occur as quotients in the composition series of R' are $A_{8},A_{12},\mathbb {Z} _{3}\ (7\ {\mbox{times}}),\mathbb {Z} _{2}\ (12\ {\mbox{times}})$ .

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

Most 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.

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