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Rubik's Cube

Rubik's Cube on a diagonal tilt
The standard 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.

 Contents

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 × 3 × 3). In its original state each side of the Rubik's Cube is a different colour, 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.

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 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 coloured 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 colours 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! × 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 there exist an algorithm that can always solve the cube in 29 moves or fewer, tough in practice often much less. Some people believe that all cubes can be solved in 20 moves or fewer tough there is no proof for that. The algorithm that always solves a cube in the shortest possible number of moves is called 'god's algorithm'.

During the Falklands War a Royal Navy Submariner found one of the quickest solution; he put it in an oven at low temperature for a couple of minutes. The glue on the back of the colored squares melted a bit and he was able to peel them off and put them all back in the finished position.

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 June 5, 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 24, 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 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.

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 [1] http://www.gap-system.org/Intro/rubik.html ).

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 the cube is not an element of Cube). 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 $Z_3^7 \times Z_2^{11}$

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 Cp = $( A_8 \times A_{12} )$ $\,\!Z_2$

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

This group can also be described as the quotient group [(Z37S8)×(Z211 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×[(Z37 S8)×(Z211 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 coloured marks on each edge, each corresponding to the colour 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

Rubik's Cube and its variants

• Twisty Puzzles Collection http://www.calormen.com/TwistyPuzzles/Photos/

Rubik's Cube art

• Rubik's Cube Art http://www.wunderland.com/WTS/Jake/CubeArt/

Solutions for the puzzle

• How to Solve the Rubik's Cube http://jeays.net/rubiks.htm
• Rubik's Cube http://cube.helm.lu
• Solving Rubik's Cube for speed http://lar5.com/cube/index.html
• Beginner Solution to the Rubik's Cube http://www.geocities.com/jasmine_ellen/RubiksCubeSolution.html
• Simple Way of Solving the Rubik's Cube http://www.freewebs.com/rubiksolution

Official world records

• Rubik's Official Online Site http://www.rubikschamps.com/

Unofficial world records

• speedcubing.com http://www.speedcubing.com

World championship 2003

• World Rubik's Games Championship 2003 http://www.speedcubing.com/events/wc2003

European championship 2004

• European Rubik's Games Championships 2004 http://www.speedcubing.com/events/euro2004

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
• Disassembled Cube (can be taken apart to see how the cube works) http://www.randelshofer.ch/rubik/virtualcubes/
• A JavaScript version to play online http://en.ludanto.org/rubik.shtml
• The version 2x2x2 with JavaScript http://en.ludanto.org/rubik2.shtml
• Magic Cube 4D http://www.superliminal.com/cube/cube.htm - 4D variant

Last updated: 02-27-2005 05:06:20