# Difference between revisions of "General Information"

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===Permutations=== | ===Permutations=== | ||

A normal (3×3×3) Rubik's Cube has eight corners and twelve edges. There are [[factorial|8!]] ways to arrange the corner cubies. Seven can be oriented independently, and the orientation of the eighth depends on the preceding seven, giving 3<sup>7</sup> possibilities. There are 12!/2 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 2<sup>11</sup> possibilities. | A normal (3×3×3) Rubik's Cube has eight corners and twelve edges. There are [[factorial|8!]] ways to arrange the corner cubies. Seven can be oriented independently, and the orientation of the eighth depends on the preceding seven, giving 3<sup>7</sup> possibilities. There are 12!/2 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 2<sup>11</sup> possibilities. | ||

− | :< | + | :<tex> {8! \cdot 3^7 \cdot 12! \cdot 2^{10}} \approx 4.33 \cdot 10^{19}</tex> |

There are exactly 43,252,003,274,489,856,000 possibilities. In other words, there are forty-three [[quintillion]] ([[short scale]]) or forty-three trillion ([[long scale]]). The puzzle is often advertised as having only "[[1000000000 (number)|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]]. | There are exactly 43,252,003,274,489,856,000 possibilities. In other words, there are forty-three [[quintillion]] ([[short scale]]) or forty-three trillion ([[long scale]]). The puzzle is often advertised as having only "[[1000000000 (number)|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]]. |

## Revision as of 01:12, 5 October 2008

The Rubik's Cube is a mechanical puzzle invented in 1974 by Hungarian sculptor and professor of architecture Ernő Rubik. Originally called the "Magic Cube" by its inventor, this puzzle was renamed "Rubik's Cube" by Ideal Toys in 1980 and won the German Game of the Year special award for Best Puzzle that year. It is said to be the world's best-selling toy, with over 300,000,000 Rubik's Cubes and imitations sold worldwide.

In a classic Rubik's Cube, each of the six faces is covered by 9 stickers, among six solid colours (traditionally being white, yellow, orange, red, blue, and green). 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.

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. There exist four widely available variations: the 2×2×2 (Pocket Cube, also Mini Cube, Junior Cube, or Ice Cube), the standard 3×3×3 cube, the 4×4×4 (Rubik's Revenge, or Master Cube), and the 5×5×5 (Professor's Cube). Recently, larger sizes are also on the market (V-Cube 6 and V-Cube 7). All of these items belong to a broad category of puzzles commonly referred to as "twisty puzzles".

For readability, 3x3x3 is frequently abbreviated 3×3 (and similarly for the other sizes) when there is no ambiguity. Common misspellings include "rubix cube", "rubics cube", "rubick's cube", and "rubiks cube".

## Contents

## Conception and development

In March 1970, Larry Nichols invented a 2×2×2 "Puzzle with Pieces Rotatable in Groups" and filed a Canadian patent application for it. Nichols's cube was held together with magnets. Nichols was granted Template:US patent on April 11, 1972, two years before Rubik invented his improved cube.

On April 9, 1970, Frank Fox applied to patent his "Spherical 3×3×3". He received his UK patent (1344259) on January 16, 1974.

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.

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.

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 and was granted patent JP55‒8192 (1976); Ishigi's is generally accepted as an independent reinvention.

Rubik applied for another Hungarian patent on October 28, 1980, and applied for other patents. In the United States, Rubik was granted Template:US patent on March 29, 1983, for the Cube.

Greek inventor Panagiotis Verdes patented a method of creating cubes beyond the 5×5×5, up to 11×11×11. His designs, which include improved mechanisms for the 3×3×3, 4×4×4, and 5×5×5, are suitable for speedcubing, whereas existing designs for cubes larger than 5×5×5 are prone to break. As of June 19, 2008, 5x5x5, 6x6x6, and 7x7x7 models are available (V-Cube Official Site.

## Workings

A standard cube measures approximately 2¼ inches (5.7 cm) on each side. The puzzle consists of the twenty-six unique miniature cubes on the surface. However, the centre cube of each face is merely a single square façade; all 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. However, as prying loose a corner cube is a good way to break off a centre cube — thus ruining the Cube — it is far safer to lever a centre cube out using a screwdriver. 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 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. 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. Hofstader, in the July 1982 *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 been produced commercially.

### Permutations

A normal (3×3×3) Rubik's Cube has eight corners and twelve edges. There are 8! ways to arrange the corner cubies. Seven can be oriented independently, and the orientation of the eighth depends on the preceding seven, giving 3^{7} possibilities. There are 12!/2 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 2^{11} possibilities.

There are exactly 43,252,003,274,489,856,000 possibilities. In other words, there are forty-three quintillion (short scale) or forty-three trillion (long scale). 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.

The preceding figure is limited to permutations that can be reached solely by turning the sides of the cube. If one considers permuations reached through disassembly of the cube, the number becomes twelve times as large:

- <math> {8! \cdot 3^8 \cdot 12! \cdot 2^{12}} \approx 5.19 \cdot 10^{20}</math>

The full number is 519,024,039,293,878,272,000 or 519 quintillion (on the short scale) possible arrangements of the pieces that make up the Cube, but only one in twelve of these are actually solveable. 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.

Despite the vast number of positions, all Cubes can be solved in twenty-five or fewer moves (see Optimal solutions for Rubik's Cube).
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The large number of permutations is often given as a measure of the Rubik's cube's complexity. However, the puzzle's difficulty does not necessarily follow from the large number of permutations. The problem of putting the 26 letters of the alphabet in alphabetical order has a larger complexity (26! = 4.03 × 10^{26} possible orderings), but is less difficult.

### Centre faces

The original (official) Rubik's Cube has 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 unshuffled 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 scramble and then unscramble the Cube yet have the markings on the centers rotated, and it becomes an additional test to "solve" the centers as well. This is known as "supercubing"Template:Fact.

Putting markings on the Rubik's Cube increases the difficulty mainly because it expands the 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 4^{6}/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 (4.3×10^{19}) to 88,580,102,706,155,225,088,000 (8.9×10^{22}).

## Solutions

Many general solutions for the Rubik's Cube have been discovered independently. The most popular method was developed by David Singmaster and published in the book *Notes on Rubik's "Magic Cube"* in 1981. This solution involves solving the Cube layer by layer, in which one layer, designated the top, is solved first, followed by the middle layer, and then the final and bottom layer. After practice, solving the Cube layer by layer can be done in under one minute. Other general solutions include "corners first" methods or combinations of several other methods.
Most tutorials teach the layer by layer method, as it gives an easy-to-understand step-by-step guide on how to solve it.

Speedcubing solutions have been developed for solving the Rubik's Cube as quickly as possible. The most common speedcubing solution was developed by Jessica Fridrich. It is a very efficient layer-by-layer method that requires a large number of algorithms (see below), especially for orienting and permuting the last layer. The first-layer corners and second layer are done simultaneously, with each corner paired up with a second-layer edge piece. Another well-known method was developed by Lars Petrus. In this method, a 2×2×2 section is solved first, followed by a 2×2×3, and then the incorrect edges are solved using a three-move algorithm, which eliminates the need for a possible 32-move algorithm later. One of the advantages of this method is that it tends to give solutions in fewer moves. For this reason, the method is also popular for fewest move competitions.

Solutions follow a series of steps and include a set of algorithms for solving each step. An algorithm, also known as a process or an operator, is a series of twists that accomplishes a particular goal. For instance, one algorithm might switch the locations of three corner pieces, while leaving the rest of the pieces in place. Basic solutions require learning as few as four or five algorithms but are generally inefficient, needing around 100 twists on average to solve an entire Cube. In comparison, Fridrich's advanced solution requires learning roughly 120 algorithms but allows the Cube to be solved in only 55 moves on average. A different kind of solution developed by Ryan Heise<ref>Ryan Heise's method</ref> uses no algorithms but rather teaches a set of underlying principles that can be used to solve in fewer than 40 moves. A number of complete solutions can also be found in any of the books listed in the bibliography, and most can be used to solve any Cube in under five minutes.

### Algorithms

In Rubik's cubists' parlance, an *algorithm* means "a memorized sequence of moves whose effect on the cube is known". This fully conforms with the mathematical and logical use of algorithm defined as *a list of well-defined instructions for completing a task from a given initial state, through well-defined successive states, to a desired end-state*. A Rubik's cube algorithm transforms the state of the cube in such a way that a small part of the cube becomes solved without "scrambling" any parts that have previously been solved, or else places the cube in a state from which the solver knows it can now be partly, or fully, solved by the application of further algorithms.

For instance, if we label the six sides of a cube like the six sides of a die, the sequence of movements 116622553344 will have a definite effect, namely, it will transform a solved cube into a cube with an "X" design in each face. More complicated sequences of movements will have more useful results, such as swapping three corners of the third layer without moving any other pieces. The sequences that are useful to solve the cube are called "algorithms".

### The search for optimal solutions

Template:Main The manual solution methods described above are intended to be easy to learn, but much effort has gone into finding even faster solutions to the Rubik's Cube.

In 1982, David Singmaster and Alexander Frey hypothesized that the number of moves needed to solve the Rubik's Cube, given an ideal algorithm, might be in "the low twenties". In 2007, Daniel Kunkle and Gene Cooperman used computer search methods to demonstrate that any 3×3×3 Rubik's Cube configuration can be solved in a maximum of 26 moves.

In 2008, Tomas Rokicki lowered the maximum to 22 moves.

Work continues to try to reduce the upper bound on optimal solutions.
The arrangement known as the super-flip, where every edge is in its correct position but flipped, requires 20 moves to be solved (Using the official notation, these are: U R2 F B R B2 R U2 L B2 R U' D' R2 F R' L B2 U2 F2). No arrangement of the Rubik's Cube has been discovered so far that requires more than 20 moves to solve.

### Competitions and record times

Many speedcubing competitions have been held to determine who can solve the Rubik's Cube in the shortest time. The number of contests is going up every year; there were 72 official competitions from 2003 to 2006; 33 were in 2006 alone.

The first world championship organized by the *Guinness Book of World Records* was held in Munich on March 13, 1981. All Cubes were moved 40 times and rubbed with petroleum jelly. The official winner, with a record of 38 seconds, was Jury Froeschl, born in Munich.

The first international world championship was held in Budapest on June 5, 1982, and was won by Minh Thai, a Vietnamese student from Los Angeles, California, with a time of 22.95 seconds.

Since 2003, competitions are decided by the best average of 5, dropping the best and worst time and averaging the middle 3 solves. The World Cube Association maintains a database of all World Cube Association official attempts. In 2004, the WCA made it mandatory to use a special timing device called a Stackmat timer.

The current world record for single time is set by Erik Akkersdijk in 2008; he set a best time of 7.08 at the Czech Open 2008. The world record average solve is by Yu Nakajima, when he set a world record average of 11.28 seconds on May 4, 2008.

### Alternative competitions

In addition, alternative competitions are held (these are official WCA recognized events). These include:

- Blindfolded solving

Rubik's 3x3x3 Cube: Blindfolded records

- Solving the Cube using a single hand

Rubik's 3x3x3 Cube: One-handed

- Solving the Cube with one's feet