# Rubik's Cube

Left mouse button turns the planes, right button - the whole cube. Use the player controls for the simple sequence
The Rubik's Cube is a 3-D mechanical puzzle invented in 1974 by Hungarian sculptor and professor of architecture Ernő Rubik. Originally called the "Magic Cube", the puzzle was licensed by Rubik to be sold by Ideal Toys in 1980 and won the German Game of the Year special award for Best Puzzle that year. As of January 2009, 350 million cubes have sold worldwide making it the world's top-selling puzzle game. It is widely considered to be the world's best-selling toy.

In a classic Rubik's Cube, each of the six faces is covered by nine stickers, among six solid colours (traditionally white, red, blue, orange, green, and yellow). 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. Similar puzzles have now been produced with various numbers of stickers, not all of them by Rubik. The original 3×3×3 version celebrates its thirtieth anniversary in 2010.

Greek inventor Panagiotis Verdes patented a method of creating cubes beyond the 5×5×5, up to 11×11×11, in 2003 although he claims he originally thought of the idea around 1985. As of June 19, 2008, the 5×5×5, 6×6×6, and 7×7×7 models are in production.

## Mathematics

The original (3×3×3) Rubik's Cube has eight corners and twelve edges. There are 8! (40,320) ways to arrange the corner cubes. Seven can be oriented independently, and the orientation of the eighth depends on the preceding seven, giving (2,187) possibilities. There are 12!/2 (239,500,800) 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 (2,048) possibilities.[1]

There are exactly 43,252,003,274,489,856,000 permutations, which is approximately forty-three quintillion. 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 permutations reached through disassembly of the cube, the number becomes twelve times as large:

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

### Centre faces

The original Rubik's Cube had 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 unscrambled 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 nominally solve a Cube yet have the markings on the centres rotated; it then becomes an additional test to solve the centers as well.

Marking the Rubik's Cube increases its difficulty because this expands its 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 = 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 (4.3×10E19) to 88,580,102,706,155,225,088,000 (8.9×10E22).[2]

### Algorithms

Example of the sequence that modifies only one (top) plane
In Rubik's cubists' parlance, a memorised sequence of moves that has a desired effect on the cube is called an algorithm. This terminology is derived from the mathematical use of algorithm, meaning a list of well-defined instructions for performing a task from a given initial state, through well-defined successive states, to a desired end-state. Each method of solving the Rubik's Cube employs its own set of algorithms, together with descriptions of what the effect of the algorithm is, and when it can be used to bring the cube closer to being solved.

Most algorithms are designed to transform only a small part of the cube without scrambling other parts that have already been solved, so that they can be applied repeatedly to different parts of the cube until the whole is solved. For example, there are well-known algorithms for cycling three corners without changing the rest of the puzzle, or flipping the orientation of a pair of edges while leaving the others intact.

Some algorithms have a certain desired effect on the cube (for example, swapping two corners) but may also have the side-effect of changing other parts of the cube (such as permuting some edges). Such algorithms are often simpler than the ones without side-effects, and are employed early on in the solution when most of the puzzle has not yet been solved and the side-effects are not important. Towards the end of the solution, the more specific (and usually more complicated) algorithms are used instead, to prevent scrambling parts of the puzzle that have already been solved.

### Optimal solutions

Although there are a significant number of possible permutations for the Rubik's Cube, there have been a number of solutions developed which allow for the cube to be solved in well under 100 moves.

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. In 1982, David Singmaster and Alexander Frey hypothesised 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 26 moves or less.[3][4][5] In 2008, Tomas Rokicki lowered that number to 22 moves.[6][7][8]

A solution commonly used by speed cubers was developed by Jessica Fridrich. It is similar to the layer-by-layer method but employs the use of a large number of algorithms, especially for orienting and permuting the last layer. The cross is done first followed by first-layer corners and second layer edges simultaneously, with each corner paired up with a second-layer edge piece. This is then followed by orienting the last layer then permuting the last layer (OLL and PLL respectively). Fridrich's solution requires learning roughly 120 algorithms but allows the Cube to be solved in only 55 moves on average.

Philip Marshall's The Ultimate Solution to Rubik's Cube is a modified version of Fridrich's method, averaging only 65 twists yet requiring the memorization of only two algorithms.[9]

A now 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. The principle behind this is that in layer by layer you must constantly break and fix the first layer; the 2×2×2 and 2×2×3 sections allow three or two layers to be turned without ruining progress. One of the advantages of this method is that it tends to give solutions in fewer moves.

In 1997, Denny Dedmore published a solution described using diagrammatic icons representing the moves to be made, instead of the usual notation.[10]

## Competitions and records

### Speedcubing competitions

Speedcubing (or speedsolving) is the practice of trying to solve a Rubik's Cube in the shortest time possible. There are a number of speedcubing competitions that take place around the world.

The first world championship organised by the Guinness Book of World Records was held in Munich on March 13, 1981. All Cubes were moved 40 times and lubricated 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, with a time of 22.95 seconds.

Since 2003, the winner of a competition is determined by taking the average time of the middle three of five attempts. However, the single best time of all tries is also recorded. The World Cube Association maintains a history of world records.[11] In 2004, the WCA made it mandatory to use a special timing device called a Stackmat timer.

### Records

The current world record for single time on a 3×3×3 Rubik's Cube was set by Erik Akkersdijk in 2008, who had a best time of 7.08 seconds at the Czech Open 2008. The world record average solve is currently held by Feliks Zemdegs; which is 9.21 seconds at the Melbourne Summer Open 2010.

On March 17, 2010, 134 school boys from Dr Challoner's Grammar School, Amersham, England broke the previous Guinness World Record for most people solving a Rubik's cube at once in 12 minutes.[12] The previous record set in December 2008 in Santa Ana, CA achieved 96 completions.

## References

1. 1 Martin Schönert "Analyzing Rubik's Cube with GAP": the permutation group of Rubik's Cube is examined with GAP computer algebra system
2. 2 Scientific American, p28, vol 246, 1982 retrieved online Jan 29, 2009.
3. 3 Template:Cite conference
4. 4 Template:Cite conference
5. 5 Cracking the Cube
6. 6 Twenty-Five Moves Suffice for Rubik's Cube
7. 7 Rubik's Cube Algorithm Cut Again, Down to 23 Moves
8. 8 Twenty-Two Moves Suffice
9. 9 Philip Marshall (2005), The Ultimate Solution to Rubik's Cube.
10. 10 Website with solutions created by Denny Dedmore
11. 11  World Cube Association Official Results
12. 12 BBC: Pupils break Rubik's Cube Record

Acknowledgements

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