Rubik's Cube

2008/9 Schools Wikipedia Selection. Related subjects: Games; Mathematics

Variations of Rubik's Cubes (from left to right: Rubik's Revenge, Rubik's Cube, Professor's Cube, & Pocket Cube (also known as "Mini-Cube")).

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 also won the 1980 German Game of the Year ( Spiel des Jahres) special award for Best Puzzle. 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 typical Cube, each face is covered by nine stickers of one of six solid colours. When the puzzle is solved, each face of the Cube is a solid colour. The Cube celebrated its twenty-fifth anniversary in 2005, when a special edition Cube in a presentation box 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.

The puzzle comes in four widely available versions: the 2×2×2 ( Pocket Cube, also Mini Cube or Ice Cube), the 3×3×3 standard cube, the 4×4×4 ( Rubik's Revenge), and the 5×5×5 ( Professor's Cube). Even larger sizes have been built and are to be launched in September of 2008.

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 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 debut at toy fairs 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 on March 29, 1983, for the Cube.

Recently, 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 1, 2007, these designs were not yet widely available, although videos of actual, working prototypes for the 6×6×6 and 7×7×7 have been released, and it was recently announced that these cubes would be released sometime in 2008.

Popularity and Pop Culture

Over 100 million Rubik's Cubes were sold in the period from 1980 to 1982. It won the BATR Toy of the Year award in 1980 and again in 1981. Ideal Toys published a Rubik's Cube Newsletter from 1982 to 1983.

In 1983, there was a Saturday morning cartoon series based on this toy, entitled Rubik, the Amazing Cube. It lasted for one season.

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 4×4×4 version of the Rubik's Cube. There are also 2×2×2 and 5×5×5 Cubes (known as the Pocket Cube and the Professor's Cube, respectively) and puzzles in other shapes, such as the Pyraminx, a tetrahedron.

In May 2005, the Greek inventor Panagiotis Verdes constructed a 6×6×6 Rubik's Cube; on May 23, 2006, Frank Morris, a world champion Rubik's Cube solver, tested this version. He had previously solved the 3×3×3 in 15 seconds, the 4×4×4 in 1 minute and 10 seconds, and the 5×5×5 in 1 minute and 46.1 seconds. The 6×6×6 took him 5 minutes and 37 seconds to solve. Morris himself thanked the inventor for making it and purportedly stated that the bigger the Cube is, the greater the pleasure. In July 2006, Verdes successfully constructed the 7×7×7 cube; on October 27, 2006, a video of Morris testing the cube was released. He solved this cube in 6 minutes and 29.31 seconds. Videos of these tests can be viewed at http://www.v-cubes.com.

In 1994, Melinda Green, Don Hatch, and Jay Berkenilt created a working software model of a 3×3×3×3 four-dimensional analogue of a Rubik's Cube called the MagicCube4D. In 2006, Roice Nelson and Charlie Nevill created a working 3×3×3×3×3 five-dimensional software analogue.

In 1981, Patrick Bossert, a twelve-year-old schoolboy from England, published his own solution in a book called You Can Do the Cube ( ISBN 0-14-031483-0). The book sold over 1.5 million copies worldwide in seventeen editions and became the number one book on The Times. He didn't reach the New York Times Best Seller list for that year .

At the height of the puzzle's popularity, separate sheets of coloured stickers were sold so that frustrated or impatient Cube owners could restore their puzzle to its original appearance.

The "Rubik's Cube" is known as such in most languages except for Hebrew, Chinese, German, Portuguese and Icelandic. In Hebrew it is known as the "Hungarian Cube", whilst in the other languages it is known as "Magic Cube" ("魔方" in Chinese, Zauberwürfel in German, Cubo Mágico in Portuguese and Töfrateningur in Icelandic).

In 1982 at the World's Fair held in Knoxville, Tennessee, a six-foot rotating cube was put on display at the World's Fair Park. After the fair, it was moved and forgotten until July 2007, when it was restored and placed in the Holiday Inn lobby that overlooks the park.

The Rubik's Cube has had numerous cameos in movies and TV shows since its heyday in the 1980s, most recently in Scrubs, Hellboy and The Pursuit of Happyness.

Workings

Rubik's Cube partially disassembled.

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 facade; 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 follow the maxim "When all else fails, use a screwdriver", and lever a centre cube out. It is a 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).

Permutations

A normal (3×3×3) Rubik's Cube can have (8! × 3⁸⁻¹) × (12! × 2¹²⁻¹)/2 = 43,252,003,274,489,856,000 different positions (permutations), or about 4.3 × 10¹⁹, 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. If they were laid side by side, it would cover the earth approximately 276 times.

In fact, there are (8! × 3⁸) × (12! × 2¹²) = 519,024,039,293,878,272,000 (about 5.2 × 10²⁰ 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 reachable. This is because there is no sequence of moves that will swap a single pair 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-six or fewer moves (see Optimal solutions for Rubik's Cube).

The large number of permutations is often given as a measure of the Rubik's cube's complexity. However, the puzzles difficulty does not necessarily follow from the large number of permutations. The problem of putting a jumbled set of encyclopedias (26 volumes) in alphabetical order has a larger complexity, 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".

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 4⁶/2 = 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 (4.3×10¹⁹) to 88,580,102,706,155,225,088,000 (8.9×10²²).

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 sites 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, 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 53+ algorithms but allows the Cube to be solved in only 55 moves on average. A different kind of solution developed by Ryan Heise 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.

The search for optimal solutions

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. Work continues to try to reduce the upper bound on optimal solutions to 25 moves, or even lower. The arrangement known as the super-flip (U R2 F B R B2 R U2 L B2 R U' D' R2 F R' L B2 U2 F2), where every edge is in its correct position but flipped, requires 20 moves to be solved. No arrangement of the Rubik's Cube has been discovered so far that requires more than 20 moves to solve.

Move notation

Rubik's Cube in a tilted state.

Rubik's Cube in solved state.

Most 3×3×3 Rubik's Cube solution guides use the same notation, originated by David Singmaster, to communicate sequences of moves. This is generally referred to as "cube notation" or in some literature "Singmaster notation" (or variations thereof). Its relative nature allows algorithms to be written in such a way that they can be applied regardless of which side is designated the top or how the colours are organized on a particular cube.

• F (Front): the side currently facing you
• B (Back): the side opposite the front
• U (Up): the side above or on top of the front side
• D (Down): the side opposite the top, underneath the Cube
• L (Left): the side directly to the left of the front
• R (Right): the side directly to the right of the front
• x (rotate): rotate the Cube up
• y (rotate): rotate the Cube to the left
• z (rotate): rotate the Cube on its side to the right

When an apostrophe follows a letter, it means to turn the face counter-clockwise a quarter-turn, while a letter without an apostrophe means to turn it a quarter-turn clockwise. Such an apostrophe mark is pronounced prime. A letter followed by a 2 (occasionally a superscript ²) means to turn the face a half-turn (the direction does not matter). Lowercase letters indicate that you should move that face and the face next to it. So R is right side counter-clockwise, but r is right side and centre side next to it counter-clockwise. When x, y or z are primed, simply rotate the cube in the opposite direction. When they are squared, rotate it twice. For 'z', you should still be viewing the same front face when rotating.

This notation can also be used on the Pocket Cube, the Revenge, and the Professor, with additional notation. They not only have the F, B, L, R, U, D notation but also f, b, l, r, u, d. For example: (Rr)' l2 f'

(Some solution guides, including Ideal's official publication, The Ideal Solution, use slightly different conventions. Top and Bottom are used rather than Up and Down for the top and bottom faces, with Back being replaced by Posterior. '+' indicates clockwise rotation and '-' counterclockwise, with '++' representing a half-turn. However, alternative notations failed to catch on, and today the Singmaster scheme is used universally by those interested in the puzzle.)

Less-often used moves include rotating the entire Cube or two-thirds of it. The letters x, y, and z are used to indicate that the entire Cube should be turned about one of its axes. The x-axis is the line that passes through the left and right faces, the y-axis is the line that passes through the up and down faces, and the z-axis is the line that passes through the front and back faces. (This type of move is used infrequently in most solutions, to the extent that some solutions simply say "stop and turn the whole cube upside-down" or something similar at the appropriate point.)

Lowercase letters f, b, u, d, l, and r signify to move the first two layers of that face while keeping the remaining layer in place. This is of course equivalent to rotating the whole cube in that direction, then rotating the opposite face back the same amount in the opposite direction, but is useful notation to describe certain triggers for speedcubing. Furthermore, M, E, and S (and respectively their lowercase for larger sized cubes) are used for inner-slice movements. M signifies turning the layer that is between L and R downward (clockwise if looking from the left side). E signifies turning the layer between U and D towards the right (counter-clockwise if looking from the top). S signifies turning the layer between F and B clockwise.

For example, the algorithm (or operator, or sequence) F2 U' R' L F2 R L' U' F2, which cycles three edge cubes in the top layer without affecting any other part of the cube, means:

1. Turn the Front face 180 degrees.
2. Turn the Up face 90 degrees counterclockwise.
3. Turn the Right face 90 degrees counterclockwise.
4. Turn the Left face 90 degrees clockwise.
5. Turn the Front face 180 degrees.
6. Turn the Right face 90 degrees clockwise.
7. Turn the Left face 90 degrees counterclockwise.
8. Turn the Up face 90 degrees counterclockwise.
9. Finally, turn the Front face 180 degrees.

For beginning students of the Cube, this notation can be daunting, and many solutions available online therefore incorporate animations that demonstrate the algorithms presented.

4×4×4 and larger cubes use slightly different notation to incorporate the middle layers. Generally speaking, uppercase letters (F B U D L R) refer to the outermost portions of the cube (called faces). Lowercase letters (f b u d l r) refer to the inner portions of the cube (called slices). Again Ideal breaks rank by describing their 4×4×4 solution in terms of layers (vertical slices that rotate about the z-axis), tables (horizontal slices), and books (vertical slices that rotate about the x-axis).

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 are 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, with a time of 22.95 seconds.

Since 2003, competitions are decided by the best average (middle three of five attempts); but the single best time of all tries is also recorded. The World Cube Association maintains a history of world records. In 2004, the WCA made it mandatory to use a special timing device called a Stackmat timer.

The current world records for both average and single times were set by Edouard Chambon in 2008, he set an average of 11.48 seconds and a best time of 9.18 on February 23, 2008 at Murcia Open 2008.

Many individuals have recorded shorter times, but these records are not accepted due to possible lack of compliance with standards.

Alternative competitions

In addition, informal alternative competitions have been held, inviting participants to solve the Cube under unusual situations. These include:

• Blindfolded solving
• Solving the Cube with one person blindfolded and the other person saying what moves to do, known as "Team Blindfold".
• Solving the Cube underwater in a single breath
• Solving the Cube using a single hand
• Solving the Cube with one's feet.

Custom built puzzles

A lot of puzzles have been built in the past resembling the Rubik's Cube or just its working (as a permutation puzzle). For example, a Cuboid is a Rubik's Cube extended with one or more extra layers, which are glued or fused onto it. Since the extra layer is not functional, the cube will function like the original Cube, although in some cases the extra pieces do place additional constraints on the moves that can be used. People often make extended cubes thanks to the unique shapes they can form. The most common extended cube is the 3×3×5 (extended) cube.

Rubik's Cube software

Several computer programs have been written to perform various functions, such as, among other things, solving the Cube or animating it. In general, these programs can be considered to fall in one of several categories:

• Timers
• Solvers
• Graphical programs
  • Animations
  • Image generators
• Analyzers

Some of the software handles not only the 3×3×3 cube, but also other puzzle types. There is even software for virtual puzzles that do not have a real life counterpart. Examples are the four-dimensional cube and the gliding cube.

In addition these programs may also record player metrics, store and generate scrambled Cube positions or offer either animations or online competition. Solvers are usually given a scramble, after which a solution is generated automatically. Graphical programs can generate a static image or animate the Cube and its motions, e.g. using Java or Flash. Programs may also analyze sequences of moves and transform them to other notations or give player metrics.

For examples of such programs see List of Rubik's Cube software.