index

Rubik's Cube in popular culture

Fri, 07 Aug 2009 16:38:29 +0100

The Rubik's Cube, a mid-1970s invention of Ernő Rubik of Hungary fascinated people around the globe and became one of the most popular games in America at the time.^[1] In just seven years worldwide sales surpassed thirty million units^[2] with a senior buyer at the New York FAO Schwarz toy emporium noting it had become "the world's most asked-for plaything".^[2] Some even argued it could lead to obsessive behavior.^[2] Pirated editions turned up in Taiwan, Hong Kong and some American cities.^[2] The cube spawned an array of sequels, spinoffs and literary works.^[2] As of January 2009 350 million cubes have sold worldwide^[3]^[4] making it the world's top-selling puzzle game^[5]. It earned a place as a permanent exhibit in New York’s Museum of Modern Art and entered the Oxford English Dictionary after just two years.^[4] The Cube retains a cult following, with almost 40,000 entries on YouTube featuring tutorials and video clips of quick solutions.^[4]

Movies

There is a Rubik's Cube in the poster of the 1982's thriller Deathtrap.
The 1989 "Weird Al" Yankovic comedy film UHF features a brief scene where a blind man tries to solve the cube. He twists the puzzle, then holds it up and asks the sighted man next to him, "Is that it?" Each time, the answer is no, so he tries again.
In the 1998 sci-fi-action film Armageddon Rockhound (character played by Steve Buscemi) solves a Rubik's cube in the psychological evaluation prior to training for the space drilling mission.
There is a scene in the 1998 comedy The Wedding Singer where a character throws it away in frustration saying: "No one will ever be able to solve that thing."
In the 2000 comedy Dude, Where's My Car? the "continuum transfunctioner" that will destroy the universe appears to be a Rubik's Cube, being activated by one of the main characters as a result of solving it.
In the 2004 comedy Anchorman: The Legend of Ron Burgundy, Brian Fantana (character played by Paul Rudd), portrayed through much of the movie to be dim-witted, has a 2x2x2 cube on his desk.
In the 2004 supernatural action-thriller film Hellboy, Abe Sapien tries to solve a Rubik's cube.
In the 2006 movie The Pursuit of Happyness^[6], Chris Gardner (character played by Will Smith) impresses his future employer by solving a Rubik's Cube^[7]. Will Smith accomplished the same trick thirteen years prior in an episode of The Fresh Prince of Bel-Air to impress a college recruiter.
In the 2006 movie Flushed Away, Rita throws a Rubik's Cube while throwing things at Roddy.
In the 2008 Disney movie, Wall-E^[8] the main character shows the Rubik's Cube to EVE among other typical objects on Earth, and she quickly solves it.
It also appears in the 2008 Swedish film, Let the Right One In.
In the 2009 movie Night at the Museum: Battle of the Smithsonian, Larry Daley temporarily distracts the evil pharaoh Kahmunrah from his plans by telling him of the "Cube of Rubik", which has the power to turn his foes into dust. He then proceeds to lead them to a box containing a giant squid.

Television

Rubik, the Amazing Cube was an animated show which aired from 1983 to 1984. It featured a magic Rubik's Cube who was granted magical powers when its colored squares were aligned.
In the popular British sporting show, I'm on Setanta Sports, puppet character Wayne Rooney is seen, blindfolded, to do kickups, solve two Rubik's Cubes, and sing the first two lines to the song Ave Maria.
The game was featured at the time of its US debut on Saturday Night Live.^[9]
The anime Tenchi Muyo features a spacecraft with a remote control modeled after a Rubik's cube. Its pilot is completely inept at using the cube, which often causes humorous malfunctions.
The cube also appeared in South Park's episode The Coon, in Professor Chaos hideout, as a "cube of chaos that can destroy the world".
In The Simpsons episode HOMR, when Homer becomes intelligent, he is shown solving a number of cubes in rapid succession.
The cube plays a large part in several episodes of Lawrence Leung's Choose Your Own Adventure

Commercials

In a commercial for EA's 2009 golf video game^[10], Tiger Woods "solves" a Rubik's Cube by putting it.^[11]^[12] His first take on camera was successful.^[13]

Art

Probably from the earliest days of the Rubik's Cube craze in the 1980s people have assembled cubes to form simple art pieces, several early 'Folk Artists' are noted for their work.^[14]^[15] Rubik’s cubes have also been the subject of several pop art installations. Owing to their popularity as a children’s toy several artists and groups have created large Rubik’s cubes.

Tony Rosenthal's Alamo ("The Astor Cube") is a spinnable statue of a Cube standing in New York City. Once the cube was covered with colored panels so that it resembled a Rubik's Cube.^[16]^[17] Similarly, the University of Michigan students covered Endover creating a large Rubik’s cube on the University of Michigan’s central campus for April fool’s day in 2008. In conjunction with the 2008 April fool’s day cube covering, a student group created a large rotating non-functional Rubik’s cube for the University of Michigan's North Campus. Built out of 600+ lbs. of steel, the cube was an entertaining addition to North Campus. Removed later the same semester, the cube reappeared in the fall of 2008 on the first day of classes. It was later removed, but in response to the cube, the university is planning on a permanent Rubik's Cube art installation on North Campus. A oversized Cube installation is also exhibited outside the Disney World 'Pop Century' Hotel ^[18].

Beyond the Folk Art of the 1980' and 1990's and the simple replication of a Rubik's Cube in oversized form, artists have developed a pointillist art style using the cubes to create portraits of Pop Culture Icons including Elton John, Mick Jagger, Jim Morrison and Marilyn Monroe among others.^[19]. This Pop Art form has even acquired its own name "Cube Art" aka Rubik's Cube Art, RubikCubism ^[20] For more detail on the origins and development of Rubik's Cube Art see Wikipedia page for Rubik's Cubism ^[20]

Speedcubing - Rubik's Cube

Fri, 07 Aug 2009 16:36:51 +0100

Speedsolving a 3×3×3 Rubik's Cube with Fridrich Method

Speedcubing (also known as speedsolving, speed cubing or speed-cubing or speed~cubing) is the activity of solving a Rubik's Cube or related puzzle as quickly as possible. Here, solving is defined as performing a series of moves that transforms an incomplete cube into a state where each of the cube's six faces is one single, solid color.

Regular cubes are sold commercially in variations of 2×2×2, 3×3×3, 4×4×4, 5×5×5, 6×6×6 and 7×7×7. Variations of the puzzle have been designed with as many as 100 layers, but the largest denomination cube that has been patented is an 11×11×11. ^[1] The current world record for a single solve of the 3×3×3 is now 7.08 seconds, set by Erik Akkersdijk at the Czech Open on July 12-13, 2008.^[2]^[3]

Speedcubing is a popular activity among international Rubik's Cube community. Members come together to hold competitions, work to develop new solving methods, and seek to perfect their technique. As a part of the community, puzzle builders try to invent new forms of permutation puzzles.

History

The Rubik's Cube was invented in 1974 by Hungarian professor of architecture Ernő Rubik. A widespread international interest in the cube began in 1980, which soon developed into a global craze. On June 5, 1982, the first world championship was held in Budapest. The height of the craze began to fade away after 1983, but with the advent of the Internet, sites relating to speedcubing began to surface. Simultaneously spreading effective speedsolving methods and teaching people new to the cube to solve it for the first time, these sites brought in a new generation of cubers, created a growing international on-line community, and raised the profile of the art. Twenty years after the first World Championship, the 2002 Dutch Open competition was the first in a new wave of organized speedcubing events, which include regular national and international competitions.^[4] There have been three more World Championships since Budapest's 1982 competition, the first held in Toronto in 2003, the second in Lake Buena Vista, Florida in 2005, and after 25 years the tournament returned to Budapest in 2007.

Solving Methods

The standard Rubik's Cube can be solved using a number of methods, not all of which are intended for speedcubing. Although some methods employ a layer-by-layer algorithm, other significant (though less widely-used) methods include corners-first methods, and the Roux method.

Fridrich method

The Fridrich method was named after its inventor Jessica Fridrich who finished 2nd in the 2003 Rubik's Cube World Championships. It first works to solve a cross-shaped arrangement of pieces on the first layer. The remainder of the first layer and all of the second layer are then solved together in what are referred to as "corner-edge pairs" or slots. Finally, the last layer is solved in two steps — first, all of the cubies in the layer are oriented to form a solid color (but without the individual pieces being in their correct places on the cube). This step is referred to as orientation and usually is performed with a single algorithm known as OLL (Orientation of Last Layer). Then, all of those cubies are permuted to their correct spots. This is also usually performed as a single algorithm known as PLL (Permutation of Last Layer).

The Fridrich method is a widely-used speedcubing method. Its popularity stems from the speed at which it can be easily performed. Besides the first step, which can be planned during the customary 15-second inspection time, the entire solve of the cube consists of executing predefined algorithms based on the state of the cube.

Petrus method

The Petrus method, named after its inventor Lars Petrus, is considered by some people to be more intuitive than the structured Fridrich method. The Petrus method works by first solving a 2×2×2 block of the cube. This block is then extended to a solved 2×2×3 block. All edges are then oriented and then the first and second layers are completed. Next, the top corners are put in the right place and then the layer is oriented correctly (all stickers facing up) and finally the last edges are permuted (moved around). Lars Petrus developed this method to address what he felt were inherent inefficiencies in layer-by-layer approaches, which he explains in his method's tutorial: "When you have completed the first layer, you can do nothing without breaking it up. So you break it, do something useful, then restore it. Break it, do something, restore it. Again and again. In a good solution you do something useful all the time. The first layer is in the way of the solution, not a part of it!". This method is often used as the basis for fewest moves competition solutions.

Roux method

The first step of the Roux method is the formation of a 3×2×1 block. The 3×2×1 block is usually placed in the lower portion of the left layer. The second step is to create another 3×2×1 on the opposite layer. The remaining four corners are then solved, which leaves six edges and four centers that are solved in the last step.

This method makes more efficient use of the standard 15 second inspection time, since one can plan the solution of 5 pieces rather than 4 for the Fridrich and Petrus method. It also isn't as dependent on algorithm memorization as the Fridrich method, since all but the third step is done with intuition as opposed to predefined sets of algorithms. Because of this, however, the solve may not be executed as quickly as a solve done with the Fridrich method. It doesn't require as many cube rotations as the Fridrich method, so it is easier to look ahead while solving i.e. solving a collection of pieces and at the same time looking for the solution to the next step.

Corners-first method

This method involves solving the corners then finishing the edges with slice turns. Corners-first solutions were common in the 1980s, with one of the most popular methods that of 1982 world champion Minh Thai. Currently corners-first solutions are less common among speedsolvers. The best corners first method was created in the cube craze by Dutch cuber Marc Waterman. He averaged 16 seconds in the mid-late 80s. First, build a face on the left. Then, solve the remaining corners. Next, solve two right edges and place one remaining right edge in the right layer OR solve three right edges. Then, solve the last right edge(s) and orient middle edges simultaneously. Finally, permute middle edges. About 7 algorithms to memorize.

ZZ method

ZZ is a modern speedcubing method, originally proposed by Zbigniew Zborowski in 2006.^[5] The method was designed specifically to achieve high turning speed by focusing on move ergonomics. The initial pre-planned step is called EOLine, and is the most distinctive hallmark of the ZZ method. It involves orienting all edges while placing two opposite down-face edges. The next step solves the remaining first two layers using only left, right and top face turns. On completion of the first two layers, the last layer's edges are all correctly orientated because of edge pre-orientation during EOLine. The last layer may be completed using a number of techniques including those used in the Fridrich method. An expert variant of this method (ZZ-a) allows the last layer to be completed in a single step with an average of just over 12 moves and knowledge of 177 algorithms.^[6]

Competitions

According to the World Cube Association, competitors (in the same round) must solve cubes that are scrambled using a consistent algorithm (as in, every competitor solves the same scramble). Currently, the official timer used in competition is the StackMat timer. This device has touch-sensitive pads that are triggered by the speedcuber lifting their hands to start the time and placing their hands back on the pads after releasing the puzzle to stop the time. In addition to the electronic timer, there are human judges with stopwatches, who act as a back-up in case the timer doesn't work properly. These judges also ensure that the competitors are following competition regulations.

Official competitions are currently being held in several categories.

Category	Cube Type
speedsolving	2×2×2, 3×3×3, 4×4×4, 5×5×5, 6×6×6, 7×7×7
one-handed solving	3×3×3
blindfolded solving	3×3×3, 4×4×4, 5×5×5
solving with feet	3×3×3
solving in fewest moves	3×3×3

Competitions will often include events for speedsolving these other puzzles, as well:

World records

These are the world records for speedsolving the six types of cubes as set during WCA-approved events. ^[2]

Cube type	Time (min:sec.sec)	Record holder
2×2×2	0:00.96	Erik Akkersdijk
3×3×3	0:07.08	Erik Akkersdijk
4×4×4	0:39.83	Erik Akkersdijk
5×5×5	1:07.25	Dan Cohen
6×6×6	2:23.63	Dan Cohen
7×7×7	3:47.36	Yu Nakajima

Lubrication

Some speedcubers will lubricate their cubes to prevent wrist and finger injury. Lubricating the cube also allows it to be manipulated more quickly than a non-lubed cube. The WCA allows lubrication for WCA-sanctioned competitions. Usually, the lubricant's main ingredient is polysiloxane.

ABS, the main plastic in Rubik's cubes, should NOT be lubricated with lubricants containing any of the following:

Methylene chloride, often used for welding ABS plastic
Acetone, a good solvent for ABS plastic
WD-40, incompatible with ABS plastic due to white spirit (also known as Stoddard solvent) content

Checking a lubricant's MSDS is often helpful in identifying cube-damaging ingredients.

Terminology

Here are some definitions generally used by the speedcubing community. For a more complete list of speedcubing terminology, see the cubefreak.net glossary.

Algorithm: A predefined sequence of moves used to effect a specific change on the cube. Often referred to as alg.
BLD: Blindfold solving, i.e. memorize, blindfold, then solve.
Center piece: One of the six centers of the faces of the cube. The centers never move relative to each other on an NxNxN cube, where N is odd.
CLL: Corners of Last Layer. This is the first of two steps to solve the last layer of the cube. In the process, edges may not be unoriented. This is used in Corners First methods for the last layer, in which the first all corners are solved, followed by the edges (see: ELL).
Corner piece: One of the 8 pieces with exactly three stickers, called a "corner" piece because a corner is exposed.
Cubie: One of the mechanically independent pieces that make up a puzzle. The cubies do not include fixed center pieces, nor the central axis to which they are attached.
Cycle: To rotate pieces' positions on the cube. E.g. a 3-cycle would make cubie set A-B-C become C-A-B.
DNF: Did Not Finish, used in competition e.g. when a piece pop occurs and the competitor decides not to continue the solving of the puzzle.
DNS: Did Not Start, used in competition when the competitor does not begin a solve, either by opting to skip it (common in Blindfold Cubing) or by not showing up when he or she is called.
Edge piece: One of the 12 pieces with exactly two stickers, called an "edge" piece because only one edge is exposed.
ELL: Edges of Last Layer. The second of two steps to solve the last layer of the cube, solving the edge pieces without disturbing the orientation of the corner pieces (see: CLL).
F2B: First two blocks.
F2L: First two layers.
F2L method: A method which solves the first and second layers simultaneously.
Layer: One section of a cube consisting of a number of cubies that turn as a unit. (e.g. a standard Rubik's cube has 3 layers)
LL: Last Layer.
Method: A combination of steps that can be used to solve a cube.
Move: A turn or double turn of one of the six faces or three slices of the cube.
N-look, also known as X-Look: Refers to the number of algorithms needed to complete a step in a particular solving method, often the last layer, e.g. '4-look LL'.
OLL: Orient Last Layer, usually used in reference to the respective step of the Fridrich method.
Orient: To flip or twist pieces so they turn 'in-place'.
PB: Personal Best - personal record time to solve a puzzle. This can either be a single attempt or a trimmed average, depending on context.
Permute: Swap or cycle two or more pieces.
PLL: Permute Last Layer. Usually used in reference to the respective step of the Fridrich method, in which case it would follow the OLL step.
Pop: When, during a speedsolve, one or more cubies come out of the puzzle. Also known as piece pop.
Prime: A counter-clockwise move popularly denoted with a ', e.g. 'R Prime', denoted as R', R-, $R - 1$ , Ri. Also known as "inverse" or "inverted".
Slice: The four center pieces and four edge pieces between two opposite faces of the cube.
Two-Second Penalty: A penalty of 2 seconds which is added to a solving time in competition when the cube is placed back on the timing pad with one or more misaligned faces.
UWR: Unofficial World Record.
WCA: World Cube Association, the international governing body for official cube competitions.
WR: World Record. Also "World Rank" when referring to the rank of a person's record in a database.

Optimal solutions for Rubik's Cube

Fri, 07 Aug 2009 16:34:43 +0100

There are many algorithms to solve scrambled Rubik's Cubes. One such method is described in Wikibooks' article How to solve the Rubik's Cube, and the notation from that article is used here as well. This is one algorithm that has the advantage of being simple enough to be memorizable by humans, however it will usually not give an optimal solution which only uses the minimum possible number of moves.

It is not known how many moves is the minimum required to solve any instance of the Rubik's cube, although the latest claims put this number at 22 ^[1]. This number is also known as the diameter of the Cayley graph of the Rubik's Cube group. An algorithm that solves a cube in the minimum number of moves is known as 'God's algorithm'.

When discussing the length of a solution, there are two common ways to measure this. The first is to count the number of quarter turns. The second is to count the number of face turns. A move like F2 (a half turn of the front face) would be counted as 2 moves in the quarter turn metric and as only 1 turn in the face metric.

Lower bounds

It can be proven by counting arguments that there exist positions needing at least 18 moves to solve. To show this, first count the number of cube positions that exist in total, then count the number of positions achievable using at most 17 moves. It turns out that the latter number is smaller.

This argument was not improved upon for many years. Also, it is not a constructive proof: it does not exhibit a concrete position that needs this many moves. It was conjectured that the so-called superflip would be a position that is very difficult. The superflip is a position on the cube where all the cubies are in their correct position, all the corners are correctly oriented but each edge is oriented the wrong way.

One indication that this might be the case is that it is the only element other than the identity that is in the center of the cube group.

In 1992 a solution for the superflip with 20 face turns was found by Dik T. Winter. In 1995, Michael Reid proved its minimality, thereby giving a new lower bound for the diameter of the cube group.

Also in 1995, a solution for superflip in 24 quarter turns was found by Michael Reid, its minimality was proven by Jerry Bryan. ^[2]

In 1998 Michael Reid found a new position requiring more than 24 quarter turns to solve. The position, named by him as 'superflip composed with four spot' needs 26 quarter turns. ^[3]

Upper bounds

Thistlethwaite's Algorithm

The first upper bounds were based on the 'human' algorithms. By combining the worst-case scenarios for each part of these algorithms, the typical upper bound was found to be around 100. The breakthrough was found by Morwen Thistlethwaite; details of Thistlethwaite's Algorithm were published in Scientific American in 1981 by Douglas R. Hofstadter. The approaches to the cube that lead to algorithms with very few moves are based on group theory and on extensive computer searches.

Thistlethwaite's idea was to divide the problem into subproblems. Where algorithms up to that point divided the problem by looking at the parts of the cube that should remain fixed, he divided it by restricting the type of moves you could execute.

In particular he divided the cube group into the following chain of subgroups:

G₀ = <L,R,F,B,U,D>
G₁ = <L,R,F,B,U2,D2>
G₂ = <L,R,F2,B2,U2,D2>
G₃ = <L2,R2,F2,B2,U2,D2>
G₄ = {I}

Next he prepared tables for each of the right coset spaces G_[i+1]\G_i. For each element he found a sequence of moves that took it to the next smaller group.

After these preparations he worked as follows. A random cube is in the general cube group G₀. Next he found this element in the right coset space G₁\G₀. He applied the corresponding process to the cube. This took it to a cube in G₁. Next he looked up a process that takes the cube to G₂, next to G₃ and finally to G₄.

Although the whole cube group G₀ is very large (~4.3×10¹⁹), the right coset spaces G₁\G₀, G₂\G₁, G₃\G₂ and G₃ are much smaller. The coset space G₂\G₁ is the largest and contains only 1082565 elements. The number of moves required by this algorithm is the sum of the largest process in each step. In the original version this was 52.

Kociemba's Algorithm

Thistlethwaites algorithm was improved by Herbert Kociemba in 1992. He reduced the number of intermediate groups to only two:

G₀ = <L,R,F,B,U,D>
G₁ = <L,R,F2,B2,U2,D2>
G₂ = {I}.

As with Thistlethwaite's Algorithm, he would search through the right coset space G₁\G₀ to take the cube to group G₁. Next he searched the optimal solution for group G₁. The searches in G₁\G₀ and G₁ were both done with a method equivalent to IDA*. The search in G₁\G₀ needs at most 12 moves and the search in G₁ at most 18 moves, as Michael Reid showed in 1995. By generating also suboptimal solutions that take the cube to group G₁ and looking for short solutions in G₁, you usually get much shorter overall solutions. Using this algorithm solutions are typically found of less than 21 moves, though there is no proof that it will always do so.

In 1995 Michael Reid proved that using these two groups every position can be solved in at most 29 face turns, or in 42 quarter turns. This result was improved by Silviu Radu in 2005 to 40.

Korf's Algorithm

Using these group solutions combined with computer searches will generally quickly give very short solutions. But these solutions do not always come with a guarantee of their minimality. To search specifically for minimal solutions a new approach was needed.

In 1997 Richard Korf^[4] announced an algorithm with which he had optimally solved random instances of the cube. Of the ten random cubes he did, none required more than 18 face turns. The method he used is called IDA* and is described in his paper "Finding Optimal Solutions to Rubik's Cube Using Pattern Databases." Korf describes this method as follows

IDA* is a depth-first search that looks for increasingly longer solutions in a series of iterations, using a lower-bound heuristic to prune branches once a lower bound on their length exceeds the current iterations bound.

It works roughly as follows. First he identified a number of subproblems that are small enough to be solved optimally. He used:

The cube restricted to only the corners, not looking at the edges
The cube restricted to only 6 edges, not looking at the corners nor at the other edges.
The cube restricted to the other 6 edges.

Clearly the number of moves required to solve any of these subproblems is a lower bound for the number of moves you will need to solve the entire cube.

Given a random cube C, it is solved as iterative deepening. First all cubes are generated that are the result of applying 1 move to them. That is C * F, C * U, … Next, from this list, all cubes are generated that are the result of applying two moves. Then three moves and so on. If at any point a cube is found that needs too many moves based on the upper bounds to still be optimal it can be eliminated from the list.

Although this algorithm will always find optimal solutions there is no worst case analysis. It is not known how many moves this algorithm might need. An implementation of this algorithm can be found here ^[5].

Further Improvements

In 2006, Silviu Radu further improved his methods to prove that every position can be solved in at most 27 face turns or 35 quarter turns^[6].

In August 2007, Daniel Kunkle and Gene Cooperman used a supercomputer to show that all unsolved cubes can be solved in no more than 26 moves (in face-turn metric). Instead of attempting to solve each of the billions of variations explicitly, the computer was programmed to bring the cube to one of 15,000 states, each of which could be solved within a few extra moves. All were proved solvable in 29 moves, with most solvable in 26. Those that could not initially be solved in 26 moves were then solved explicitly, and shown that they too could be solved in 26 moves. ^[7] ^[8]

In 2008, Tomas Rokicki was reported to have devised a computational proof that all unsolved cubes could be solved in 25 moves or fewer.^[9] This was later reduced to 23 moves.^[10] In August 2008 Rokicki announced that he had a proof for 22 moves.^[11]

Rubik's Cube

Fri, 07 Aug 2009 16:32:39 +0100

A classic Rubik's Cube, solved.

The Rubik's Cube is a 3-D mechanical puzzle invented in 1974^[1] by Hungarian sculptor and professor of architecture Ernő Rubik. Originally called the "Magic Cube",^[2] the puzzle was licensed by Rubik to be sold by Ideal Toys in 1980^[3] 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^[4]^[5] making it the world's top-selling puzzle game.^[6]^[7] It is widely considered to be the world's best-selling toy.^[8]

In a classic Rubik's Cube, each of the six faces is covered by 9 stickers, among six solid colours (traditionally white, red, blue, orange, green, and yellow).^[9] 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 celebrated its twenty-fifth anniversary in 2005.

Conception and development

Prior attempts

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 U.S. Patent 3,655,201 on April 11, 1972, two years before Rubik invented his 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's invention

In the mid-1970s, Ernő Rubik worked at the Department of Interior Design at the Academy of Applied Arts and Crafts in Budapest.^[10] He sought to find a teaching tool to help his students understand 3D objects. 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 prevented the puzzle being easily pulled apart, unlike 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.

Packaging of Rubik's Cube, Toy of the year 1980- Ideal Toy Corp 1980, Made in Hungary.

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 imitations appeared.

Patent disputes

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.^[11]

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, which was granted in 1976 (Japanese patent publication JP55-008192). Until 1999, when an amended Japanese patent law was enforced, Japan's patent office granted Japanese patents for non-disclosed technology within Japan without requiring worldwide novelty^[12]^[13]. Hence, Ishigi's patent is generally accepted as an independent reinvention at that time.^[14]^[15]^[16]

Rubik applied for another Hungarian patent on October 28, 1980, and applied for other patents. In the United States, Rubik was granted U.S. Patent 4,378,116 on March 29, 1983, for the Cube.

Greek inventor Panagiotis Verdes patented^[17] 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.^[18] As of June 19, 2008, the 5×5×5, 6×6×6, and 7×7×7 models are in production.

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.

Mechanics

Rubik's Cube partially disassembled.

A standard Rubik's cube measures 5.7 cm (approximately 2¼ inches) on each side. The puzzle consists of twenty-six unique miniature cubes, also called "cubies" or "cubelets". However, the centre cube of each of the six faces is merely a single square façade; all six 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 rotating the top layer by 45° and then prying one of its edge cubes away from the other two layers. Consequently 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. 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. Hofstadter, in the July 1982 issue of 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 become popular.

Mathematics

Permutations

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 3⁷ (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 2¹¹ (2,048) possibilities.^[19]

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. Alternatively, if laid out on the ground, this is enough to cover the earth with 273 layers of cubes, recognizing the fact that the radius of the earth sphere increases by 57 mm with each layer of cubes.

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 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).^[20]

Algorithms

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.

Solutions

Move notation

Many 3×3×3 Rubik's Cube enthusiasts use a notation developed by David Singmaster to denote a sequence of moves, referred to as "Singmaster notation".^[21] 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 organised 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
ƒ (Front two layers): the side facing you and the corresponding middle layer
b (Back two layers): the side opposite the front and the corresponding middle layer
u (Up two layers) : the top side and the corresponding middle layer
d (Down two layers) : the bottom layer and the corresponding middle layer
l (Left two layers) : the side to the left of the front and the corresponding middle layer
r (Right two layers) : the side to the right of the front and the corresponding middle layer
x (rotate): rotate the entire Cube on R
y (rotate): rotate the entire Cube on U
z (rotate): rotate the entire Cube on F

When a prime symbol ( ′ ) follows a letter, it denotes face counter-clockwise, while a letter without a prime symbol denotes a clockwise turn. A letter followed by a 2 (occasionally a superscript ²) denotes two turns, or a 180-degree turn. R is right side clockwise, but R' is right side counter-clockwise. The letters x, y, and z are used to indicate that the entire Cube should be turned about one of its axes. When x, y or z are primed, it is an indication that the cube must be rotated in the opposite direction. When they are squared, the cube must be rotated twice.

For methods using middle-layer turns (particularly corners-first methods) there is a generally accepted "MES" extension to the notation where letters M, E, and S denote middle layer turns. It was used e.g. in Marc Waterman's Algorithm.^[22]

M (Middle): the layer between L and R, turn direction as L (top-down)
E (Equator): the layer between U and D, turn direction as D (left-right)
S (Standing): the layer between F and B, turn direction as F

The 4×4×4 and larger cubes use an extended notation to refer to the additional middle layers. Generally speaking, uppercase letters (F B U D L R) refer to the outermost portions of the cube (called faces). Lowercase letters (ƒ b u d ℓ r) refer to the inner portions of the cube (called slices). An asterisk (L*), a number in front of it (2L), or two layers in parenthesis (Lℓ), means to turn the two layers at the same time (both the inner and the outer left faces) For example: (Rr)' ℓ2 ƒ' means to turn the two rightmost layers counterclockwise, then the left inner layer twice, and then the inner front layer counterclockwise.

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.^[23]^[24]^[25] In 2008, Tomas Rokicki lowered that number to 22 moves.^[26]^[27]^[28]

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 respectivly). 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.^[29]
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 constanly break and fix the first layer; the 2×2×2 and 2×2×3 sections allow three or two layers to be turned with out 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.^[30]

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 ^[31]. In 2004, the WCA made it mandatory to use a special timing device called a Stackmat timer.

In addition to official competitions, informal alternative competitions have been held which invite participants to solve the Cube in unusual situations. Some such situations include:

Blindfolded solving^[32]
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^[33]
Solving the Cube using a single hand^[34]
Solving the Cube with one's feet^[35]

Of these informal competitions, the World Cube Association only sanctions blindfolded, one-handed, and feet solving as official competition events.^[36]

In blindfolded solving, the contestant first studies the scrambled cube (i.e., looking at it normally with no blindfold), and is then blindfolded before beginning to turn the cube's faces. Their recorded time for this event includes both the time spent examining the cube and the time spent manipulating it.

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 Tomasz Zolnowski; which is 10.63 seconds at the Warsaw Open 2009.

On December 20, 2008, 96 people in Santa Ana, CA broke the Guinness World Record for most people solving a Rubik's cube at once.^[37] The previous record was 75 people by a group in Atlanta, GA.

Variations

Variations of Rubik's Cubes, clockwise from upper left: V-Cube 7, Professor's Cube, V-Cube 6, Pocket Cube, original Rubik's Cube, Rubik's Revenge.

There are different variations of Rubik's Cubes with up to seven layers: the 2×2×2 (Pocket/Mini Cube), the standard 3×3×3 cube, the 4×4×4 (Rubik's Revenge/Master Cube), and the 5×5×5 (Professor's Cube), the 6×6×6 (V-Cube 6), and 7×7×7 (V-Cube 7).

CESailor Tech's E-cube.

CESailor Tech's E-cube is an electronic variant of the 3x3x3 cube, made with RGB LEDs and switches.^[38] There are two switches on each row and column. Pressing the switches indicates the direction of rotation, which causes the LED display to change colors, simulating real rotations. The product was demonstrated at the Taiwan government show of College designs on 30 October 2008.

Another electronic variation of the 3×3×3 Cube is the Rubik's TouchCube. Sliding a finger across its faces causes its patterns of colored lights to rotate the same way they would on a mechanical cube. The TouchCube was introduced at the American International Toy Fair in New York on February 15, 2009.^[39]^[40]

The Cube has inspired an entire category of similar puzzles, commonly referred to as twisty puzzles, which includes the cubes of different sizes mentioned above as well as various other geometric shapes. Some such shapes include the tetrahedron (Pyraminx), the octahedron (Skewb Diamond), the dodecahedron (Megaminx), the icosahedron (Dogic). There are also puzzles that change shape such as Rubik's Snake and the Square One.

Custom-built puzzles

In the past, puzzles have been built resembling the Rubik's Cube or based on its inner workings. For example, a cuboid is a puzzle based on the Rubik's Cube, but with different functional dimensions, such as, 2×3×4, 3×3×5, or 2×2×4. Many cuboids are based on 4×4×4 or 5×5×5 mechanisms, via building plastic extensions or by directly modifying the mechanism itself.

Some custom puzzles are not derived from any existing mechanism, such as the Gigaminx v1.5-v2, Bevel Cube, SuperX, Toru, Rua, and 1×2×3. These puzzles usually have a set of masters 3D printed, which then are copied using molding and casting techniques to create the final puzzle.^{[citation needed]}

Other Rubik's Cube modifications include cubes that have been extended or truncated to form a new shape. An example of this is the Trabjer's Octahedron, which can be built by truncating and extending portions of a regular 3×3. Most shape mods can be adapted to higher-order cubes. In the case of Tony Fisher's Rhombic Dodecahedron, there are 3×3, 4×4, 5×5, and 6×6 versions of the puzzle.

Rubik's Cube software

Magic Cube 4D, a 4×4×4×4 virtual puzzle,

and Magic Cube 5D, a 3x3x3x3x3 virtual puzzle

Puzzles like the Rubik's Cube can be simulated by computer software, which provide functions such as recording of player metrics, storing scrambled Cube positions, conducting online competitions, analyzing of move sequences, and converting between different move notations. Software can also simulate very large puzzles that are impractical to build, such as 100×100×100 and 1,000×1,000×1,000 cubes, as well as virtual puzzles that cannot be physically built, such as 4- and 5-dimensional analogues of the cube.^[41]^[42]