Rubik's Cube


The Rubik's Cube is a 3-D combination 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 Toy Corp. in 1980 via businessman Tibor Laczi and Seven Towns founder Tom Kremer. Rubik's Cube won the 1980 German Game of the Year special award for Best Puzzle., 350 million cubes had been sold worldwide, making it the world's top-selling puzzle game. It is widely considered to be the world's best-selling toy.
On the original classic Rubik's Cube, each of the six faces was covered by nine stickers, each of one of six solid colours: white, red, blue, orange, green, and yellow. Some later versions of the cube have been updated to use coloured plastic panels instead, which prevents peeling and fading. In currently sold models, white is opposite yellow, blue is opposite green, and orange is opposite red, and the red, white, and blue are arranged in that order in a clockwise arrangement. On early cubes, the position of the colours varied from cube to cube. An internal pivot mechanism enables each face to turn independently, thus mixing up the colours. For the puzzle to be solved, each face must be returned to have only one colour. Similar puzzles have now been produced with various numbers of sides, dimensions, and stickers, not all of them by Rubik.
Although Rubik's Cube reached its height of mainstream popularity in the 1980s, it is still widely known and used. Many speedcubers continue to practice it and similar puzzles; they also compete for the fastest times in various categories. Since 2003, the World Cube Association, the international governing body of Rubik's Cube, has organised competitions worldwide and recognises world records.

Conception and development

Similar precursors

In March 1970, Larry D. 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 by magnets. Nichols was granted on 11 April 1972, two years before Rubik invented his Cube.
On 9 April 1970, Frank Fox applied to patent an "amusement device", a type of sliding puzzle on a spherical surface with "at least two 3×3 arrays" intended to be used for the game of noughts and crosses. He received his UK patent on 16 January 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. Although it is widely reported that the Cube was built as a teaching tool to help his students understand 3D objects, his actual purpose was solving the structural problem of moving the parts independently without the entire mechanism falling apart. He did not realise that he had created a puzzle until the first time he scrambled his new Cube and then tried to restore it. Rubik applied for a patent in Hungary for his "" on 30 January 1975, and HU170062 was granted later that year.
The first test batches of the Magic Cube were produced in late 1977 and released in 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. With Ernő Rubik's permission, businessman Tibor Laczi took a Cube to Germany's Nuremberg Toy Fair in February 1979 in an attempt to popularise it. It was noticed by Seven Towns founder Tom Kremer, and they signed a deal with Ideal Toys in September 1979 to release the Magic Cube worldwide. Ideal wanted at least a recognisable name to trademark; of course, that arrangement put Rubik in the spotlight because the Magic Cube was renamed after its inventor in 1980. 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 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.

Subsequent history

1980s Cube craze

After the first batches of Rubik's Cubes were released in May 1980, initial sales were modest, but Ideal began a television advertising campaign in the middle of the year which it supplemented with newspaper adverts. At the end of 1980, Rubik's Cube won a German Game of the Year special award and won similar awards for best toy in the UK, France, and the US. By 1981, Rubik's Cube had become a craze, and it is estimated that in the period from 1980 to 1983 around 200 million Rubik's Cubes were sold worldwide. In March 1981, a speedcubing championship organised by the Guinness Book of World Records was held in Munich, and a Rubik's Cube was depicted on the front cover of Scientific American that same month. In June 1981, The Washington Post reported that Rubik's Cube is "a puzzle that's moving like fast food right now... this year's Hoola Hoop or Bongo Board", and by September 1981, New Scientist noted that the cube had "captivated the attention of children of ages from 7 to 70 all over the world this summer."
As most people could solve only one or two sides, numerous books were published including David Singmaster's Notes on Rubik's "Magic Cube" and Patrick Bossert's You Can Do the Cube. At one stage in 1981, three of the top ten best selling books in the US were books on solving Rubik's Cube, and the best-selling book of 1981 was James G. Nourse's The Simple Solution to Rubik's Cube which sold over 6 million copies. In 1981, the Museum of Modern Art in New York exhibited a Rubik's Cube, and at the 1982 World's Fair in Knoxville, Tennessee a six-foot Cube was put on display. ABC Television even developed a cartoon show called Rubik, the Amazing Cube. In June 1982, the First Rubik's Cube World Championship took place in Budapest and would become the only competition recognized as official until the championship was revived in 2003.
In October 1982, The New York Times reported that sales had fallen and that "the craze has died", and by 1983 it was clear that sales had plummeted. However, in some Communist countries, such as China and USSR, the craze had started later and demand was still high because of a shortage of Cubes.

21st-century revival

Rubik's Cubes continued to be marketed and sold throughout the 1980s and ’90s, but it was not until the early 2000s that interest in the Cube began increasing again. In the US, sales doubled between 2001 and 2003, and The Boston Globe remarked that it was "becoming cool to own a Cube again". The 2003 World Rubik's Games Championship was the first speedcubing tournament since 1982. It was held in Toronto and was attended by 83 participants. The tournament led to the formation of the World Cube Association in 2004. Annual sales of Rubik branded cubes were said to have reached 15 million worldwide in 2008. Part of the new appeal was ascribed to the advent of Internet video sites, such as YouTube, which allowed fans to share their solving strategies. Following the expiration of Rubik's patent in 2000, other brands of cubes appeared, especially from Chinese companies. Many of these Chinese branded cubes have been engineered for speed and are favoured by speedcubers.

Imitations

Taking advantage of an initial shortage of Cubes, many imitations and variations appeared, many of which may have violated one or more patents. Today, the patents have expired and many Chinese companies produce copies of—and in nearly all cases, improvements upon—the Rubik and V-Cube designs.

Patent history

Nichols assigned his patent to his employer Moleculon Research Corp., which sued Ideal 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, which was granted in 1976. 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. Hence, Ishigi's patent is generally accepted as an independent reinvention at that time. Rubik applied for more patents in 1980, including another Hungarian patent on 28 October. In the United States, Rubik was granted on 29 March 1983, for the Cube. This patent expired in 2000.

Trademarks

Rubik's Brand Ltd. also holds the registered trademarks for the word "Rubik" and "Rubik's" and for the 2D and 3D visualisations of the puzzle. The trademarks have been upheld by a ruling of the General Court of the European Union on 25 November 2014 in a successful defence against a German toy manufacturer seeking to invalidate them. However, European toy manufacturers are allowed to create differently shaped puzzles that have a similar rotating or twisting functionality of component parts such as for example Skewb, Pyraminx or Impossiball.
On 10 November 2016, Rubik's Cube lost a ten-year battle over a key trademark issue. The European Union's highest court, the Court of Justice, ruled that the puzzle's shape was not sufficient to grant it trademark protection.

Mechanics

A standard Rubik's Cube measures on each side. The puzzle consists of 26 unique miniature cubes, also called "cubies" or "cubelets". Each of these includes a concealed inward extension that interlocks with the other cubes while permitting them to move to different locations. However, the centre cube of each of the six faces is merely a single square façade; all six are affixed to the core mechanism. These provide structure for the other pieces to fit into and rotate around. Hence, there are 21 pieces: a single core piece consisting of three intersecting axes holding the six centre squares in place but letting them rotate, and 20 smaller plastic pieces which fit into it to form the assembled puzzle.
Each of the six centre pieces pivots on a screw held by the centre piece, a "3D cross". A spring between each screw head and its corresponding piece tensions the piece inward, so that collectively, the whole assembly remains compact but can still be easily manipulated. The screw can be tightened or loosened to change the "feel" of the Cube. Newer official Rubik's brand cubes have rivets instead of screws and cannot be adjusted.
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 six central pieces which show one coloured face, twelve edge pieces which show two coloured faces, and eight corner pieces which show three coloured faces. Each piece shows a unique colour combination, but not all combinations are present. The location of these cubes relative to one another can be altered by twisting an outer third of the Cube by increments of 90 degrees, but the location of the coloured sides relative to one another in the completed state of the puzzle cannot be altered; it is fixed by the relative positions of the centre squares. However, Cubes with alternative colour arrangements also exist; for example, with the yellow face opposite the green, the blue face opposite the white, and red and orange remaining opposite each other.
Douglas 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

The puzzle was originally advertised as having "over 3,000,000,000 combinations but only one solution". Depending on how combinations are counted, the actual number can be significantly higher.

Permutations

The original Rubik's Cube has eight corners and twelve edges. There are 8! ways to arrange the corner cubes. Each corner has three possible orientations, although only seven can be oriented independently; the orientation of the eighth corner depends on the preceding seven, giving 37 possibilities. There are 12!/2 ways to arrange the edges, restricted from 12! because edges must be in an even permutation exactly when the corners are. Eleven edges can be flipped independently, with the flip of the twelfth depending on the preceding ones, giving 211 possibilities.
which is approximately 43 quintillion. To put this into perspective, if one had one standard-sized Rubik's Cube for each permutation, one could cover the Earth's surface 275 times, or stack them in a tower 261 light-years high.
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 larger:
which is approximately 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 12 possible sets of reachable configurations, sometimes called "universes" or "orbits", into which the Cube can be placed by dismantling and reassembling it.
The preceding numbers assume the center faces are in a fixed position. If one considers turning the whole cube to be a different permutation, then each of the preceding numbers should be multiplied by 24. A chosen colour can be on one of six sides, and then one of the adjacent colours can be in one of four positions; this determines the positions of all remaining colours.

Centre faces

The original Rubik's Cube had no orientation markings on the centre faces, and therefore solving it does not require any attention to orienting those faces correctly. However, with marker pens, 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; a cube marked in this way is referred to as a "supercube". 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. Cubes have also been produced where the nine stickers on a face are used to make a single larger picture, and centre orientation matters on these as well. Thus one can nominally solve a Cube yet have the markings on the centres rotated; it then becomes an additional test to solve the centres as well.
Marking Rubik's Cube's centres increases its difficulty, because this expands the set of distinguishable possible configurations. There are 46/2 ways to orient the centres since an even permutation of the corners implies an even number of quarter turns of centres as well. In particular, when the Cube is unscrambled apart from the orientations of the central squares, there will always be an even number of centre squares requiring a quarter turn. Thus orientations of centres increases the total number of possible Cube permutations from 43,252,003,274,489,856,000 to 88,580,102,706,155,225,088,000.
When turning a cube over is considered to be a change in permutation then we must also count arrangements of the centre faces. Nominally there are 6! ways to arrange the six centre faces of the cube, but only 24 of these are achievable without disassembly of the cube. When the orientations of centres are also counted, as above, this increases the total number of possible Cube permutations from 88,580,102,706,155,225,088,000 to 2,125,922,464,947,725,402,112,000.

Algorithms

In Rubik's cubers' 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 Cube employs its own set of algorithms, together with descriptions of what effect the algorithm has, and when it can be used to bring the cube closer to being solved.
Many algorithms are designed to transform only a small part of the cube without interfering with 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 do have a certain desired effect on the cube but may also have the side-effect of changing other parts of the cube. 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. Most are long and difficult to memorise. Towards the end of the solution, the more specific algorithms are used instead.

Relevance and application of mathematical group theory

Rubik's Cube lends itself to the application of mathematical group theory, which has been helpful for deducing certain algorithms – in particular, those which have a commutator structure, namely XYX−1Y−1, or a conjugate structure, namely XYX−1, often referred to by speedcubers colloquially as a "setup move". In addition, the fact that there are well-defined subgroups within the Rubik's Cube group enables the puzzle to be learned and mastered by moving up through various self-contained "levels of difficulty". For example, one such "level" could involve solving cubes which have been scrambled using only 180-degree turns. These subgroups are the principle underlying the computer cubing methods by Thistlethwaite and Kociemba, which solve the cube by further reducing it to another subgroup.

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". 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.
When a prime symbol follows a letter, it denotes an anticlockwise face turn; while a letter without a prime symbol denotes a clockwise turn. These directions are as one is looking at the specified face. A letter followed by a 2 denotes two turns, or a 180-degree turn. R is right side clockwise, but R′ is right side anticlockwise. The letters x, y, and z are used to indicate that the entire Cube should be turned about one of its axes, corresponding to R, U, and F turns respectively. 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 180 degrees.
The most common deviation from Singmaster notation, and in fact the current official standard, is to use "w", for "wide", instead of lowercase letters to represent moves of two layers; thus, a move of Rw is equivalent to one of r.
For methods using middle-layer turns, 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.
The 4×4×4 and larger cubes use an extended notation to refer to the additional middle layers. Generally speaking, uppercase letters refer to the outermost portions of the cube. Lowercase letters refer to the inner portions of the cube. An asterisk, a number in front of it, or two layers in parentheses, means to turn the two layers at the same time For example: ' l2 f' means to turn the two rightmost layers anticlockwise, then the left inner layer twice, and then the inner front layer anticlockwise. By extension, for cubes of 6×6×6 and larger, moves of three layers are notated by the number 3, for example, 3L.
An alternative notation, Wolstenholme notation, is designed to make memorising sequences of moves easier for novices. This notation uses the same letters for faces except it replaces U with T, so that all are consonants. The key difference is the use of the vowels O, A, and I for clockwise, anticlockwise, and twice turns, which results in word-like sequences such as LOTA RATO LATA ROTI. Addition of a C implies rotation of the entire cube, so ROC is the clockwise rotation of the cube around its right face. Middle layer moves are denoted by adding an M to corresponding face move, so RIM means a 180-degree turn of the middle layer adjacent to the R face.
Another notation appeared in the 1981 book The Simple Solution to Rubik's Cube. Singmaster notation was not widely known at the time of publication. The faces were named Top, Bottom, Left, Right, Front, and Posterior, with + for clockwise, – for anticlockwise, and 2 for 180-degree turns.
Another notation appeared in the 1982 "The Ideal Solution" book for Rubik's Revenge. Horizontal planes were noted as tables, with table 1 or T1 starting at the top. Vertical front to back planes were noted as books, with book 1 or B1 starting from the left. Vertical left to right planes were noted as windows, with window 1 or W1 starting at the front. Using the front face as a reference view, table moves were left or right, book moves were up or down, and window moves were clockwise or anticlockwise.

Optimal solutions

Although there are a significant number of possible permutations for Rubik's Cube, a number of solutions have been developed which allow solving the cube in well under 100 moves.
Many general solutions for the Cube have been discovered independently. David Singmaster first published his solution in the book Notes on Rubik's "Magic Cube" in 1981. This solution involves solving the Cube layer by layer, in which one layer is solved first, followed by the middle layer, and then the final and bottom layer. After sufficient 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 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 fewer.
In 2008, Tomas Rokicki lowered that number to 22 moves, and in July 2010, a team of researchers including Rokicki, working with Google, proved the so-called "God's number" to be 20. This is optimal, since there exist some starting positions which require a minimum of 20 moves to solve. More generally, it has been shown that an n×n×n Rubik's Cube can be solved optimally in Θ moves.

Speedcubing methods

A solution commonly used by speedcubers was developed by Jessica Fridrich. This method is called CFOP standing for "cross, F2L, OLL, PLL". 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, thus completing the first two layers. This is then followed by orienting the last layer, then permuting the last layer. Fridrich's solution requires learning roughly 120 algorithms but allows the Cube to be solved in only 55 moves on average.
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, one must constantly break and fix the completed 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. For this reason, the method is also popular for fewest move competitions.
The Roux Method, developed by Gilles Roux, is similar to the Petrus method in that it relies on block building rather than layers, but derives from corners-first methods. In Roux, a 3×2×1 block is solved, followed by another 3×2×1 on the opposite side. Next, the corners of the top layer are solved. The cube can then be solved using only moves of the U layer and M slice.

Beginners' methods

Most beginner solution methods involve solving the cube one layer at a time, using algorithms that preserve what has already been solved. The easiest layer by layer methods require only 3–8 algorithms.
In 1981, thirteen-year-old Patrick Bossert developed a solution for solving the cube, along with a graphical notation, designed to be easily understood by novices. It was subsequently published as You Can Do The Cube and became a best-seller.
In 1997, Denny Dedmore published a solution described using diagrammatic icons representing the moves to be made, instead of the usual notation.
Philip Marshall's The Ultimate Solution to Rubik's Cube takes a different approach, averaging only 65 twists yet requiring the memorisation of only two algorithms. The cross is solved first, followed by the remaining edges, then five corners, and finally the last three corners.

Rubik's Cube solver program

The most move optimal online Rubik's Cube solver programs use Herbert Kociemba's Two-Phase Algorithm which can typically determine a solution of 20 moves or fewer. The user has to set the colour configuration of the scrambled cube, and the program returns the steps required to solve it.

Competitions and records

Speedcubing competitions

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.
A speedcubing championship organised by the Guinness Book of World Records was held in Munich on 13 March 1981. The contest used standardised scrambling and fixed inspection times, and the winners were Ronald Brinkmann and Jury Fröschl with times of 38.0 seconds. The first world championship was the 1982 World Rubik's Cube Championship held in Budapest on 5 June 1982, which 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.
In 2004, the WCA made it mandatory to use a special timing device called a Stackmat timer.
In addition to the main 3x3x3 event, the WCA also holds events where the cube is solved in different ways:
In Blindfolded Solving, the contestant first studies the scrambled cube, and is then blindfolded before beginning to turn the cube's faces. Their recorded time for this event includes both the time spent memorizing the cube and the time spent manipulating it.
In Multiple Blindfolded, all of the cubes are memorised, and then all of the cubes are solved once blindfolded; thus, the main challenge is memorising many – often ten or more – separate cubes. The event is scored not by time but by the number of points achieved after the one hour time limit has elapsed. The number of points achieved is equal to the number of cubes solved correctly, minus the number of cubes unsolved after the end of the attempt, where a greater number of points is better. If multiple competitors achieve the same number of points, rankings are assessed based on the total time of the attempt, with a shorter time being better.
In Fewest Moves solving, the contestant is given one hour to find a solution and must write it down.

Records

Competition records

Top 10 solvers by average of 5 solves[World Cube Association] https://www.worldcubeassociation.org/results/events.php?eventId=333&regionId=&years=&show=100%2BPersons&average=Average Official 3x3x3 Ranking Average

Variations

There are different variations of Rubik's Cubes with up to thirty-three layers: the 2×2×2, the standard 3×3×3 cube, the 4×4×4, and the 5×5×5 being the most well known. As of 1981, the official Rubik's Brand has licensed twisty puzzle cubes only up to the 5×5×5. The 17×17×17 "Over The Top" cube was until December 2017 the largest commercially sold cube. A mass-produced 17×17×17 was later introduced by the Chinese manufacturer YuXin. A working design for a 22×22×22 cube exists and was demonstrated in January 2016, and a 33×33×33 in December 2017. Chinese manufacturer ShengShou has been producing cubes in all sizes from 2×2×2 to 15×15×15, and have also come out with a 17×17×17.
Non-licensed physical cubes as large as 17×17×17 based on the V-Cube patents are commercially available to the mass-market; these represent about the limit of practicality for the purpose of "speed-solving" competitively.
There are many variations of the original cube, some of which are made by Rubik. The mechanical products include Rubik's Magic, 360, and Twist. Also, electronics like Rubik's Revolution and Slide, were also inspired by the original. One of the newest 3×3×3 Cube variants is Rubik's TouchCube. Sliding a finger across its faces causes its patterns of coloured lights to rotate the same way they would on a mechanical cube. The TouchCube also has buttons for hints and self-solving, and it includes a charging stand. The TouchCube was introduced at the American International Toy Fair in New York on 15 February 2009.
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, the octahedron, the dodecahedron, and the icosahedron. There are also puzzles that change shape such as Rubik's Snake and the Square One.
In 2011, Guinness World Records awarded the "largest order Rubiks magic cube" to a 17×17×17 cube, made by Oskar van Deventer. On 2 December 2017, Grégoire Pfennig announced that he had broken this record, with a 33×33×33 cube, and that his claim had been submitted to Guinness for verification. On 8 April 2018, Grégoire Pfennig announced another world record, the 2x2x50 cube. Whether this is a replacement for the 33x33x33 record, or an additional record, remains to be seen.
Some puzzles have also been created in the shape of the Kepler-Poinsot polyhedra, such as Alexander's Star. Grégoire Pfennig has also created at least one puzzle in the shape of a small stellated dodecahedron.

Custom-built puzzles

Puzzles have been built resembling Rubik's Cube, or based on its inner workings. For example, a cuboid is a puzzle based on Rubik's Cube, but with different functional dimensions, such as 2×2×4, 2×3×4, and 3×3×5. 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 moulding and casting techniques to create the final puzzle.
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 3×3×3. Most shape mods can be adapted to higher-order cubes. In the case of Tony Fisher's Rhombic Dodecahedron, there are 3×3×3, 4×4×4, 5×5×5, and 6×6×6 versions of the puzzle.

Rubik's Cube software

s, like Rubik's Cube, can be simulated by computer software, which provides functions such as recording of player metrics, storing scrambled Cube positions, conducting online competitions, analysing 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.

Chrome Cube Lab

Google has released the Chrome Cube Lab in association with Ernő Rubik. The site has various interactive objects based on Rubik's Cube. Customised versions of Rubik's Cube can be created and uploaded.

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