Rubik's Cube

This article covers the 3 × 3 × 3 Rubik's Cube , also known as Rubik's Cube . For other magic cubes see rotating puzzle # N × N × N rotating puzzles .

Rubik's cube in basic position

Rubik's cube with partially turned side

Movement of the Rubik's Cube

The Rubik's Cube (sometimes, as in the English-speaking space Rubik's Cube , Rubik's cube is called) a rotation puzzle , that of the 1974 Hungarian civil engineer and architect Erno Rubik invented. In 1980 it was awarded the special prize Best Solitaire Game of the Critics' Prize Game of the Year . It was particularly popular in the early 1980s, and the speedcubing community has been growing steadily since the 2000s.

description

A standard-size Rubik's Cube is a cube with an edge length of 57 mm, measured on the central axes. However, there are also larger or smaller versions such as those with an edge length of 54.4 mm. The cube is divided into three layers each in height, width and depth, which can be rotated around their respective spatial axes by 90-degree rotations. This means that the position and location of 20 of the total of 26 stones (the central stones are permanently installed) can be changed almost at will. Small areas of color are glued to the surfaces of the stones that are visible to the outside, or the stones themselves are colored. In the basic position, the stones are arranged in such a way that each side of the cube has the same color, but is different from side to side. The basic color of the standard cube is black and the color of the surfaces corresponds to white versus yellow, blue versus green and red versus orange. The orientation of the colors when looking at the white-blue-red corner stone corresponds to white above, blue on the right and red on the left. In the case of a cube with the basic color white, the white surface is often exchanged for a black one.

The goal is usually to move the cube back to its home position after the sides have been turned to a random position. At first glance, this task appears extremely difficult, but strategies were developed at an early stage, the knowledge of which allows a relatively easy solution.

Structure and components

The Rubik's Cube has a total of 26 individual stones:

• Center stone: The six stones in the middle of the cube face sit on the axis cross inside the cube and are therefore always in the same relative position due to the design . The color of the center stone determines which other stones belong on this side and which orientation they have to have. Center stones are monochrome.
• Curb stone: The twelve curb stones each connect two adjacent surfaces and are held in place by the central stones of the two surfaces. Curb stones have two colors.
• Corner stone: The eight corner stones each connect three adjoining surfaces in the corners. They are held in place by the three adjacent curb stones and each have three colors.

• 

  Opened Rubik's Cube

• 

  Axis cross with six center stones (one color)

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  Twelve curb stones (two-tone)

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  Eight corner stones (three-colored)

history

In the program The Big Prize , the inventor explained that he wanted to give his students the opportunity to practice their spatial thinking skills by means of a three-dimensional puzzle when he noticed that they had poor knowledge of geometry from school. Even before that, Rubik brought his interests in sculpture, design and geometry into harmony and made imaginative, three-dimensional wooden figures.

Within a few weeks in 1974, Rubik constructed the first magic cube, which consisted of 27 small wooden blocks. To enable the stones to move, he first experimented with elastic bands, but they tore too easily. Finally he came up with the idea of ​​integrating a center piece, a kind of star made of three intersecting axes, into the prototype. He arranged the edges and corner pieces so that they could be moved around the center of the cube. Finally, Rubik covered each side of the small cubes with paper of different colors, thus completing the teaching material for his students. But when he began to turn the cube, he suddenly had problems restoring it to its original state. Rubik later said: "It was like a secret code that I had invented myself, but could no longer decipher!" When he rearranged his cube, he felt a sense of freedom. Then Rubik understood that there was much more to his invention than just a teaching aid. Research showed that there was no similar toy in the world. After Rubik was granted Hungarian patent no. 170062 for the cube on October 28, 1976, the cube found its way into the “capitalist world” in December 1977 when a copy of the cube was sent to the UK- based company Pentangle . This company then acquired the license to sell the cube in Great Britain. However, in 1979 the Hungarian government granted the worldwide sales rights for the cube to the American manufacturer Ideal Toy Corporation (also known as Arxon in Europe ). In breach of the Treaty, this also included the rights for the United Kingdom . Ideal Toy Corporation allowed Pentangle to sell the cube to gift stores, but not toy stores. At first, Rubik's idea made the rounds among scientists. At an international mathematics congress in Helsinki, professors spun their toys for hours. In 1979 the "Rubik's Cube" was presented at the toy fair in Nuremberg. From June 2, 1980 it was available for sale in the Federal Republic of Germany .

In 1981 the demand for the mechanical puzzle had its peak. Ideal Toy Corporation failed to meet demand, which allowed cheap Far Eastern products to flood the market. A total of around 160 million cubes were sold by the height of the boom alone. At the beginning of 1982 the demand for the dice collapsed and with it the demand for many other puzzles .

Ernő Rubik wasn't the first to deal with the subject of a game of this kind. As early as 1957, the chemist Larry D. Nichols developed a similar cube, which only consisted of 2 × 2 × 2 parts and was held together by magnets. He patented his design in 1972 . In 1984, Nichols won a patent lawsuit against the company that sold the Rubik's Cube in the United States . However, this judgment was partially overturned in 1986, so that only the 2 × 2 × 2 large pocket cube . 'Pocket Cube' concerned.

At CeBIT 2009, a digital version of the cube was also presented, which was equipped with light-emitting diodes and touch fields .

In November 2006, the German toy manufacturer Simba Toys applied for the trademark cancellation for the relevant European 3D trademark. The nullity division and the Board of Appeal of the former OHIM rejected the application as clearly unfounded, the court of first instance of the European Union (CFI) upheld this decision in 2014. With its judgment of 10 November 2016, the European Court of Justice (ECJ) overturned the decisions of the Board of Appeal and the CFI make another decision. The decision of the European Court of Justice certifies that the cancellation request has good reasons for it, which the lower courts did not take into account; a deletion was expected.

On October 24, 2019, the General Court of the European Union (EGC) ruled again and declared the EU trade mark “Rubik's Cube” to be void. The CFI stated that this shape should never have been registered as an EU trade mark, since the essential features of this shape are necessary to achieve the technical effect, which consists in the ability to rotate the Rubik's Cube.

Solution strategy for the Rubik's Cube

Play media file

Solve the 3 × 3 × 3 cube within 26.59 seconds using the Fridrich method

→ Main article : Methods for solving the Rubik's Cube

Strategies that get by with as few movements of the cube as possible can usually only be implemented with the help of a computer or extensive position tables. Other, easier-to-remember strategies use fewer basic moves, but generally require more movements.

Algorithms for solving the cube are written down using various notations. The most common approach, in which the three levels of the cube are ordered one after the other, is called the "layer-by-layer" method. They are similar to the published solution published by Spiegel (No. 4/1981). In the area of speed cubing , where speed is particularly important, other variants are used to solve the magic cube, such as the Jessica-Fridrich method or the one according to Lars Petrus .

Letter notation

In order to note combinations of moves for the die, each action is assigned a letter.

abbreviation		page
German	engl.
V	Front)	front
H	B (ack)	back
R.	R (ight)	right
L.	L (eft)	Left
O	U (p)	above
U	D (own)	below
x		Rotation of the whole cube when looking at the right side
y		Rotation of the whole cube when looking at the top
z		Rotation of the whole cube when looking at the front side
M.		Rotation of the plane between L and R. Direction as Left
S.		Rotation of the plane between F and B. Direction as front
E.		Rotation of the plane between up and down. Direction like Down

A letter always means a rotation of the page by 90 ° clockwise, a ′ or ⁻¹ counterclockwise relative to the page being viewed. For example, rotating the bottom 90 ° clockwise (D) is exactly the opposite of rotating the top 90 ° clockwise (U). A 2 stands for a rotation of the plane by 180 °. Lower case letters or letters with a small “w” appended to them that refer to pages mean the rotation of two planes from the corresponding side; for example for r and Rw the right and parallel middle plane. Sometimes more letters are used for middle class trains. To describe finger techniques or solves , 2 ′ is sometimes used to indicate a 180 ° counterclockwise rotation of the right side. In order to be able to remember move sequences better, several moves are sometimes put in brackets.

Example: The following combination tilts two curb stones and leaves all others unchanged:

K ₁ = B ′ R2 B2 RB ′ R ′ B ′ R2 FDBD ′ F ′

Graphic notation

Alternatively, some instructions also use graphical forms of notation , e.g. B. as three-dimensional cube representations or as a 3 × 3 view of the front with arrows that indicate the rotation of the cube faces. The latter have the disadvantage that operations of the middle and rear cube plane (seen from the front) are difficult to represent, for example by an additional development of the upper side. It is also possible to dispense with the representation of a cube and only use arrows.

Optimal solutions

The “super flip” is the most popular position that cannot be solved in less than 20 moves (quarter and half turns)

One of the three well-known positions that cannot be solved in less than 26 turns (quarter turns)

In order to move the Rubik's Cube from a given position to its original starting position, you need a certain minimum number of moves. A path that only consists of this minimum number of steps is therefore an optimal solution. (There can be several different but equally short paths between the two positions.)

The way to find out of any position of such a shortest path is called God's Algorithm (Engl. God's Algorithm ) refers. This name comes from the English group theorist John Conway or one of his colleagues in Cambridge. Based on this, the number of moves that you need at least to solve the Rubik's Cube from any position - i.e. the length of the optimal paths for the positions "furthest" from the starting position - is called God's number .

There are two ways to count the dice moves:

• Quarter turns (± 90 °) and half turns (180 °) of side faces are considered as a single move
• The quarter turns are counted individually.

If you only count quarter turns, you can already tell by evaluating the position of the cube whether an even or odd number of turns is necessary to solve.

The first algorithm for finding an optimal solution was formulated by Richard E. Korf , who showed in 1997 that the average optimal solution requires 18 moves ( with half turns). He also assumed that it never took more than 20 moves, but couldn't prove it. As early as 1992, Dik T. Winter had found a position (the so-called super flip ) that required 20 moves. Michael Reid provided evidence in 1995 that this position cannot actually be resolved in fewer moves.

In March 2008, the American computer scientist Tomas Rokicki was able to show with a huge amount of calculations that the maximum number of moves that you need to turn a Rubik's Cube from any position into its starting position with the right strategy can be at most 25, which it does in August thanks to improved computer support (by software engineer John Welborn from Sony Pictures ) to 22.

In July 2010 Tomas Rokicki together with Morley Davidson, John Dethridge and Herbert Kociemba proved the assumption that never more than 20 moves are necessary. 12,000,000 positions were found that cannot be resolved in less than 20 moves. Presumably there are a total of 490,000,000 such positions.

In August 2014, the calculation was carried out if you only count in quarter turns. You never need more than 26 quarter turns to solve the problem. The position that can be solved in no less than 26 moves was found back in 1998. If the cube is in this maximum position, all corners are correctly placed, but the edges are turned. In addition, two (of the three) pairs of opposite centers are swapped. This means that three possible maximum positions are known, but they do not differ mathematically. The proof that these are the only ones is still pending.

Speed ​​cubing

→ Main article : Speedcubing

Speedcubers can solve any twisted Rubik's Cube with 45 to 60 movements. When Speedcubing , so releasing on time, it depends on the rapid recognition of of positions, internalizing a large number of algorithms that advance planning and dexterity. In speed cubing, national, continental and world championships are held by the World Cube Association ( WCA ).

Normal loosening

The first world championship, organized by the Guinness Book of Records , took place on March 13, 1981 in Munich. The cubes were twisted 40 times and rubbed with petroleum jelly . The winner of the championship was Jury Fröschl from Munich with a record time of 38 seconds.

The current world record for a 3 × 3 × 3 cube is 3.47 seconds and was set by Yusheng Du at the Wuhu Open 2018 .

One-handed release

The Rubik's Cube is the only rotating puzzle for which competitions in one-handed solving are organized by the WCA. If both hands have touched the cube during the solving process (this does not have to happen at the same time), the attempt is considered a DNF ( Did not finish ), i.e. H. Unrated. In the inspection phase, however, both hands are allowed to touch the cube.

The current world record, set by Max Park at Bay Area Speedcubin '20 2019 , is 6.82 seconds.

Blindfold cubing

Play media file

Demonstration: Blind solving the 3 × 3 × 3 cube in 49.83 seconds

Another popular discipline is blindfold cubing . First you memorize the twisted Rubik's Cube and then loosen it blindfolded without seeing it again. The inspection time and the release time are included in the time. In fact, you don't memorize the whole cube, but often only the order of the algorithms. To solve this, “beginners” usually use methods that change as few other stones as possible per algorithm.

The current world record, set by Max Hilliard at CubingUSA Nationals 2019 , is 15.50 seconds.

Multiple blindfold cubing

There is also multiple blindfold cubing , an increase in blindfold cubing . First you memorize as many dice as possible in order to then solve them all blindly with your eyes closed. Points are not awarded for the time, but for the number of solved dice minus the number of unsolved dice that remain after one hour.

The current world record, set by Graham Siggins at OSU Blind Weekend 2019 , is 59/60 in 59:46 minutes.

Solve with as few moves as possible (Fewest Moves)

In this discipline, the participants try to solve the dice in as few moves as possible. According to the official WCA rules, they have 60 minutes to do this. Then they must have worked out a solution comprising a maximum of 80 moves, which they hand over to the judge for examination.

The world record of 16 trains was set by Sebastiano Tronto at FMC 2019 .

As shown above , each die can be solved in 20 or fewer moves; most positions even in 18 moves.

Machine solution

Robot built by vocational school students to solve the Rubik's Cube

There are a number of machines that can solve the cube using image recognition and automated mechanics. In 2011, for example, the official human record was beaten for the first time by robotics: CubeStormer 2 solved the cube in 5.27 seconds - the record set by a human (Feliks Zemdegs) was 5.66 seconds. In 2014, CubeStormer 3 solved the cube in 3.25 seconds using a Galaxy S4 and eight Lego Mindstorms EV3s .

In January 2016, a video was released in which a robot could solve the Rubik's Cube in 1.047 seconds. Further attempts at a solution remained consistently under 1.2 seconds. The robot "Sub1" analyzes the dice with four USB - Webcams , it is rotated by means of stepping motors .

In November 2016, the “Sub1 reloaded” robot solved a magic cube in 0.637 seconds at the Electronica trade fair in Munich . The Infineon Aurix microcontroller developed for autonomous driving was installed .

In March 2018, Ben Katz and Jared Di Carlo presented another machine that solves the cube in a record time of 0.38 seconds.

Create a pattern

In addition to the usual solving of the Rubik's Cube, another popular way of playing is to create regular and irregular patterns with the Rubik's Cube .

With many patterns, only the cubes on the opposite sides are swapped (" Pepita basic pattern", "Quadruple cross pattern", "Sixfold T-pattern"). With other patterns only the cubes of three adjacent sides ("center point pattern", " Six-fold cross pattern "," Cube-in-the cube "(also" 2 × 2 "in" 3 × 3 × 3 ")," Circulating worm "/" Snake ").

In addition, there are color-mixed patterns such as the Superflip (all curb stones tilted) or a circumferential diagonal with two different “three-cornered corners”.

In principle, there are three different approaches to creating patterns:

1. Create a pattern with a special sequence of moves or a combination of several sequences of moves - starting from a cube in its original starting position with six colored areas.
2. Create a pattern with a special sequence of moves or a combination of several sequences of moves - based on a cube that has already been rotated in a pattern position ("pattern change").
3. Create a pattern according to a template or your own idea with the known sequence of moves - starting from a cube in its original starting position with six colored areas or a randomly twisted cube.

The phenomenon with some invented patterns is that, due to the construction of the cube, not all patterns can actually be realized. Often at the end a corner cube is not in the right position or two edge cubes are in the wrong position (examples: six-fold circumferential diagonal, various Pepita variants with adjacent sides). With other patterns, a different combination of the colored areas of the corner or edge cubes would be required or one edge cube would be required twice.

Another variation in this context is to restore the original starting position of the magic cube with the six colored areas from a cube turned in the pattern with just a few sequences of moves.

Prime examples

• 

  Floral pattern

• 

  Checkerboard pattern

• 

  extended checkerboard pattern

variants

→ Main article : rotating puzzle

There are several variations on this mechanical puzzle. A cube with pictures printed on it is somewhat more difficult, as the generally known solution strategies mean that the colored areas are in the right place, but the middle areas do not always have the correct orientation. With the Rubiks calendar cube (date cube) the surfaces are provided with numbers and texts, from which the current date with weekday, month and day can be put together on the front surface. There are simpler cubes that consist of only two levels in each spatial direction like the Pocket Cube , and more complicated variants that consist of four levels ( Rubik's Revenge , also known as Rubik's Revenge or Rubik's Master Cube ), five levels ( Professor's Cube or 5 × 5 × 5 Cube or Rubiks Wahn ) or two or more cubes integrated into one another (Rubik's Fusion) . The largest n × n × n mass-produced magic cube is the 17 × 17 from Yuxin (as of 2019). This is called Yuxin Huanglong 17 × 17. There are also cuboid and dodecahedral rotating puzzles. There are also rotating puzzles in barrel or pyramid shape and balls ( masterball ), also in different levels of difficulty. The Fisher Cube has been around since the mid-1980s .

In 2005 a cube with six levels was presented for the first time. The underlying mechanism also allows cubes with up to eleven levels. However, these must be barrel-shaped - the centers of the surfaces facing outwards - so that the fastening of the corner stones is still completely within the cube. This distortion, together with the necessary size and weight, will require a certain amount of skill from the player. The solution methods for these large cubes do not require any moves that are not already known from the four or five level cube.

Since June 2008 6 × 6 × 6 and 7 × 7 × 7 magic cubes have also been on the market. There are now larger Rubik's Cubes, but since February 2009 the official championships have only been held for a maximum of 7 × 7 × 7 Rubik's Cubes.

A mechanical puzzle that is very popular because of its star-shaped shape is the 4D8 Rubik's Cube. This is derived from a star tetrahedron , (Stella octangula) also Kepler Stern called. However, its tips are cut off, leaving truncated pyramids .

With computer programs that simulate the Rubik's Cube , even more levels can be set.

As a result of the boom in the 1980s, mechanical puzzles emerged, which was a different mechanism underlying such as Rubik's Magic , the Devil ton , Back to Square One , Rubik's triamide , Rubik's Clock , Alexander's Star or the magic tower . The mechanically most demanding puzzle of this kind is probably the Dogic in the form of an icosahedron (twenty surfaces ).

• 

  Rubik's cube variants

• 

  Rubik's Cube in a random position

• 

  Floppy cube

• 

  Pyraminx

• 

  Megaminx

• 

  Gigaminx

• 

  Teraminx

• 

  Special form of the Rubik's Cube

• 

  4D8 Rubik's Cube

mathematics

The cube as a mathematical group

The cube can be seen as a mathematical group . For this, each position is viewed as a combination of the six possible basic permutations . All possible permutations (positions) form the set . Each position can be reached by linking the six basic permutations , which are linked with the two-digit link . ${\displaystyle B=\{V,H,R,L,O,U\}}$ ${\displaystyle G_{W}}$ ${\displaystyle B_{W}=\{V,H,R,L,O,U\}\subset G_{W}}$ ${\displaystyle \circ \colon G\times G\rightarrow G}$

In addition, both exists a neutral element , the basic position (equivalent to a "no-op" running dissolved cubes), because for all possible permutations (group elements) applies , as well as an inverse element because every permutation of an element with exists, for example, or . Furthermore applies to everyone . ${\displaystyle i}$ ${\displaystyle p}$ ${\displaystyle p\circ i=i\circ p=p}$ ${\displaystyle p}$ ${\displaystyle p^{-1}}$ ${\displaystyle p\circ p^{-1}=p^{-1}\circ p=i}$ ${\displaystyle R\circ R^{-1}=i}$ ${\displaystyle H^{-2}\circ U^{-1}\circ U\circ H^{2}=i}$ ${\displaystyle X\in B_{W}:X^{2}=X^{-2}}$

The triple therefore forms a group in the sense of algebra. This is not commutative, because the link is not commutative: For example . ${\displaystyle (G,\circ ,i)}$ ${\displaystyle \circ }$ ${\displaystyle R\circ H\neq H\circ R}$

Solutions of the cube

Given a permutation (a twisted cube), the task is to find a finite sequence of permutations from the set that produces exactly this permutation : ${\displaystyle s\in G_{W}}$ ${\displaystyle (\sigma _{i})}$ ${\displaystyle B_{W}}$ ${\displaystyle s}$

${\displaystyle \sigma _{1}\circ \sigma _{2}\circ \dotsb \circ \sigma _{n}=s}$

The solution is not clear, that is, there are many solutions, the shortest of which is sought. The diameter of the groups, i.e. the maximum length of a permutation with which all elements from can be reached, is 20. ${\displaystyle G}$ ${\displaystyle G_{W}}$

In July 2010, the three Americans Tomas Rokicki, Morley Davidson and John Dethridge and Herbert Kociemba from Darmstadt calculated that each position can be solved in a maximum of 20 moves ( with half turns). In August 2014, Tomas Rokicki and Morley Davidson showed that a maximum of 26 moves are necessary if a move only allows quarter turns (half turns are then two turns).

Order of group G

The order of a group corresponds to the thickness of its carrier set . Since there is only a finite number of possible positions, this corresponds to the number of possible positions: ${\displaystyle (G,\circ ,i)}$ ${\displaystyle |G|}$

${\displaystyle |G_{W}|}$ = ${\displaystyle {\frac {8!\cdot 3^{8}\cdot 12!\cdot 2^{12}}{3\cdot 2\cdot 2}}=43.252.003.274.489.856.000\approx 4{,}3\cdot 10^{19}}$

These result from:

• 8 positions where the corner cubes can be. The first can occupy all 8 positions, the second 7 and so on, whereby the number of combinations corresponds to the factorial of 8 (8!).
• 3 orientations that every corner cube can take (3 ⁸ ).
• 12 positions where the edge cubes can be (12!).
• 2 orientations that each edge can adopt (2 ¹² ).

The denominator results from three conditions that apply when the cube is twisted but not taken apart:

• Seven of the eight corner cubes can be oriented at will, while the orientation of the eighth is forced (3).
• Eleven of the twelve edge cubes can be oriented at will, while the orientation of the twelfth is enforced (2).
• You can neither swap two corner cubes, nor can you swap two edges. The number of pair exchanges must always be even (2).

Subgroups

If the set of generating permutations is limited, carrier sets of lesser thickness arise that are subsets of . These subgroups are crucial for solving the cube with computers. ${\displaystyle G_{W}}$

Trivia

• On the occasion of its 40th birthday, Google honored the Rubik's Cube with an interactive doodle .
• The Rubik's Cube appears several times in the Simpsons . Two examples that did not go well for the Simpsons: In the episode Der Ernstfall , Homer Simpson tries to remember the briefing in his control panel in the nuclear power plant before the impending meltdown . Back then, instead of listening, he was looking at the Rubik's Cube. In the episode The Mad Ned , Marge Simpson takes out the Rubik's Cube while the family is in the basement during a hurricane to pass the time. The whole family gets into an argument, so that Marge, disappointed, puts the die back with the words: "Now I know again why I put it here."
• In the 80s show with Oliver Geissen , an oversized fragment of the cube is a typical symbol of the 1980s. The base of the coffee table at which the guests and Oliver Geissen are sitting.
• In the show Who Wants to Be a Millionaire? On December 7th, 2015 the following question was asked for one million euros: “How many stones is the classic Rubik's cube made of?” The candidate Leon Windscheid answered the question correctly with “26” and thus became the tenth regular millionaire on the show.
• In the sitcom The Big Bang Theory , an occasional paper towel dispenser is seen in the shape of an oversized Rubik's cube.
• The artist Invader recreates works of art from Rubik's cubes, such as the Mona Lisa . This was auctioned in February 2020 for € 480,000.

See also

• GNU Gnubik

literature

• Matthias Stolz: The return of magic. In: Die Zeit , No. 4/2009, pp. 10–15 (Life, About the comeback of the Rubik's Cube, people and the inventor of the Rubik's Cube. Photos, interviews).
• Shout hurray! Throw a round! In: Der Spiegel . No. 4 , 1981 ( online solution).

Introductions and instructions

• We unravel the Rubik's Cube. In: Mathematical Cabinet. Image of science . Munich 1980, November, pp. 174-177.
• Douglas R. Hofstadter : On the magic of the Rubik's Cube. In: Mathematical games. Spectrum of science . Heidelberg 1981, May, p. 16 ff. (Original: Scientific American , March 1981) ISSN  0170-2971 (including instructions for professional cube assembly, solution strategy, graphic patterns and variations).
• Kurt Endl: Rubik's Riddle of the Century. Würfel-Verlag, Giessen 1981, ISBN 3-923210-15-9 .
• Josef Trajber: The Cube (Rubik's Cube). Falken, Niedernhausen / Ts. 1981, ISBN 3-8068-0565-2 , ISBN 3-8068-0585-7 .
• Josef Trajber: The cube for advanced players . Falken, Niedernhausen / Ts. 1981, ISBN 3-8068-0590-3 .
• Tom Werneck : The Rubik's Cube. Heyne, Munich 1982, ISBN 3-453-41449-7 .
• Tom Werneck: The magic cube for experts. Heyne, Munich 1982, ISBN 3-453-41478-0 .
• Tom Werneck: The magic ball . Foreword by Martin Gardner . Heyne, Munich 1982, ISBN 3-453-41505-1 (Rubik authorized solution book).

mathematics

The following titles deal with the mathematical properties of Rubik's Cube, but also contain instructions that may help. U. are easier to understand than the informal introductions.

• David Singmaster : Notes on Rubik's Magic Cube . Enslow, Hillside NJ 1981. (classic study, the 5th and final edition is twice the size of the first from 1979)
• Alexander H. Frey jr., David Singmaster: Handbook of Cubik Math . Enslow, Hillside NJ 1982.
• Wolfgang Hintze: The Hungarian Rubik's Cube. Deutscher Verlag der Wissenschaften VEB, Berlin OST 1982 (partly based on Singmaster's book).
• Christoph Bandelow : Introduction to Cubology . Vieweg, Braunschweig / Wiesbaden 1981, ISBN 3-528-08499-5 .
• Christoph Bandelow: Inside Rubik's Cube and Beyond . Birkhäuser, Basel / Boston 1982. (extended English version of the above)
• Ernő Rubik , Tamas Varga, Gerzson Keri, Gyorgy Marx, Tamas Vekerdy: Rubik's Cubic Compendium . English translation by A. Buvös Kocka, with an afterword by David Singmaster. Oxford University Press, London 1987 (from the inventor of the Rubik's Cube).
• David Joyner: Adventures in Group Theory: Rubik's Cube, Merlin's Machine, and Other Mathematical Toys . Johns Hopkins University Press, Baltimore MD 2002 (An Introduction to Group Theory Using Rubik's Cube).

Web links

Commons : Rubik's Cube  - Album with Pictures

Wiktionary: Rubik's Cube  - explanations of meanings, word origins, synonyms, translations

Wikibooks: Rubik's Cube  - learning and teaching materials

• World records with the Rubik's Cube
• Literature on Rubik's Cube in the catalog of the German National Library
• Wolfgang Blum: God's number is 20 . In: Zeit online , August 10, 2010 (How many moves do you need to crack every imaginable combination?)
• Detailed Rubik's Cube instructions
• Online rubiks cube solver

Individual evidence

1. ^ ^{A b} William Fotheringham: Fotheringham's Sporting Pastimes . Anova Books, 2007, ISBN 1-86105-953-1 , p. 50.
2. ^ ^{A b} Tom de Castella: The people who are still addicted to the Rubik's Cube . In: BBC News Magazine . bbc.com. Retrieved April 28, 2014.
3. ^ Editing of the school youth magazine FLOH (Hrsg.): Hello world - The great youth yearbook . Domino Verlag Günther Brinek GmbH, Munich 1990, p. 48 .
4. ↑ Patent HU170062: Térbeli logikai játék. Published on October 28, 1976 , inventor: Rubik Ernő ( https://www.jaapsch.net/puzzles/patents/hu170062.pdf ). ‌
5. ↑ Court ruling on patent infringement on digital-law-online.info
6. ↑ Community trademark No. 162784, registered on April 6, 1999,
7. ↑ EGC, judgment of November 25, 2014 - T-450/09 = GRUR-Prax 2014, 546.
8. ↑ ECJ, judgment of November 10, 2016 - C-30/15 P = GRUR 2017, 66 - Simba Toys / EUIPO [Rubik's Cube].
9. ↑ Annette Kur: "Rubik's Cube - Magic Cube at the End?". In: GRUR 2017, pp. 134–141.
10. ↑ EGC, judgment of October 24, 2019 - T-601/17 - Rubik's Brand / EUIPO
11. ↑ Rubik's Cube - disenchanted as a brand , in: Rechtslupe.de, November 20, 2019
12. ↑ graphical notation on a speedcubing website
13. ↑ Jerry Slocum: The Cube. The Ultimate Guide to the World's Bestselling Puzzle. Secrets - Stories - Solutions. Black Dog & Leventhal, New York 2009, p. 26.
14. ^ Korf: Optimal Solutions to Rubik's Cube . (PDF; 122 kB)
15. ↑ arxiv : 0803.3435
16. ↑ The last riddle of the cube . In: Der Spiegel . No. 23 , 2010, p. 103 ( online ). Homepage of Rokicki
17. ↑ ^{a b c} God’s Number is 20
18. ^ Rokicki: Twenty-two moves suffice for Rubik's cube . ( Memento of December 22, 2008 in the Internet Archive ) In: Mathematical Intelligencer , 2010, No. 1, p. 33
19. ^ The Diameter of the Rubik's Cube Group Is Twenty . In: SIAM J. Discrete Math. , 27 (2), pp. 1082-1105, doi: 10.1137 / 120867366
20. ↑ ^{a b} God’s Number is 26 in the Quarter-Turn Metric
21. ↑ Annette Schär: Rubik-Würfel: "It's complicated, I want to be able to do that too!" In: Neue Zürcher Zeitung . June 10, 2019 (The Rubik cube, the cult toy of the 1980s, is still fascinating today. In Lucerne, the best Swiss Speedcubers competed in 18 categories over Whitsun.). 
22. ↑ List of future competitions on the WCA website
23. ↑ ^{a b c d e} List of all disciplines and records on the World Cube Association website
24. ^ Official WCA rules
25. ↑ http://www.cube20.org/ - Presentation of the frequencies according to the steps necessary to solve them
26. ↑ Duncan Geere: Video: CubeStormer II robot beats Rubik's Cube speed record. (No longer available online.) In: wired.co.uk. November 11, 2011, archived from the original on November 13, 2011 ; accessed on May 20, 2014 (English). Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice.  
27. ↑ The CubeStormer 2 - World Record Rubik's Cube Solver made from LEGO NXT Mindstorms. In: YouTube . legobuildingblocks, November 12, 2012, accessed March 17, 2014 . 
28. ↑ Ingo Pakalski: Robots solve Rubik's cube faster than humans. Golem.de, March 16, 2014, accessed on March 17, 2014 . 
29. ↑ Henning van Lil: Record turning on the Rubiks Cube: 1.04 seconds - the disenchanted cube. In: tagesschau.de. Norddeutscher Rundfunk, January 28, 2016, accessed on January 28, 2016 . 
30. ↑ Jay Flatland, Paul Rose: World's Fastest Rubik's Cube Solving Robot. In: youtube.com. January 11, 2016, accessed January 29, 2016 . 
31. ↑ Robot with Infineon chip solves Rubik's cube in record time . In: VDI nachrichten No. 46, November 18, 2016, p. 2, section: This week.
32. ↑ [1] . In: Heise News
33. ↑ lichtuchender.wordpress.com
34. ↑ History of the 6 × 6 × 6 Rubik's Cube Records on the official World Cube Association website
35. ↑ History of the 7 × 7 × 7 Rubik's Cube Records on the official World Cube Association website
36. ^ Rubik's Cube . University of Mannheim, seminar on computer algebra with GAP
37. ↑ The interactive Google Doodle for the 40th birthday
38. ↑ The Rubik's Cube at the Simpsons
39. ↑ mka / dpa: "Who will be a millionaire?": Student wins the million - after 20 minutes of pondering. In: spiegel.de. Spiegel Online GmbH, December 7, 2015, accessed on July 6, 2020 . 
40. ↑ Screenshot from The Big Bang Theory
41. ↑ bam / AFP: Mona Lisa made of Rubik's Cubes auctioned for 480,000 euros. In: spiegel.de. Spiegel Online GmbH, February 24, 2020, accessed on July 5, 2020 .