Rubik's Cube

The Rubik's Cube (sometimes, as in the English-speaking world Rubik 's Cube, Eng. F Rubik's cube, called ) is a mechanical puzzle invented by the Hungarian engineer and architect Ernő Rubik and in 1980 awarded the special prize Best Solitaire Game of the critics prize game of the year. It enjoyed particular in the early 1980s great popularity.

• 3.1 Graphical Notation
• 3.2 letter notation
• 3.3 Optimal Solutions

• 4.1 The cube as a mathematical group
• 4.2 Solutions of the cube
• 4.3 order of the group G
• 4.4 Subgroups

• 5.1 Speedcubing
• 5.2 Blindfold Cubing 5.2.1 Multiple Blindfold Cubing

Description

It is a cube (measured at the center axes and in the standard size ), which is divided in height, width and depth in three layers, which each by 90 degree rotations about their having an edge length of 57.5 mm can bring spatial axis to cover. This allows the position and orientation of the various stones are changed almost at will. In the externally visible surfaces of the stones small areas of color are glued. In the initial position the stones are so arranged that each side of the cube has a uniform, but from side to side different color.

The goal is usually, the dice to move back to its normal position after the previous pages have been rotated in a random position. This task would be extremely difficult at first glance, but strategies were developed early on, their knowledge allows a relatively easy release.

Construction and components

• Means stone: the six blocks in the center of the cube faces sitting on the coordinate system in the interior of the cube, and therefore to each other by design always have the same relative position. The color of the center stone determines what other pieces belong to this page and what orientation they must have.
• Curb: The twelve stones edges connecting two adjacent surfaces and are held by the central stones of the two surfaces.
• Eckstein: The eight cornerstones connect three adjacent faces in the corners. They are held by the three adjacent edges of bricks in position.

Coordinate system with six resources stones ( one color)

Twelve stones edges (two colors)

Eight cornerstones ( tri-color)

History

In the show The Big Prize, the inventor stated that he had would like to give a three-dimensional puzzle his students an opportunity to practice their spatial thinking skills.

After Rubik, the Hungarian Patent No. 170062 was issued for the cube on 28 October 1976 of the dice 1977 collection held in December in the " capitalist world ", as a copy of the cube of the UK-based company Pentangle was sent. This company acquired then the license for distribution of the cube in the UK. The government in Hungary awarded but in 1979 the worldwide sales rights for the cube to the U.S. company Ideal Toy Corporation ( in Europe also Arxon known). In it were in breach of contract and the rights for the United Kingdom included. Ideal Toy Corporation Pentangle allowed the sale of the cube of gift, but not to Toy Shops. From 2 June 1980, he was available in the FRG.

In 1981 the demand for the mechanical puzzle peaked. Ideal Toy Corporation could not satisfy, which allowed far eastern cheap products to flood the market demand. Overall, probably about 160 million cubes have been sold alone until the height of the boom. In early 1982, broke a demand for the cube and with it the demand for many other puzzle games.

Ernő Rubik was not the first, which dealt with the issue of a game of this type. Already in 1957 by the chemist Larry Nichols a similar cube, but the only consisted of 2 × 2 × 2 parts and was held together by magnets. He had his design patented in 1972. 1984 Nichols won a patent lawsuit against the company that drove the Rubik 's Cube in the USA. However, this judgment was partially lifted in 1986, making it only the 2 × 2 × 2 Pocket Cube large, Eng. f pocket cube, concerned.

At CeBIT 2009, a digital version of the cube was introduced, which is equipped with LEDs and touch fields.

Solution strategy for the Rubik's Cube

Strategies which get by with as few movements of the cube, are usually realized only with the help of a computer or extensive position tables. Other, more mnemonic strategies come out with a few basic traits, but generally require a higher number of movements.

Algorithms for the solution of the cube are written by different notations. The most common approach, in which the three levels of the cube are arranged one after the other, is often referred to as the Beginner Method. They are similar to the published solution, which published the mirror ( No. 4/ 1981). In the area Speedcubing where it matters most to the speed of other variants of solving the Rubik's Cube used to call are Jessica Fridrich method or by Lars Petrus.

Graphical notation

Alternatively, use some instructions and graphical notation forms, either as a three-dimensional cube representations or 3 × 3- supervision of the front with arrows that indicate the rotation of the cube faces. The latter have the disadvantage that the operations (front ) middle and rear cube plane are difficult to be represented, eg, by an additional handling of the top. An advantage of this notation is, however, that it can represent rotations of the other mid-planes as individual traits.

Letter notation

To write trailer combinations for the cube, each action, a letter is assigned.

A letter always here means a rotation of 90 ° clockwise, a ' -1 or counterclockwise relative to the currently viewed page. Thus, the rotation of the bottom by 90 ° in the clockwise direction (D ), for example exactly opposite to the rotation of the upper side by 90 degrees in a clockwise direction (U). Small letters that refer to pages that rotation of two levels of the relevant page from mean considered; for r, for example, the right and parallel to middle level. Sometimes even more letters for middle class trains are used.

Example: The following combination flips two edge stones and leaves any remaining unchanged:

This means B 'is a rotation of the rear page 90 ° counter-clockwise rotation R2 of the right side by 180 ° and R is a rotation of the right-hand side by 90 ° clockwise.

Optimal Solutions

In order to convert the Rubik's Cube from a given position to the original starting position, one needs a certain minimum number of moves. A path from a given initial position according to which only consists of the number of steps, thus provides an optimal solution dar. ( Between the two positions, although there may be several different, but equally short distances. )

The way to find out of any position of such a shortest path is referred to as God's Algorithm ( engl. God's Algorithm). This name comes from the English group theorist John Conway or one of his colleagues in Cambridge. In line with this, is that number moves you more than necessary to solve the Rubik's Cube from any position out - that is the length of the optimal paths for the " farthest " away from the initial position positions - God called number.

There are several ways to count the cubes movements: In general, both quarter-turns (± 90 ° ) and half rotations are considered (180 ° ) of faces as a single train; rare only the quarter-turns are counted. The information in the following paragraphs go also by the variation of half- twists.

The first algorithm for finding an optimal solution formulated Richard E. Korf, who showed in 1997 that the average optimal solution requires 18 moves. He also assumed that no more than 20 trains are required, but he could not prove it. As early as 1992 Dik T. Winter had found a position that requires 20 trains. The proof that this position is not actually solve in fewer strokes, Michael Reid rendered in 1995.

In August 2008, the American computer Tomas Rokicki showed with tremendous computational effort, that the number of moves you maximum required with proper strategy to Rubik's cube to turn back any position to its initial position, can be at most 25, which he in 2010 through improved computer support could be reduced to 22 ( from the software engineer John Welborn of Sony Pictures ).

In July 2010, Tomas Rokicki proved together with Morley Davidson, John Dethridge and Herbert Kociemba the assumption that no more than 20 moves are necessary.

Mathematics

The cube as a mathematical group

The cube can be considered as a mathematical group.

For each position is considered as a combination of the six possible base - permutations.

All possible permutations ( positions ) form the set. Each position is catch me by linking the six Grundpermutationen that are connected to the two -digit shortcut.

Furthermore, both a neutral element, the position ( corresponding to a " no operation " is carried out on the dissolved cube), for all possible permutations ( array elements ), there is considered, and an inverse element, since for each permutation of the element there is, for example, or. Continue to apply to all.

The triples thus forms a group in the sense of algebra. This is not commutative, because the link is not commutative ().

Solutions of the cube

Be now given a permutation ( a twisted cubes), so the task is to find a finite sequence of permutations of the set that generates exactly this permutation:

The solution is not unique, ie, there are many solutions, of which the shortest is searched. The diameter of the group, that is the maximum length of a permutation of all the elements are obtained from, for 20

In June 2007, Gene Cooperman and Dan Kunkle from Northeastern University in Boston have shown that 26 trains are always sufficient. In April 2008, this barrier has again reduced to 23 (see above). In July 2010, the three Americans Morley Davidson, John and Tomas Rokicki Dethridge and the Darmstadt Herbert Kociemba have finally found the minimum number of 20. This bound is the smallest possible, which is sufficient for all the cubes positions.

Order of the group G

The order of a group is the cardinality of its support amount. Since there are only a finite number of possible positions, that is the number of possible states:

These arise from itself

• 8 points at which the corner cube can be located ( 8! )
• 3 rotation positions of which can adopt each of corner cubes (38),
• 12 points to spread the edges of cubes ( 12! )
• 2 rotational positions, which can take any edge (212).

The denominators here are three conditions that apply when the cube rotates, but will not be taken apart:

• Seven of the eight corner cubes can be freely oriented, while the orientation of the eighth thereby forced ( 3)
• Eleven of the twelve edges of cube can be oriented as desired, while the orientation of the twelfth is enforced by ( 2)
• It can neither be alone two corner cubes swap, nor let alone swap two edges. The number of pairwise Zweiertäusche must always be straight (2).

Subgroups

If one limits the amount of generating permutations arise carrier quantities of lower cardinality, which are subsets of. These subgroups are for solving the cube with computers crucial.

Competitions

Speedcubing

Some people who call themselves speedcubers have found strategies that enable them with 45 to 60 moves to solve an arbitrarily twisted cubes. When Speedcubing, so releasing on time, it is beyond but also on dexterity and internalizing a high number of prefabricated headways. In Speedcubing national, continental and world championships are held.

The first world championship organized by the Guinness Book of World Records was held in Munich on March 13, 1981. The dice were rolled 40 times and rubbed with Vaseline. Winner of the championship was a jury Fröschl from Munich with a record time of 38 seconds.

The current world record, set during the Zonhoven Open 2013 in Belgium by the Dutchman Mats Valk, is 5.55 seconds for a 3 × 3 × 3 cube.

Blindfold Cubing

Another well-known discipline called Blindfold Cubing. At first, one grinds into the twisted cube and then solves it blindfolded without seeing him one more time. The current world record in the Blindfold Cubing is 23.80 seconds, set up by Poland Marcin Zalewski at the Polish Championships 2013. Herein both the Einprägzeit and the solution time are included.

Multiple Blindfold Cubing

In addition, there is also the Multiple Blindfold Cubing, an increase of Blindfold cubing. Here so many dice you stamps itself as possible in a row to them afterwards, without seeing them again, blind to solve. Methods are usually used to solve here, change as few other stones per step. Similar to memory sport you memorize the sequence of steps using a Memoriersystems to put them in the corresponding finger movements later blindfolded. The world record is 41 to cubes dissolved in a time of 54:14 minutes from Poland Marcin Kowalczyk.

Mechanical solution

There are a number of machines that can solve the cube by means of image recognition and automated mechanics. Thus, the human record in 2011 was first undercut by a robotics: CubeStormer 2 replaced the cube in 5.27 seconds - the fastest at this time man needed 5.66 seconds. 2014 sparked CubeStormer 3 by a Galaxy S4 and eight Lego Mindstorms EV3 the cube in 3.25 seconds.

Create a pattern

In addition to the usual solving the Rubik's Cube is another popular variety, to create regular and irregular pattern with the Rubik's Cube.

With many patterns only cubes of opposite sides are reversed ( eg " Pepita basic pattern ", " quadruple cross pattern ", " Six- T- pattern " ) are in other patterns only the cube of three abutting sides (eg. "center pattern ", " Six- cross pattern ", " cube in the cube " (or " 2 × 2 " to " 3 × 3 × 3 "), " All-round worm " / " snake ").

In addition, there are mixed color patterns such as all twelve edges Cube simply tilted in its initial position or positioned a circumferential diagonal by two different colored " three corners ".

In principle, one can distinguish three approaches when creating patterns:

The phenomenon in some imaginary patterns is that can not be due to the construction of the cube actually realize all patterns. Often at the end is a corner cube on its position not in the correct position or they are two edges cubes in the wrong position (examples: six times circumferential diagonal, various Pepita variants at adjacent sides ). For other patterns you needed a different combination of color surfaces of the corner or edge cube or a cube edges would doubly needed.

Another variety in this context is just a few headways restore from a rotated cube pattern in the original starting position of the cube with the six areas of color.

Variants

There are some versions of this mechanical puzzle. More difficult is printed with pictures dice, since although the areas of color coming through the solution strategies generally known to be in the right place, however, the mean areas not always in the correct orientation. So there are simpler cube consisting of only two levels in each spatial direction as the Pocket Cube and more complicated variants 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 Rubik's Cube or delusions ) or two or more puts into each integrated cubes ( Rubik 's Fusion ) exist. Also, there was a 2 × 3 × 3 cube: Rubiks Magic Domino and a dodecahedron ( Megaminx ). Furthermore, there are Rubik puzzles in tonnes or pyramid shape and balls ( eg Master Ball), also in various difficulty levels.

2005, the first cube was presented with six levels. The mechanism also allows for dice with up to eleven layers. These must Barrel - the centers of the palms outward - are distorted, so that the attachment of the cornerstones completely still lies within the cube. This distortion along with the necessary size and weight will demand the player some skill in handling. The solution methods for these large cubes do not require features that are not already on comprehensive four or five levels cubes ago known.

Since June 2008, are also 6 × 6 × 6 - and 7 × 7 × 7 Rubik's Cube on the market. On 27 January 2011 a cube with 17 levels was presented, which was produced with a 3D printer and the unofficial world record "largest Puzzle" represents.

A because of its star-shaped form very popular mechanical puzzle is the 4D8 - Rubik's Cube. This is derived from a star tetrahedron, (Stella Octangula ) also called Kepler star. However, while his tips are cut off, it will remain so-called truncated pyramids ( Truncated Pyramids ).

In computer programs that simulate the Rubik's Cube, often can be adjusted even more levels.

When Rubik's Cube Calendar (date cube ) the areas with numbers and texts are provided, from which the current date, day, month and day can be assembled on the front surface.

As a result of the boom in the 1980s emerged and mechanical puzzles, which was another mechanism to reason, for example, Rubik's Magic, the Devil ton, Back to Square One, Rubik's triamide, Rubik 's Clock, Alexander 's Star and the Magic Tower. The mechanically demanding puzzle of this kind is probably the Dogic in the form of an icosahedron ( icosahedral ).