Rubiks Cube is a mechanical puzzle invented in by Hungarian sculptor and professor of architecture Erno Rubik. Originally called the Magic Cube by its inventor, this puzzle was renamed Rubiks Cube by Ideal Toys in and also won the German Game of the Year Spiel des Jahres special award for Best Puzzle. It is said to be the worlds bestselling toy, with over ,, Rubiks Cubes and imitations sold worldwide.In a typical Cube, each face is covered by nine stickers of one of six solid colours. When the puzzle is solved, each face of the Cube is a solid colour. The Cube celebrated its twentyfifth anniversary in , when a special edition Cube in a presentation box was released, featuring a sticker in the centre of the reflective face which replaced the white face with a Rubiks Cube logo.The puzzle comes in four widely available versions the Pocket Cube, also Mini Cube, Junior Cube, or Ice Cube, the standard cube, the Rubiks Revenge, and the Professors Cube. Larger sizes of the cubes, xx and xx, are planned for release in .In March , Larry Nichols invented a Puzzle with Pieces Rotatable in Groups and filed a Canadian patent application for it. Nicholss cube was held together with magnets. Nichols was granted U.S. Patent ,, on April , , two years before Rubik invented his improved cube.On April , , Frank Fox applied to patent his Spherical . He received his UK patent on January , .Rubik invented his Magic Cube in and obtained Hungarian patent HU for the Magic Cube in but did not take out international patents. The first test batches of the product were produced in late and released to Budapest toy shops. Magic Cube was held together with interlocking plastic pieces that were less expensive to produce than the magnets in Nicholss design. In September , 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 .
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 Rubiks Cube, and the first batch was exported from Hungary in May . Taking advantage of an initial shortage of Cubes, many cheap imitations appeared.Nichols assigned his patent to his employer Moleculon Research Corp., which sued Ideal Toy Company in . In , Ideal lost the patent infringement suit and appealed. In , the appeals court affirmed the judgment that Rubiks Pocket Cube infringed Nicholss patent, but overturned the judgment on Rubiks Cube.Even while Rubiks patent application was being processed, Terutoshi Ishigi, a selftaught engineer and ironworks owner near Tokyo, filed for a Japanese patent for a nearly identical mechanism and was granted patent JP? Ishigis is generally accepted as an independent nvention.Rubik applied for another Hungarian patent on October , , and applied for other patents. In the United States, Rubik was granted U.S. Patent ,, on March , , for the Cube.Recently, Greek inventor Panagiotis Verdes patented a method of creating cubes beyond the , up to . His designs, which include improved mechanisms for the , , and , are suitable for speedcubing, whereas existing designs for cubes larger than are prone to break. As of June , , these designs were not yet widely available, although videos of actual, working prototypes for the and have been released, and it was recently announced that these cubes would be released sometime in .
A standard cube measures approximately ¼ inches . cm on each side. The puzzle consists of the twentysix unique miniature cubes on the surface. However, the centre cube of each face is merely a single square facade all are affixed to the core mechanisms. These provide structure for the other pieces to fit into and rotate around. So there are twentyone pieces a single core piece consisting of three intersecting axes holding the six centre squares in place but letting them rotate, and twenty smaller plastic pieces which fit into it to form the assembled puzzle. The Cube can be taken apart without much difficulty, typically by turning one side through a ° angle and prying an edge cube away from a centre cube until it dislodges. However, as prying loose a corner cube is a good way to break off a centre cube — thus ruining the Cube — it is far safer to lever a centre cube out using a screwdriver. 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 °, ° or °, but the location of the coloured sides relative to one another in the completed state of the puzzle cannot be altered it is fixed by the relative positions of the centre squares and the distribution of colour combinations on edge and corner pieces.For most recent Cubes, the colours of the stickers are red opposite orange, yellow opposite white, and green opposite blue. However, Cubes with alternative colour arrangements also exist for example, they might have the yellow face opposite the green, and the blue face opposite the white with red and orange opposite faces remaining unchanged.
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 touchsensitive 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 backup in case the timer doesnt work properly. These judges also ensure that the competitors are following competition regulations.The Petrus method works by first solving a xx block of the cube. This block is then extended to a solved xx block. All edges are then oriented, and then the remaining two sides of the cube are then solved using only a few algorithms. Lars Petrus developed this method to address what he felt were inherent inefficiencies in layerbylayer approaches, which he explains in his methods 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 uses significantly fewer turns than a layers approach, and is often used as the basis for fewest moves competition solutions.The first step of the Roux method is the formation of a xx block. The xx block is usually placed in the lower portion of the left layer. The second step is to create another xx 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 second inspection time, since one can plan the solution of pieces rather than for the Fridrich and Petrus method. It also isnt as dependent on algorithm memorizing 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 doesnt 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.
Speedcubing also known as speedsolving, speed cubing or speedcubing is the activity of solving a Rubiks 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 cubes six faces is one single, solid color.Regular cubes are sold commercially in variations of xx, xx, xx, and xx. Variations of the puzzle have been designed with as many as layers, but the largest denomination cube that has been physically produced is a xx. The current world record for a single solve of the xx stands at . seconds, set by Yu Nakajima at the Kashiwa Open competition Japan on May , .Speedcubing is the most prominent activity of the international Rubiks 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.The Rubiks Cube was invented in by Hungarian professor of architecture Erno Rubik. A widespread international interest in the cube began in , which soon developed into a global craze. On June , , the first world championship was held in Budapest. The height of the craze began to fade away after , 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 online community, and raised the profile of the art. Twenty years after the first World Championship, the Dutch Open competition was the first in a new wave of organized speedcubing events, which include regular national and international competitions. There have been three more World Championships since Budapests competition, the first held in Toronto in , the second in Lake Buena Vista, Florida in , and after years the tournament returned to Budapest in .
The standard Rubiks cube can be solved using a number of methods, not all of which are suited for speedcubing. One of the mostused speedcubing methods is the Fridrich method, named after its inventor, Jessica Fridrich, who finished nd in the Rubiks Cube World Championships. Another popular method is the Petrus system also commonly known as the Lars Method, named after its inventor, Lars Petrus, a method that is considered by some to be more intuitive than the structured Fridrich method. Other significant though less widelyused methods are various cornersfirst methods, simpler layerbylayer approaches, and the Roux method.The Fridrich method first works to solve a crossshaped 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 corneredge 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 widelyused 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 second inspection time, the entire solve of the cube consists of executing predefined algorithms based on the state of the cube.
In fact, there are ! ! = ,,,,,, about . or quintillion on the short scale possible arrangements of the pieces that make up the Cube, but only one in twelve of these are actually reachable. This is because there is no sequence of moves that will swap a single pair or rotate a single corner or edge cube. Thus there are twelve possible sets of reachable configurations, sometimes called universes or orbits, into which the Cube can be placed by dismantling and reassembling it.Despite the vast number of positions, all Cubes can be solved in twentyfive or fewer moves see Optimal solutions for Rubiks Cube. The large number of permutations is often given as a measure of the Rubiks cubes complexity. However, the puzzles difficulty does not necessarily follow from the large number of permutations. The problem of putting a jumbled set of encyclopedias volumes in alphabetical order has a larger complexity ! = . , but is less difficult.Names of numbers larger than a quadrillion are almost never used, for reasons discussed further below. It is debatable which of them should be considered real working English vocabulary and which are merely trivia, curiosities, or coinages. The following table lists those names of numbers which are found in many English dictionaries and thus have a special claim to being real words. The Traditional British values shown are unused in American English and are largely obsolete in British English, but are dominant in many nonEnglishspeaking areas, including continental Europe and Spanishspeaking countries in Latin America see Long and short scales.In mathematics, a symmetry group describes all symmetries of objects. This is formalized by the notion of a group action every element of the group acts like a bijective map or symmetry on some set. In this case, the group is also called a permutation group especially if the set is finite or not a vector space or transformation group especially if the set is a vector space and the group acts like linear transformations of the set. A permutation representation of a group G is a representation of G as a group of permutations of the set usually if the set is finite, and may be described as a group representation of G by permutation matrices, and is usually considered in the finitedimensional case—it is the same as a group action of G on an ordered basis of a vector space.
Note that the difference between the two scales grows as numbers get larger. Million is the same in both scales, but the longscale billion is a thousand times larger than the shortscale billion, the longscale trillion is a million times larger than the shortscale trillion, and so on.For most of the th and th centuries, the United Kingdom uniformly used the long scale, while the United States of America used the short scale, so that usage of the two systems was often referred to as British and American respectively. In the government of the UK abandoned the long scale, so that the UK now applies the short scale interpretation exclusively in mass media and official usage. Although some residual longscale usage still continues, the terms British and American no longer represent accurate terminology.The existence of the different scales means that care must be taken when comparing large numbers between languages or countries, or when using old documents in countries where the dominant scale has changed over time. For example, British English documents from used long scale values, which are different from current British short scale usage. Both scales were used in France and Italy at various times in their history, but these countries and most other European countries now officially use long scale. For example, the French word billion, the German word Billion and the Dutch word biljoen all refer to . This translates to the shortscale term trillion , not billion in the short scale. See Current usage below.A lightyear or light year symbol ly is a unit of length, equal to just under ten trillion kilometres. As defined by the International Astronomical Union which is the body which has the jurisdictional authority to promulgate the definition, a lightyear is the distance that light travels in a vacuum in one Julian year.
The original official Rubiks Cube has no orientation markings on the center faces, although some carried the words Rubiks Cube on the centre square of the white face, and therefore solving it does not require any attention to orienting those faces correctly. However, if one has a marker pen, one could, for example, mark the central squares of an unshuffled Cube with four coloured marks on each edge, each corresponding to the colour of the adjacent face. Some Cubes have also been produced commercially with markings on all of the squares, such as the Lo Shu magic square or playing card suits. Thus one can scramble and then unscramble the Cube yet have the markings on the centers rotated, and it becomes an additional test to solve the centers as well. This is known as supercubingcitation needed.Putting markings on the Rubiks Cube increases the difficulty mainly because it expands the set of distinguishable possible configurations. When the Cube is unscrambled apart from the orientations of the central squares, there will always be an even number of squares requiring a quarter turn. Thus there are = , possible configurations of the centre squares in the otherwise unscrambled position, increasing the total number of possible Cube permutations from ,,,,,, . to ,,,,,,, ..In recreational mathematics, a magic square of order n is an arrangement of n² numbers, usually distinct integers, in a square, such that the n numbers in all rows, all columns, and both diagonals sum to the same constant. A normal magic square contains the integers from to n². The term magic square is also sometimes used to refer to any of various types of word square.Chinese literature dating from as early as BC tells the legend of Lo Shu or scroll of the river Lo. In ancient China, there was a huge flood. The people tried to offer some sacrifice to the river god of one of the flooding rivers, the Lo river, to calm his anger. Then, there emerged from the water a turtle with a curious figurepattern on its shell there were circular dots of numbers that were arranged in a three by three ninegrid pattern such that the sum of the numbers in each row, column and diagonal was the same . This number is also equal to the number of days in each of the cycles of the Chinese solar year. This pattern, in a certain way, was used by the people in controlling the river.
The Lo Shu Square, as the magic square on the turtle shell is called, is the unique normal magic square of order three in which is at the bottom and is in the upper right corner. Every normal magic square of order three is obtained from the Lo Shu by rotation or reflection.The Square of Lo Shu is also referred to as the Magic Square of Saturn or Cronos. Its numerical value is obtained from the workings of the I Ching when the Trigrams are placed in an order given in the first river map, the Ho Tu or Yellow River. The Ho Tu produces squares of Hexagrams x in its outer values of to , to , to , and to , and these outer squares can then be symmetrically added together to give an inner central square of to . The central values of the Ho Tu are those of the Lo Shu so they work together, since in the total value of x light and dark is found the number of years in the cycle of equinoctial precession , x = ,. The Ho Tu produces a total of light and dark numbers called the days and nights the alternations of light and dark, and a total of x x Hexagrams whose opposite symmetrical addition equals , therefore each value of a square is called a season as it equals . is the number of hours in a day year, and years equals an aeon aeons = , yrs.To validate the values contained in the river maps Ho Tu and Lo Shu the I Ching provides numbers of Heaven and Earth that are the Original Trigrams father and mother from to . Heaven or a Trigram with all unbroken lines light lines yang have odd numbers ,,,,, and Earth a Trigram with all broken lines have even numbers ,,,,. If each of the Trigrams lines is given a value by multiplying the numbers of Heaven and Earth, then the value of each line in Heaven would be + + = , and its partner in the Ho Tu of Earth would be + + = , these Original Trigrams thereby produce more Trigrams or children in all their combinations and when the sequences of Trigrams are placed at right angles to each other they produce an x square of Hexagrams or cubes that each have lines of values. From this simple point the complex structure of the maths evolves as a hexadecimal progression, and it is the hexagon that is the link to the turtle or tortoise shell. In Chinese texts of the I Ching the moon is symbolic of water darkness whose transformations or changes create the light or fire the dark value creates the light when its number is increased by . This same principle can be found in ancient calendars such as the Egyptian, as the day year of hrs was divided by to produce the extra days or hours on which the gods were born. It takes years for the heavens to move degree through its Precession.
Many general solutions for the Rubiks Cube have been discovered independently. The most popular method was developed by David Singmaster and published in the book Notes on Rubiks Magic Cube in . This solution involves solving the Cube layer by layer, in which one layer, designated the top, is solved first, followed by the middle layer, and then the final and bottom layer. After practice, solving the Cube layer by layer can be done in under one minute. Other general solutions include corners first methods or combinations of several other methods. Most tutorials teach the layer by layer method, as it gives an easytounderstand stepbystep guide on how to solve it.Speedcubing solutions have been developed for solving the Rubiks Cube as quickly as possible. The most common speedcubing solution was developed by Jessica Fridrich. It is a very efficient layerbylayer method that requires a large number of algorithms, especially for orienting and permuting the last layer. The firstlayer corners and second layer are done simultaneously, with each corner paired up with a secondlayer edge piece. Another wellknown method was developed by Lars Petrus. In this method, a section is solved first, followed by a , and then the incorrect edges are solved using a threemove algorithm, which eliminates the need for a possible move algorithm later. One of the advantages of this method is that it tends to give solutions in fewer moves. For this reason, the method is also popular for fewest move competitions.Solutions follow a series of steps and include a set of algorithms for solving each step. An algorithm, also known as a process or an operator, is a series of twists that accomplishes a particular goal. For instance, one algorithm might switch the locations of three corner pieces, while leaving the rest of the pieces in place. Basic solutions require learning as few as four or five algorithms but are generally inefficient, needing around twists on average to solve an entire Cube. In comparison, Fridrichs advanced solution requires learning roughly algorithms but allows the Cube to be solved in only moves on average. A different kind of solution developed by Ryan Heise uses no algorithms but rather teaches a set of underlying principles that can be used to solve in fewer than moves. A number of complete solutions can also be found in any of the books listed in the bibliography, and most can be used to solve any Cube in under five minutes.
In mathematics, computing, linguistics and related disciplines, an algorithm is a sequence of instructions, often used for calculation, data processing. It is formally a type of effective method in which a list of welldefined instructions for completing a task will, when given an initial state, proceed through a welldefined series of successive states, eventually terminating in an endstate. The transition from one state to the next is not necessarily deterministic some algorithms, known as probabilistic algorithms, incorporate randomness. partial formalization of the concept began with attempts to solve the Entscheidungsproblem the decision problem posed by David Hilbert in . Subsequent formalizations were framed as attempts to define effective calculability Kleene or effective method Rosser those formalizations included the GödelHerbrandKleene recursive functions of , and , Alonzo Churchs lambda calculus of , Emil Posts Formulation I of , and Alan Turings Turing machines of and .AlKhwarizmi, Persian astronomer and mathematician, wrote a treatise in Arabic in AD, On Calculation with Hindu Numerals. See algorism. It was translated into Latin in the th century as Algoritmi de numero Indorum alDaffa , which title was likely intended to mean Book by Algoritmus on the numbers of the Indians, where Algoritmi was the translators rendition of the authors name in the genitive case but people misunderstanding the title treated Algoritmi as a Latin plural and this led to the word algorithm Latin algorismus coming to mean calculation method. The intrusive th is most likely due to a false cognate with the Greek a???µ?? arithmos meaning number.The words enumerably infinite mean countable using integers perhaps extending to infinity. Thus Boolos and Jeffrey are saying that an algorithm implies instructions for a process that creates output integers from an arbitrary input integer or integers that, in theory, can be chosen from to infinity. Thus we might expect an algorithm to be an algebraic equation such as y = m + n — two arbitrary input variables m and n that produce an output y. As we see in Algorithm characterizations — the word algorithm implies much more than this, something on the order of for our addition exampleThe concept of algorithm is also used to define the notion of decidability. That notion is central for explaining how formal systems come into being starting from a small set of axioms and rules. In logic, the time that an algorithm requires to complete cannot be measured, as it is not apparently related with our customary physical dimension. From such uncertainties, that characterize ongoing work, stems the unavailability of a definition of algorithm that suits both concrete in some sense and abstract usage of the term.
There are many algorithms to solve scrambled Rubiks Cubes. One such method is described in Wikibooks article How to solve the Rubiks Cube. 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.Note Notation from How to solve the Rubiks Cube is used in this article.It is not known how many moves is the minimum required to solve any instance of the Rubiks cube. This number is also known as the diameter of the Cayley graph of the Rubiks Cube group. An algorithm that solves a cube in the minimum number of moves is known as Gods 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 F a half turn of the front face would be counted as moves in the quarter turn metric and as only turn in the face metric.In mathematics, the Cayley graph, also known as the Cayley colour graph, is the graph that encodes the structure of a discrete group. Its definition is suggested by Cayleys theorem named after Arthur Cayley and uses a particular, usually finite, set of generators for the group. It is a central tool in combinatorial and geometric group theory.Gods algorithm is a notion originating in discussions of ways to solve the Rubiks Cube puzzle, but which can also be applied to other combinatorial puzzles and mathematical games. It stands for any practical algorithm that produces a solution having the least possible number of moves, the idea being that an omniscient being would know an optimal step from any given configuration.
The notion applies to puzzles that can assume a finite number of configurations, with a relatively small, welldefined arsenal of moves that may be applicable to configurations and then lead to a new configuration. Solving the puzzle means to reach a specific designated final configuration or one of a collection of final configurations by applying a sequence of moves, starting from some arbitrary initial configuration.Some wellknown puzzles fitting this description are mechanical puzzles like Rubiks Cube, Towers of Hanoi, and the puzzle. The oneperson game of peg solitaire is also covered, as well as many logic puzzles, such as the missionaries and cannibals problem. These have in common that they can be modelled mathematically as a directed graph, in which the configurations are the vertices, and the moves the arcs.An algorithm can be considered to solve such a puzzle if it takes as input an arbitrary initial configuration and produces as output a sequence of moves leading to a final configuration, if the puzzle is solvable from that initial position, otherwise signals the impossibility of a solution. A solution is optimal if the sequence of moves is as short as possible. Gods algorithm, then, for a given puzzle, is an algorithm that solves the puzzle and produces only optimal solutions.For an algorithm to be properly referred to as Gods algorithm, it should also be practical, meaning that the algorithm does not require extraordinary amounts of memory or time. For example, using a giant lookup table indexed by initial configurations would allow solutions to be found very quickly, but would require an extraordinary amount of memory.Instead of asking for a full solution, one can equivalently ask for a single move from an initial but not final configuration, where the move is the first of some optimal solution. An algorithm for the singlemove version of the problem can be turned into an algorithm for the original problem by invoking it repeatedly while applying each move reported to the present configuration, until a final one is reached. Conversely, any algorithm for the original problem can be turned into an algorithm for the singlemove version by truncating its output to its first move.