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106-531: The Rubik's Cube is a 3D 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 Pentangle Puzzles in the UK in 1978 , and then by Ideal Toy Corp in 1980 via businessman Tibor Laczi and Seven Towns founder Tom Kremer . The cube was released internationally in 1980 and became one of

212-414: A = 0 {\displaystyle a=0} . While not explicitly studied by Hamilton, this indirectly introduced notions of basis, here given by the quaternion elements i , j , k {\displaystyle i,j,k} , as well as the dot product and cross product , which correspond to (the negative of) the scalar part and the vector part of the product of two vector quaternions. It

318-479: A 2x2x2 "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 U.S. patent 3,655,201 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

424-487: A parallelogram , and hence are coplanar. A sphere in 3-space (also called a 2-sphere because it is a 2-dimensional object) consists of the set of all points in 3-space at a fixed distance r from a central point P . The solid enclosed by the sphere is called a ball (or, more precisely a 3-ball ). The volume of the ball is given by V = 4 3 π r 3 , {\displaystyle V={\frac {4}{3}}\pi r^{3},} and

530-611: 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 the 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 (T), Bottom (B), Left (L), Right (R), Front (F), and Posterior (P), with + for clockwise, – for anticlockwise, and 2 for 180-degree turns. Another notation appeared in

636-482: 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

742-522: A choice of basis, corresponding to a set of axes. But in rotational symmetry, there is no reason why one set of axes is preferred to say, the same set of axes which has been rotated arbitrarily. Stated another way, a preferred choice of axes breaks the rotational symmetry of physical space. Computationally, it is necessary to work with the more concrete description R 3 {\displaystyle \mathbb {R} ^{3}} in order to do concrete computations. A more abstract description still

848-463: A clear plastic cylinder but cardboard versions were also used. The cube itself had slightly different variations in the order of the colours (Western vs. Japanese colour scheme where blue/yellow are switched) and some of the cubes did not have a white piece logo. 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

954-736: A cube which is initially in solved state will eventually return the cube back to its solved state: the smallest number of iterations required is the period of the sequence. For example, the 180-degree turn of any side has period 2 (e.g. {U} ); the 90-degree turn of any side has period 4 (e.g. {R} ). The maximum period for a move sequence is 1260: for example, allowing for full rotations, {F x} or {R y} or {U z} ; not allowing for rotations, {D R' U M} , or {B E L' F} , or {S' U' B D} ; only allowing for clockwise quarter turns, {U R S U L} , or {F L E B L} , or {R U R D S} ; only allowing for lateral clockwise quarter turns, {F B L F B R F U} , or {U D R U D L U F} , or {R L D R L U R F} . Although there are

1060-528: A field , which is not commutative nor associative , but is a Lie algebra with the cross product being the Lie bracket. Specifically, the space together with the product, ( R 3 , × ) {\displaystyle (\mathbb {R} ^{3},\times )} is isomorphic to the Lie algebra of three-dimensional rotations, denoted s o ( 3 ) {\displaystyle {\mathfrak {so}}(3)} . In order to satisfy

1166-412: A given plane, intersect that plane in a unique point, or be parallel to the plane. In the last case, there will be lines in the plane that are parallel to the given line. A hyperplane is a subspace of one dimension less than the dimension of the full space. The hyperplanes of a three-dimensional space are the two-dimensional subspaces, that is, the planes. In terms of Cartesian coordinates, the points of

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1272-400: A hyperplane satisfy a single linear equation , so planes in this 3-space are described by linear equations. A line can be described by a pair of independent linear equations—each representing a plane having this line as a common intersection. Varignon's theorem states that the midpoints of any quadrilateral in R 3 {\displaystyle \mathbb {R} ^{3}} form

1378-459: 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

1484-419: A plane curve about a fixed line in its plane as an axis is called a surface of revolution . The plane curve is called the generatrix of the surface. A section of the surface, made by intersecting the surface with a plane that is perpendicular (orthogonal) to the axis, is a circle. Simple examples occur when the generatrix is a line. If the generatrix line intersects the axis line, the surface of revolution

1590-501: A prime symbol denotes a clockwise turn. These directions are as one is looking at the specified face. A letter followed by a 2 (occasionally a superscript ) denotes two turns, or a 180-degree turn. For example, R means to turn the right side clockwise, but R′ means to turn the 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

1696-447: A quarter turn. Thus orientations of centres increases the total number of possible Cube permutations from 43,252,003,274,489,856,000 (4.3×10) to 88,580,102,706,155,225,088,000 (8.9×10). 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

1802-399: A sequence of moves, referred to as "Singmaster notation" or simple "Cube 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 indicates an anticlockwise face turn; while a letter without

1908-553: 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. Three-dimensional space Technically, a tuple of n numbers can be understood as the Cartesian coordinates of a location in a n -dimensional Euclidean space. The set of these n -tuples is commonly denoted R n , {\displaystyle \mathbb {R} ^{n},} and can be identified to

2014-442: A subtle way. By definition, there exists a basis B = { e 1 , e 2 , e 3 } {\displaystyle {\mathcal {B}}=\{e_{1},e_{2},e_{3}\}} for V {\displaystyle V} . This corresponds to an isomorphism between V {\displaystyle V} and R 3 {\displaystyle \mathbb {R} ^{3}} :

2120-445: A unique plane, so skew lines are lines that do not meet and do not lie in a common plane. Two distinct planes can either meet in a common line or are parallel (i.e., do not meet). Three distinct planes, no pair of which are parallel, can either meet in a common line, meet in a unique common point, or have no point in common. In the last case, the three lines of intersection of each pair of planes are mutually parallel. A line can lie in

2226-477: A vector A is denoted by || A || . The dot product of a vector A = [ A 1 , A 2 , A 3 ] with itself is which gives the formula for the Euclidean length of the vector. Without reference to the components of the vectors, the dot product of two non-zero Euclidean vectors A and B is given by where θ is the angle between A and B . The cross product or vector product

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2332-627: Is a binary operation on two vectors in three-dimensional space and is denoted by the symbol ×. The cross product A × B of the vectors A and B is a vector that is perpendicular to both and therefore normal to the plane containing them. It has many applications in mathematics, physics , and engineering . In function language, the cross product is a function × : R 3 × R 3 → R 3 {\displaystyle \times :\mathbb {R} ^{3}\times \mathbb {R} ^{3}\rightarrow \mathbb {R} ^{3}} . The components of

2438-477: 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 ( F B U D L R ) refer to the outermost portions of the cube (called faces). Lowercase letters ( f b u d l r ) refer to

2544-758: Is a right circular cone with vertex (apex) the point of intersection. However, if the generatrix and axis are parallel, then the surface of revolution is a circular cylinder . In analogy with the conic sections , the set of points whose Cartesian coordinates satisfy the general equation of the second degree, namely, A x 2 + B y 2 + C z 2 + F x y + G y z + H x z + J x + K y + L z + M = 0 , {\displaystyle Ax^{2}+By^{2}+Cz^{2}+Fxy+Gyz+Hxz+Jx+Ky+Lz+M=0,} where A , B , C , F , G , H , J , K , L and M are real numbers and not all of A , B , C , F , G and H are zero,

2650-400: Is a simple process to "solve" a Cube by taking it apart and reassembling it in a solved state. There are six central pieces that show one coloured face, twelve edge pieces that show two coloured faces, and eight corner pieces that show three coloured faces. Each piece shows a unique colour combination, but not all combinations are present (for example, if red and orange are on opposite sides of

2756-577: Is an Ancient Greek legend associated with Alexander the Great in Gordium in Phrygia , regarding a complex knot that tied an oxcart. Reputedly, whoever could untie it would be destined to rule all of Asia. In 333 BC Alexander was challenged to untie the knot. Instead of untangling it laboriously as expected, he dramatically cut through it with his sword, thus exercising another form of mental genius. It

2862-466: Is called a quadric surface . There are six types of non-degenerate quadric surfaces: The degenerate quadric surfaces are the empty set, a single point, a single line, a single plane, a pair of planes or a quadratic cylinder (a surface consisting of a non-degenerate conic section in a plane π and all the lines of R through that conic that are normal to π ). Elliptic cones are sometimes considered to be degenerate quadric surfaces as well. Both

2968-406: Is found in linear algebra , where the idea of independence is crucial. Space has three dimensions because the length of a box is independent of its width or breadth. In the technical language of linear algebra, space is three-dimensional because every point in space can be described by a linear combination of three independent vectors . A vector can be pictured as an arrow. The vector's magnitude

3074-425: Is its length, and its direction is the direction the arrow points. A vector in R 3 {\displaystyle \mathbb {R} ^{3}} can be represented by an ordered triple of real numbers. These numbers are called the components of the vector. The dot product of two vectors A = [ A 1 , A 2 , A 3 ] and B = [ B 1 , B 2 , B 3 ] is defined as: The magnitude of

3180-488: Is primed, it is an indication that the cube must be rotated in the opposite direction. When x , y , or z is squared, the cube must be rotated 180 degrees. One of the most common deviations 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 (particularly corners-first methods), there

3286-513: 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

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3392-669: Is the Kronecker delta . Written out in full, the standard basis is E 1 = ( 1 0 0 ) , E 2 = ( 0 1 0 ) , E 3 = ( 0 0 1 ) . {\displaystyle E_{1}={\begin{pmatrix}1\\0\\0\end{pmatrix}},E_{2}={\begin{pmatrix}0\\1\\0\end{pmatrix}},E_{3}={\begin{pmatrix}0\\0\\1\end{pmatrix}}.} Therefore R 3 {\displaystyle \mathbb {R} ^{3}} can be viewed as

3498-534: Is the Levi-Civita symbol . It has the property that A × B = − B × A {\displaystyle \mathbf {A} \times \mathbf {B} =-\mathbf {B} \times \mathbf {A} } . Its magnitude is related to the angle θ {\displaystyle \theta } between A {\displaystyle \mathbf {A} } and B {\displaystyle \mathbf {B} } by

3604-468: Is thus used as a metaphor for a seemingly intractable problem which is solved by exercising brute force. Turn him to any cause of policy, The Gordian Knot of it he will unloose, Familiar as his garter The Phrygians were without a king , but an oracle at Telmissus (the ancient capital of Lycia ) decreed that the next man to enter the city driving an ox-cart should become their king. A peasant farmer named Gordias drove into town on an ox-cart and

3710-418: Is to model physical space as a three-dimensional affine space E ( 3 ) {\displaystyle E(3)} over the real numbers. This is unique up to affine isomorphism. It is sometimes referred to as three-dimensional Euclidean space. Just as the vector space description came from 'forgetting the preferred basis' of R 3 {\displaystyle \mathbb {R} ^{3}} ,

3816-565: 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

3922-746: 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

4028-523: 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. The repetition of any given move sequence on

4134-513: The 19th century, developments of the geometry of three-dimensional space came with William Rowan Hamilton 's development of the quaternions . In fact, it was Hamilton who coined the terms scalar and vector , and they were first defined within his geometric framework for quaternions . Three dimensional space could then be described by quaternions q = a + u i + v j + w k {\displaystyle q=a+ui+vj+wk} which had vanishing scalar component, that is,

4240-418: The Cube. Newer official Rubik's brand cubes have rivets instead of screws and cannot be adjusted. Inexpensive clones do not have screws or springs, all they have is a plastic clip to keep the centre piece in place and freely rotate. 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

4346-523: The Magic Cube worldwide. Ideal wanted at least a recognisable name to trademark; 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

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4452-498: The Rubik's Cube brand. Taking advantage of an initial shortage of cubes, many imitations and variations appeared, many of which may have violated one or more patents. In 2000 the patents expired, and since then, many Chinese companies have produced copies, modifications, and improvements upon the Rubik and V-Cube designs. Nichols assigned his patent to his employer Moleculon Research Corp., which sued Ideal in 1982. In 1984, Ideal lost

4558-593: The United States, Rubik was granted U.S. patent 4,378,116 on 29 March 1983 for the Cube. This patent expired in 2000. 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 were upheld by a ruling of the General Court of the European Union on 25 November 2014 in a successful defence against

4664-466: 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. The packaging had a few variations depending on the country, most popular being

4770-695: 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. On 27 October 2020, Spin Master said it will pay $ 50 million to buy

4876-399: The above-mentioned systems. Two distinct points always determine a (straight) line . Three distinct points are either collinear or determine a unique plane . On the other hand, four distinct points can either be collinear, coplanar , or determine the entire space. Two distinct lines can either intersect, be parallel or be skew . Two parallel lines, or two intersecting lines , lie in

4982-495: The abstract vector space, together with the additional structure of a choice of basis. Conversely, V {\displaystyle V} can be obtained by starting with R 3 {\displaystyle \mathbb {R} ^{3}} and 'forgetting' the Cartesian product structure, or equivalently the standard choice of basis. As opposed to a general vector space V {\displaystyle V} ,

5088-474: The affine space description comes from 'forgetting the origin' of the vector space. Euclidean spaces are sometimes called Euclidean affine spaces for distinguishing them from Euclidean vector spaces. This is physically appealing as it makes the translation invariance of physical space manifest. A preferred origin breaks the translational invariance. Gordian Knot The cutting of the Gordian Knot

5194-406: The application of mathematical group theory , which has been helpful for deducing certain algorithms – in particular, those which have a commutator structure, namely XYX Y (where X and Y are specific moves or move-sequences and X and Y are their respective inverses), or a conjugate structure, namely XYX , often referred to by speedcubers colloquially as a "setup move". In addition,

5300-405: The assembled puzzle. Each of the six centre pieces pivots on a fastener held by the centre piece, a "3D cross". A spring between each fastener and its corresponding piece tensions the piece inward, so that collectively, the whole assembly remains compact but can still be easily manipulated. The older versions of the official Cube used a screw that can be tightened or loosened to change the "feel" of

5406-775: The axioms of a Lie algebra, instead of associativity the cross product satisfies the Jacobi identity . For any three vectors A , B {\displaystyle \mathbf {A} ,\mathbf {B} } and C {\displaystyle \mathbf {C} } A × ( B × C ) + B × ( C × A ) + C × ( A × B ) = 0 {\displaystyle \mathbf {A} \times (\mathbf {B} \times \mathbf {C} )+\mathbf {B} \times (\mathbf {C} \times \mathbf {A} )+\mathbf {C} \times (\mathbf {A} \times \mathbf {B} )=0} One can in n dimensions take

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5512-399: 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 that fit into it to form

5618-415: The centre faces (although some carried the "Rubik's Cube" mark on the centre square of the white face), 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

5724-462: The centres as well. Marking Rubik's Cube's centres increases its difficulty, because this expands the set of distinguishable possible configurations. There are 4/2 (2,048) 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

5830-630: The construction for the isomorphism is found here . However, there is no 'preferred' or 'canonical basis' for V {\displaystyle V} . On the other hand, there is a preferred basis for R 3 {\displaystyle \mathbb {R} ^{3}} , which is due to its description as a Cartesian product of copies of R {\displaystyle \mathbb {R} } , that is, R 3 = R × R × R {\displaystyle \mathbb {R} ^{3}=\mathbb {R} \times \mathbb {R} \times \mathbb {R} } . This allows

5936-491: The construction of the five regular Platonic solids in a sphere. In the 17th century, three-dimensional space was described with Cartesian coordinates , with the advent of analytic geometry developed by René Descartes in his work La Géométrie and Pierre de Fermat in the manuscript Ad locos planos et solidos isagoge (Introduction to Plane and Solid Loci), which was unpublished during Fermat's lifetime. However, only Fermat's work dealt with three-dimensional space. In

6042-880: The cross product are A × B = [ A 2 B 3 − B 2 A 3 , A 3 B 1 − B 3 A 1 , A 1 B 2 − B 1 A 2 ] {\displaystyle \mathbf {A} \times \mathbf {B} =[A_{2}B_{3}-B_{2}A_{3},A_{3}B_{1}-B_{3}A_{1},A_{1}B_{2}-B_{1}A_{2}]} , and can also be written in components, using Einstein summation convention as ( A × B ) i = ε i j k A j B k {\displaystyle (\mathbf {A} \times \mathbf {B} )_{i}=\varepsilon _{ijk}A_{j}B_{k}} where ε i j k {\displaystyle \varepsilon _{ijk}}

6148-487: The cube (for example, swapping two corners) but may also have the side-effect of changing other parts of the cube (such as permuting some edges). Such algorithms are often simpler than the ones without side effects and are employed early on in the solution when most of the puzzle has not yet been solved and the side effects are not important. Towards the end of the solution, the more specific (and usually more complicated) algorithms are used instead. Rubik's Cube lends itself to

6254-495: The cube by further reducing it to another subgroup. The Rubik's group can be endowed with a unitary representation : such a description allows the Rubik's Cube to be mapped into a quantum system of few particles, where the rotations of its faces are implemented by unitary operators. The rotations of the faces act as generators of the Lie group . Many 3×3×3 Rubik's Cube enthusiasts use a notation developed by David Singmaster to denote

6360-496: The cube can be placed by dismantling and reassembling it. The preceding numbers assume the centre 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. The original Rubik's Cube had no orientation markings on

6466-417: 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

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6572-430: 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 (8.9×10) to 2,125,922,464,947,725,402,112,000 (2.1×10). In Rubik's cubers' parlance, a memorised sequence of moves that have a desired effect on the cube is called an "algorithm". This terminology is derived from the mathematical use of algorithm , meaning

6678-516: The definition of canonical projections, π i : R 3 → R {\displaystyle \pi _{i}:\mathbb {R} ^{3}\rightarrow \mathbb {R} } , where 1 ≤ i ≤ 3 {\displaystyle 1\leq i\leq 3} . For example, π 1 ( x 1 , x 2 , x 3 ) = x {\displaystyle \pi _{1}(x_{1},x_{2},x_{3})=x} . This then allows

6784-493: The definition of the standard basis B Standard = { E 1 , E 2 , E 3 } {\displaystyle {\mathcal {B}}_{\text{Standard}}=\{E_{1},E_{2},E_{3}\}} defined by π i ( E j ) = δ i j {\displaystyle \pi _{i}(E_{j})=\delta _{ij}} where δ i j {\displaystyle \delta _{ij}}

6890-425: 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 that have been scrambled using only 180-degree turns. These subgroups are the principle underlying the computer cubing methods by Thistlethwaite and Kociemba , which solve

6996-465: The flip of the twelfth depending on the preceding ones, giving 2 (2,048) 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

7102-573: The game of noughts and crosses . He received his UK patent (1344259) on 16 January 1974. 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

7208-504: The height of its mainstream popularity in the 1980s, it is still widely known and used. Many speedcubers continue to practice it and similar puzzles, and to compete for the fastest times in various categories. Since 2003, the World Cube Association (WCA), the international governing body of the Rubik's Cube, has organised competitions worldwide and has recognised world records. In March 1970, Larry D. Nichols invented

7314-421: The hyperboloid of one sheet and the hyperbolic paraboloid are ruled surfaces , meaning that they can be made up from a family of straight lines. In fact, each has two families of generating lines, the members of each family are disjoint and each member one family intersects, with just one exception, every member of the other family. Each family is called a regulus . Another way of viewing three-dimensional space

7420-460: The identity ‖ A × B ‖ = ‖ A ‖ ⋅ ‖ B ‖ ⋅ | sin ⁡ θ | . {\displaystyle \left\|\mathbf {A} \times \mathbf {B} \right\|=\left\|\mathbf {A} \right\|\cdot \left\|\mathbf {B} \right\|\cdot \left|\sin \theta \right|.} The space and product form an algebra over

7526-405: The ineffable name of Dionysus that, knotted like a cipher, would have been passed on through generations of priests and revealed only to the kings of Phrygia. Unlike popular fable , genuine mythology has few completely arbitrary elements. This myth taken as a whole seems designed to confer legitimacy to dynastic change in this central Anatolian kingdom: thus Alexander's "brutal cutting of

7632-474: The inner portions of the cube (called slices). An asterisk (L*), a number in front of it (2L), or two layers in parentheses (Ll), means to turn the two layers at the same time (both the inner and the outer left faces) For example: ( Rr )'  l 2  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

7738-522: The knot ... ended an ancient dispensation." The ox-cart suggests a longer voyage, rather than a local journey, perhaps linking Alexander the Great with an attested origin-myth in Macedon , of which Alexander is most likely to have been aware. Based on this origin myth, the new dynasty was not immemorially ancient, but had widely remembered origins in a local, but non-priestly "outsider" class, represented by Greek reports equally as an eponymous peasant or

7844-477: The knot was loosed. Sources from antiquity disagree on his solution. In one version of the story, he drew his sword and sliced it in half with a single stroke. However, Plutarch and Arrian relate that, according to Aristobulus , Alexander pulled the linchpin from the pole to which the yoke was fastened, exposing the two ends of the cord and allowing him to untie the knot without having to cut through it. Some classical scholars regard this as more plausible than

7950-462: The locally attested, authentically Phrygian in his ox-cart. Roller (1984) separates out authentic Phrygian elements in the Greek reports and finds a folk-tale element and a religious one, linking the dynastic founder (with the cults of "Zeus" and Cybele ). Other Greek myths legitimize dynasties by right of conquest (compare Cadmus ), but in this myth the stressed legitimising oracle suggests that

8056-533: The middle of the year which it supplemented with newspaper advertisements. 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

8162-428: The most compelling and useful way to model the world as it is experienced, it is only one example of a 3-manifold. In this classical example, when the three values refer to measurements in different directions ( coordinates ), any three directions can be chosen, provided that these directions do not lie in the same plane . Furthermore, if these directions are pairwise perpendicular , the three values are often labeled by

8268-479: The most recognized icons in popular culture. It won the 1980 German Game of the Year special award for Best Puzzle. As of January 2024, around 500 million cubes had been sold worldwide, making it the world's bestselling puzzle game and bestselling toy. The Rubik's Cube was inducted into the US National Toy Hall of Fame in 2014. On the original, classic Rubik's Cube, each of the six faces

8374-583: 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 (top), so that all are consonants. The key difference is the use of the vowels O, A, and I for cl o ckwise, a nticlockwise, and tw i ce (180-degree) turns, which results in word-like sequences such as LOTA RATO LATA ROTI (equivalent to LU′ R′ U L′ U′ R U2 in Singmaster notation). The addition of

8480-483: The orientation of the eighth (final) corner depends on the preceding seven, giving 3 (2,187) possibilities. There are 12!/2 (239,500,800) ways to arrange the edges, restricted from 12! because edges must be in an even permutation exactly when the corners are. (When arrangements of centres are also permitted, as described below, the rule is that the combined arrangement of corners, edges, and centres must be an even permutation.) Eleven edges can be flipped independently, with

8586-453: The pair formed by a n -dimensional Euclidean space and a Cartesian coordinate system . When n = 3 , this space is called the three-dimensional Euclidean space (or simply "Euclidean space" when the context is clear). In classical physics , it serves as a model of the physical universe , in which all known matter exists. When relativity theory is considered, it can be considered a local subspace of space-time . While this space remains

8692-513: The palace of the former kings of Phrygia at Gordium in the fourth century BC when Alexander the Great arrived, at which point Phrygia had been reduced to a satrapy , or province, of the Persian Empire . An oracle had declared that any man who could unravel its elaborate knots was destined to become ruler of all of Asia. Alexander the Great wanted to untie the knot but struggled to do so before reasoning that it would make no difference how

8798-525: 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 "Magic Cube" ( Hungarian : bűvös kocka ) 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 toy shops in Budapest . Magic Cube

8904-403: 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

9010-580: The popular account. Literary sources of the story include Arrian ( Anabasis Alexandri 2.3), Quintus Curtius (3.1.14), Justin 's epitome of Pompeius Trogus (11.7.3), and Aelian 's De Natura Animalium 13.1. Alexander the Great later went on to conquer Asia as far as the Indus and the Oxus , thus fulfilling the prophecy. The knot may have been a religious knot-cipher guarded by priests and priestesses. Robert Graves suggested that it may have symbolised

9116-555: The position of any point in three-dimensional space is given by an ordered triple of real numbers , each number giving the distance of that point from the origin measured along the given axis, which is equal to the distance of that point from the plane determined by the other two axes. Other popular methods of describing the location of a point in three-dimensional space include cylindrical coordinates and spherical coordinates , though there are an infinite number of possible methods. For more, see Euclidean space . Below are images of

9222-406: 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 having only one colour. The Cube has inspired other designers to create a number of similar puzzles with various numbers of sides, dimensions, and mechanisms. Although the Rubik's Cube reached

9328-472: The product of n − 1 vectors to produce a vector perpendicular to all of them. But if the product is limited to non-trivial binary products with vector results, it exists only in three and seven dimensions . It can be useful to describe three-dimensional space as a three-dimensional vector space V {\displaystyle V} over the real numbers. This differs from R 3 {\displaystyle \mathbb {R} ^{3}} in

9434-407: The puzzle's shape was not sufficient to grant it trademark protection. A standard Rubik's Cube measures 5.6 centimetres ( 2 + 1 ⁄ 4  in) on each side. The puzzle consists of 26 unique miniature cubes, also known as "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,

9540-533: 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-twelfth 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

9646-469: 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 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

9752-530: The space R 3 {\displaystyle \mathbb {R} ^{3}} is sometimes referred to as a coordinate space. Physically, it is conceptually desirable to use the abstract formalism in order to assume as little structure as possible if it is not given by the parameters of a particular problem. For example, in a problem with rotational symmetry, working with the more concrete description of three-dimensional space R 3 {\displaystyle \mathbb {R} ^{3}} assumes

9858-418: The surface area of the sphere is A = 4 π r 2 . {\displaystyle A=4\pi r^{2}.} Another type of sphere arises from a 4-ball, whose three-dimensional surface is the 3-sphere : points equidistant to the origin of the euclidean space R . If a point has coordinates, P ( x , y , z , w ) , then x + y + z + w = 1 characterizes those points on

9964-404: The terms width /breadth , height /depth , and length . Books XI to XIII of Euclid's Elements dealt with three-dimensional geometry. Book XI develops notions of orthogonality and parallelism of lines and planes, and defines solids including parallelpipeds, pyramids, prisms, spheres, octahedra, icosahedra and dodecahedra. Book XII develops notions of similarity of solids. Book XIII describes

10070-446: The unit 3-sphere centered at the origin. This 3-sphere is an example of a 3-manifold: a space which is 'looks locally' like 3-D space. In precise topological terms, each point of the 3-sphere has a neighborhood which is homeomorphic to an open subset of 3-D space. In three dimensions, there are nine regular polytopes: the five convex Platonic solids and the four nonconvex Kepler-Poinsot polyhedra . A surface generated by revolving

10176-491: The work of Hermann Grassmann and Giuseppe Peano , the latter of whom first gave the modern definition of vector spaces as an algebraic structure. In mathematics, analytic geometry (also called Cartesian geometry) describes every point in three-dimensional space by means of three coordinates. Three coordinate axes are given, each perpendicular to the other two at the origin , the point at which they cross. They are usually labeled x , y , and z . Relative to these axes,

10282-479: 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" (1980) and Patrick Bossert's You Can Do the Cube (1981). 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,

10388-472: 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. The puzzle

10494-429: Was covered by nine stickers, with each face in 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. Since 1988, the arrangement of colours has been standardised, with white opposite yellow, blue opposite green, and orange opposite red, and with the red, white, and blue arranged clockwise, in that order. On early cubes,

10600-488: Was granted in 1976 (Japanese patent publication JP55-008192). Until 1999, when an amended Japanese patent law was enforced, Japan's patent office granted Japanese patents for non-disclosed technology within Japan without requiring worldwide novelty . 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

10706-583: Was held together with interlocking plastic pieces that prevented the puzzle from 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

10812-558: Was immediately declared king. Out of gratitude, his son Midas dedicated the ox-cart to the Phrygian god Sabazios (whom the Greeks identified with Zeus ) and tied it to a post with an intricate knot of cornel bark ( Cornus mas ). The knot was later described by Roman historian Quintus Curtius Rufus as comprising "several knots all so tightly entangled that it was impossible to see how they were fastened". The ox-cart still stood in

10918-473: Was not until Josiah Willard Gibbs that these two products were identified in their own right, and the modern notation for the dot and cross product were introduced in his classroom teaching notes, found also in the 1901 textbook Vector Analysis written by Edwin Bidwell Wilson based on Gibbs' lectures. Also during the 19th century came developments in the abstract formalism of vector spaces, with

11024-456: 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

11130-434: Was originally advertised as having "over 3,000,000,000 (three billion ) combinations but only one solution". Depending on how combinations are counted, the actual number is significantly higher. The original (3×3×3) Rubik's Cube has eight corners and twelve edges. There are 8! (40,320) ways to arrange the corner cubes. Each corner has three possible orientations, although only seven (of eight) can be oriented independently;

11236-456: 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 countries such as China and the USSR, the craze had started later and demand was still high because of a shortage of Cubes. Rubik's Cubes continued to be marketed and sold throughout the 1980s and 1990s, but it

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