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Rubik's Cube
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Rubik's Cube is a mechanical puzzle invented in 1974 by Hungarian sculptor and professor of architecture Erno Rubik.
Originally called the "Magic Cube" by its inventor, this puzzle was renamed "Rubik's Cube" by Ideal Toys in 1980 and also won the 1980 German Game of the Year special award for Best Puzzle. It is said to be the world's best-selling toy, with over 300,000,000 Rubik's Cubes and imitations sold worldwide.
In a typical Cube, each face is covered by nine stickers of one of six solid colours. A pivot mechanism enables each face to turn, thus mixing up the colours. When the puzzle is solved, each face of the Cube is a solid colour. The Cube celebrated its twenty-fifth anniversary in 2005, 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 "Rubik's Cube 1980-2005" logo.
The puzzle comes in four widely available versions: the 2×2×2, the 3×3×3 standard cube, the 4×4×4, and the 5×5×5.
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Rubik's Cube is a mechanical puzzle invented in 1974 by Hungarian sculptor and professor of architecture Erno Rubik.
Originally called the "Magic Cube" by its inventor, this puzzle was renamed "Rubik's Cube" by Ideal Toys in 1980 and also won the 1980 German Game of the Year special award for Best Puzzle. It is said to be the world's best-selling toy, with over 300,000,000 Rubik's Cubes and imitations sold worldwide.
In a typical Cube, each face is covered by nine stickers of one of six solid colours. A pivot mechanism enables each face to turn, thus mixing up the colours. When the puzzle is solved, each face of the Cube is a solid colour. The Cube celebrated its twenty-fifth anniversary in 2005, 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 "Rubik's Cube 1980-2005" logo.
The puzzle comes in four widely available versions: the 2×2×2, the 3×3×3 standard cube, the 4×4×4, and the 5×5×5. Larger sizes of the cubes, 6×6×6 and 7×7×7, are also available.
Conception and development In March 1970, Larry Nichols invented a 2×2×2 "Puzzle with Pieces Rotatable in Groups" and filed a Canadian patent application for it. Nichols's cube was held together with magnets. Nichols was granted
On April 9, 1970, Frank Fox applied to patent his "Spherical 3×3×3". He received his UK patent (1344259) on January 16, 1974.
Rubik invented his "" in 1974 and obtained Hungarian patent HU170062 for the Magic Cube in 1975 but did not take out international patents. The first test batches of the product were produced in late 1977 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 Nichols's design. In September 1979, 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 1980.
After its international debut, the progress of the Cube towards the toy shop shelves of the West was briefly halted so that it could be manufactured to Western safety and packaging specifications. A lighter Cube was produced, and Ideal Toys 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. 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 1982. In 1984, Ideal lost the patent infringement suit and appealed. In 1986, the appeals court affirmed the judgment that Rubik's 2×2×2 Pocket Cube infringed Nichols's patent, but overturned the judgment on Rubik's 3×3×3 Cube.
Even while Rubik's patent application was being processed, Terutoshi Ishigi, a self-taught engineer and ironworks owner near Tokyo, filed for a Japanese patent for a nearly identical mechanism and was granted patent JP55?8192 (1976); Ishigi's is generally accepted as an independent reinvention.
Rubik applied for another Hungarian patent on October 28, 1980, and applied for other patents. In the United States, Rubik was granted
Greek inventor Panagiotis Verdes patented a method of creating cubes beyond the 5×5×5, up to 11×11×11. His designs, which include improved mechanisms for the 3×3×3, 4×4×4, and 5×5×5, are suitable for speedcubing, whereas existing designs for cubes larger than 5×5×5 are prone to break. As of June 19, 2008, 5x5x5, 6x6x6, and 7x7x7 models are available.
Workings
A standard cube measures approximately 2¼ inches (5.7 cm) on each side.
The puzzle consists of the twenty-six unique miniature cubes on the surface. However, the centre cube of each face is merely a single square façade; all are affixed to the core mechanisms. These provide structure for the other pieces to fit into and rotate around. So there are twenty-one 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 45° 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 very 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 90°, 180° or 270°, 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).
Douglas R. Hofstader, in the July 1982 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 been produced commercially.
PermutationsA normal (3×3×3) Rubik's Cube has eight corners and twelve edges. There are 8! ways to arrange the corner cubies. Seven can be oriented independently, and the orientation of the eighth depends on the preceding seven, giving 37 possibilities. There are 12!/2 ways to arrange the edges, since an odd permutation of the corners implies an odd permutation of the edges as well. Eleven edges can be flipped independently, with the flip of the twelfth depending on the preceding ones, giving 211 possibilities.
There are exactly 43,252,003,274,489,856,000 possibilities. In other words, there are forty-three quintillion or forty-three trillion. The puzzle is often advertised as having only "billions" of positions, as the larger numbers could be regarded as incomprehensible to many. To put this into perspective, if every permutation of a 57-millimeter Rubik's Cube were lined up end to end, it would stretch out approximately 261 light years.
The preceding figure is limited to permutations that can be reached solely by turning the sides of the cube. If one considers permuations reached through disassembly of the cube, the number becomes twelve times larger.
The full number is 519,024,039,293,878,272,000 or 519 quintillion (on the short scale) possible arrangements of the pieces that make up the Cube, but only one in twelve of these are actually solveable. 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 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 twenty-five or fewer moves (see Optimal solutions for Rubik's Cube).
The large number of permutations is often given as a measure of the Rubik's cube's complexity. However, the puzzle's difficulty does not necessarily follow from the large number of permutations. The problem of putting the 26 letters of the alphabet in alphabetical order has a larger complexity (26! = 4.03 × 1026 possible orderings), but is less difficult.
Centre facesThe original (official) Rubik's Cube has no orientation markings on the centre faces, although some carried the words "Rubik's 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 "supercubing".
Putting markings on the Rubik's 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 46/2 = 2,048 possible configurations of the centre squares in the otherwise unscrambled position, increasing the total number of possible Cube permutations from 43,252,003,274,489,856,000 (4.3×1019) to 88,580,102,706,155,225,088,000 (8.9×1022).
Algorithms In Rubik cubists parlance, an "algorithm" means "a memorized sequence of moves whose effect on the cube is known". This is vaguely related to the proper sense of the word algorithm, that vaguely means "a list of instructions needed to do some calculation".
As for instance, if we label the six sides of a cube like the six sides of a die, the sequence of movements 116622553344 will have a definite effect, namely, it will transform a solved cube into a cube with an "X" design in each face. More complicated sequences of movements will have more useful results, such as swapping two corners of the third layer without moving any other pieces. The sequences that are useful to solve the cube are called "algorithms".
The search for optimal solutions The manual solution methods described above are intended to be easy to learn, but much effort has gone into finding even faster solutions to the Rubik's Cube.
In 1982, David Singmaster and Alexander Frey hypothesized that the number of moves needed to solve the Rubik's Cube, given an ideal algorithm, might be in "the low twenties". In 2007, Daniel Kunkle and Gene Cooperman used computer search methods to demonstrate that any 3×3×3 Rubik's Cube configuration can be solved in a maximum of 26 moves.
In 2008, Tomas Rokicki lowered the maximum to 23 moves.
Work continues to try to reduce the upper bound on optimal solutions.
The arrangement known as the super-flip, where every edge is in its correct position but flipped, requires 20 moves to be solved (Using the notations explained below, these are: U R2 F B R B2 R U2 L B2 R U' D' R2 F R' L B2 U2 F2.). No arrangement of the Rubik's Cube has been discovered so far that requires more than 20 moves to solve.
Move notation
Most 3×3×3 Rubik's Cube solution guides use the same notation, originated by David Singmaster, to communicate sequences of moves. This is generally referred to as "cube notation" or in some literature "Singmaster notation" (or variations thereof), or sometimes (but rarely) it is called "direction inferred notation" or "DIN". 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 organized on a particular cube.
- F (Front): the side currently facing you
- B (Back): the side opposite the front
- U (Up): the side above or on top of the front side
- D (Down): the side opposite the top, underneath the Cube
- L (Left): the side directly to the left of the front
- R (Right): the side directly to the right of the front
- f (Front two layers): the side facing you and the corresponding middle layer
- b (Back two layers): the side opposite the front and the corresponding middle layer
- u (Up two layers) : the top side and the corresponding middle layer
- d (Down two layers) : the bottom layer and the corresponding middle layer
- l (Left two layers) : the side to the left of the front and the corresponding middle layer
- r (Right two layers) : the side to the right of the front and the corresponding middle layer
- x (rotate): rotate the Cube up
- y (rotate): rotate the Cube to the left
- z (rotate): rotate the Cube on its side to the right
When an apostrophe follows a letter, it means to turn the face counter-clockwise a quarter-turn, while a letter without an apostrophe means to turn it a quarter-turn clockwise. Such an apostrophe mark is pronounced prime. A letter followed by a 2 (occasionally a superscript ²) means to turn the face a half-turn (the direction does not matter). So R is right side clockwise, but R is right side counter-clockwise. When x, y or z are primed, simply rotate the cube in the opposite direction. When they are squared, rotate it twice. For 'z', you should still be viewing the same front face when rotating.
This notation can also be used on the Pocket Cube, the Revenge, and the Professor, with additional notation. They not only have the F, B, L, R, U, D notation but also f, b, l, r, u, d. For example: (Rr)' l2 f
(Some solution guides, including Ideal's official publication, The Ideal Solution, use slightly different conventions. Top and Bottom are used rather than Up and Down for the top and bottom faces, with Back being replaced by Posterior. '+' indicates clockwise rotation and '-' counter-clockwise, with '++' representing a half-turn. However, alternative notations failed to catch on, and today the Singmaster scheme is used universally by those interested in the puzzle.)
Less-often used moves include rotating the entire Cube or two-thirds of it. The letters x, y, and z are used to indicate that the entire Cube should be turned about one of its axes. The x-axis is the line that passes through the left and right faces, the y-axis is the line that passes through the up and down faces, and the z-axis is the line that passes through the front and back faces. (This type of move is used infrequently in most solutions, to the extent that some solutions simply say "stop and turn the whole cube upside-down" or something similar at the appropriate point.)
However there is another (less common) system of move notation. It is very similar to cube notation, but has a key difference that makes it less daunting to new cube solvers. It is called "direction displayed notation" or "DDN". Each move is represented by two letters. The first
indicates which side is to be moved, the second indicates which direction that side is turned. from the F point of view.
- F (Front): The side facing you. "R" means turn it right or clockwise. "L" means turn it left or counter-clockwise.
- U (Up): The side on top. "R" means turn it right (from the F perspective). "L" means turn it left (from the F perspective).
- D (Down): The side on the bottom. "R" means turn it right (from the F perspective). "L" means turn it left (from the F perspective).
- R (Right): The side to right. "D" means turn it downward (from the F perspective). "U" means turn it upward (from the F perspective).
- L (Left): "D" means turn it downward (from the F perspective). "U" means turn it upward (from the F perspective).
- B (Back): The side opposite from the side facing you. This side is hardly ever used in algorithms of any notation let alone direction displayed notation. However if you need to know, first turn the entire cube so that the B side faces you then rotate it as if you were rotating the F side. "R" means turn it right or clockwise. "L" means turn it left or counter-clockwise.
To indicate a half move just put a 2 at the end of the first letter. To indicate rotation of the cube as a whole, use the same notation for direction displayed notation as one would for Singmaster notation. (x y z)
Lowercase letters f, b, u, d, l, and r signify to move the first two layers of that face while keeping the remaining layer in place. This is of course equivalent to rotating the whole cube in that direction, then rotating the opposite face back the same amount in the opposite direction, but is useful notation to describe certain triggers for speedcubing. Furthermore, M, E, and S (and respectively their lowercase for larger sized cubes) are used for inner-slice movements. M signifies turning the layer that is between L and R downward (clockwise if looking from the left side). E signifies turning the layer between U and D towards the right (counter-clockwise if looking from the top). S signifies turning the layer between F and B clockwise.
For example, the algorithm (or operator, or sequence) F2 U' R' L F2 R L' U' F2, which cycles three edge cubes in the top layer without affecting any other part of the cube, means:
- Turn the Front face 180 degrees.
- Turn the Up face 90 degrees counter-clockwise.
- Turn the Right face 90 degrees counter-clockwise.
- Turn the Left face 90 degrees clockwise.
- Turn the Front face 180 degrees.
- Turn the Right face 90 degrees clockwise.
- Turn the Left face 90 degrees counter-clockwise.
- Turn the Up face 90 degrees counter-clockwise.
- Finally, turn the Front face 180 degrees.
For beginning students of the Cube, this notation can be daunting, and many solutions available online therefore incorporate animations that demonstrate the algorithms presented.
4×4×4 and larger cubes use slightly different notation to incorporate the 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 the inner portions of the cube (called slices). Again Ideal breaks rank by describing their 4×4×4 solution in terms of layers (vertical slices that rotate about the z-axis), tables (horizontal slices), and books (vertical slices that rotate about the x-axis).
Competitions and record times
Many speedcubing competitions have been held to determine who can solve the Rubik's Cube in the shortest time. The number of contests is going up every year; there were 72 official competitions from 2003 to 2006; 33 were in 2006 alone.
The first world championship organized by the Guinness Book of World Records was held in Munich on March 13, 1981. All Cubes were moved 40 times and rubbed with petroleum jelly. The official winner, with a record of 38 seconds, was Jury Froeschl, born in Munich.
The first international world championship was held in Budapest on June 5, 1982, and was won by Minh Thai, a Vietnamese student from Los Angeles, with a time of 22.95 seconds.
Since 2003, competitions are decided by the best average (middle three of five attempts); but the single best time of all tries is also recorded.
The World Cube Association maintains a history of world records
.
In 2004, the WCA made it mandatory to use a special timing device called a Stackmat timer.
The for single time is set by Erik Akkersdijk in 2008, he set a best time of 7.08 at the Czech Open 2008. The world record average solve is by Yu Nakajima, when he set a world record average of 11.28 seconds on May 4, 2008 at .
Alternative competitions
In addition, informal alternative competitions have been held, inviting participants to solve the Cube under unusual situations. These include:
- Blindfolded solving
- Solving the Cube with one person blindfolded and the other person saying what moves to do, known as "Team Blindfold"
- Solving the Cube underwater in a single breath
- Solving the Cube using a single hand
- Solving the Cube with one's feet
Of these informal competitions, the World Cube Association only sanctions blindfolded, one-handed, and feet solving as official competition events.
Custom built puzzles
A lot of puzzles have been built in the past resembling the Rubik's Cube or just its working (as a permutation puzzle). For example, a Cuboid is a puzzle based off the Rubiks Cube, but with different fully functional dimensions, such as 2x3x4, 3x3x5, or 2x2x4. Many cuboids are based off of 4x4x4 or 5x5x5 mechanisms, via building plastic extensions or by directly modifying the mechanism itself.
Some custom puzzles are not based off of any existing mechanism, such as the Gigaminx v1.5-v2, Bevel Cube, SuperX, Toru, Rua, and 1x2x3. These puzzles usually have a set of masters 3D printed, which then are copied using molding and casting techniques to create a full puzzle.
Other Rubiks Cube modifications include cubes that have been extended or truncated to form a new shape. An example of this is the Trabjer's Octahedron, which can be built by truncating and extending portions of a regular 3x3x3 Rubiks cube. Most shape mods can be adapted to higher-order cubes. An example of this is in the case of Tony Fisher's Rhombic Dodecahedron puzzles, where there are versions of the puzzle for 3x3, 4x4, 5x5, and 6x6.
Meffert's Puzzles
Meffert's puzzles is another well known distributor of twisty puzzles. Founded by Uwe Meffert, Meffert's Puzzles has carried many unique puzzles over the years.
Meffert's products include the Pyramorphix, which is isomorphic with the Pocket Cube but has the shape of a tetrahedron (four of the Pocket Cube's corners have been turned into tetrahedra, the other four into flat triangles). This makes the Pyramorphix somewhat more difficult to solve than the Pocket Cube, as the isomorphism is not obvious. He also has been a long time producer of the Pyraminx, which is a tetrahedral twisty puzzle. More unique puzzles he has produced over the years include the Pyraminx Crystal, Impossiball, Megaminx, Skewb, and variations of the Skewb.
Rubik's Cube softwareSeveral computer programs have been written to perform various functions, such as, among other things, solving the Cube or animating it. In general, these programs can be considered to fall in one of several categories:
- Timers
- Solvers
- Graphical programs
- Animations
- Image generators
- Analyzers
Some of the software handles not only the 3×3×3 cube, but also other puzzle types. There is even software for virtual puzzles that do not have a real life counterpart. Examples are the four-dimensional cube, the five-dimensional cube and the gliding cube.
In addition these programs may also record player metrics, store and generate scrambled Cube positions or offer either animations or online competition. Solvers are usually given a scramble, after which a solution is generated automatically. Graphical programs can generate a static image or animate the Cube and its motions, e.g. using Java or Flash. Programs may also analyze sequences of moves and transform them to other notations or give player metrics.
See also
External links
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