In particular he divided the cube group into the following chain of subgroups: Where algorithms up to that point divided the problem by looking at the parts of the cube that should remain fixed, he divided it by restricting the type of moves that could be executed. Thistlethwaite's idea was to divide the problem into subproblems. The approaches to the cube that led to algorithms with very few moves are based on group theory and on extensive computer searches. The breakthrough, known as "descent through nested sub-groups" was found by Morwen Thistlethwaite details of Thistlethwaite's algorithm were published in Scientific American in 1981 by Douglas Hofstadter. Soon after, Conway’s Cambridge Cubists reported that the cube could be restored in at most 94 moves. Guy had come up with a different algorithm that took at most 160 moves. Later, Singmaster reported that Elwyn Berlekamp, John Conway, and Richard K. He simply counted the maximum number of moves required by his cube-solving algorithm. Perhaps the first concrete value for an upper bound was the 277 moves mentioned by David Singmaster in early 1979. By combining the worst-case scenarios for each part of these algorithms, the typical upper bound was found to be around 100. The first upper bounds were based on the 'human' algorithms. The position, which was called a 'superflip composed with four spot' needs 26 quarter turns. In 1998, a new position requiring more than 24 quarter turns to solve was found. Also in 1995, a solution for superflip in 24 quarter turns was found by Michael Reid, with its minimality proven by Jerry Bryan. Winter, of which the minimality was shown in 1995 by Michael Reid, providing a new lower bound for the diameter of the cube group. In 1992, a solution for the superflip with 20 face turns was found by Dik T. A Rubik's Cube is in the superflip pattern when each corner piece is in the correct position, but each edge piece is incorrectly oriented. It was conjectured that the so-called superflip would be a position that is very difficult. Also, it is not a constructive proof: it does not exhibit a concrete position that needs this many moves. This argument was not improved upon for many years. It turns out that the latter number is smaller. To show this, first count the number of cube positions that exist in total, then count the number of positions achievable using at most 17 moves starting from a solved cube. It can be proven by counting arguments that there exist positions needing at least 18 moves to solve. Note that these letters are usually lowercase instead. As with regular turns, a 2 signifies a half rotation and a prime (') indicates a quarter rotation in the opposite direction. These cube rotations are used in algorithms to make the algorithms smoother and faster. Z signifies the rotation of the cube on the F direction. Y signifies the rotation of the cube in the U direction. X signifies rotating the cube in the R direction. The letters X, Y and Z are used to signify cube rotations. As with regular turns, a 2 signifies a double turn and a prime (') indicates a quarter turn anticlockwise. E represents turning the layer between the U and D faces 1 quarter turn clockwise left to right. S represents turning the layer between the F and B faces 1 quarter turn clockwise, as seen from the front. M represents turning the layer between the R and L faces 1 quarter turn top to bottom. The letters M, S and E are used to denote the turning of a middle layer. A counterclockwise quarter turn is indicated by appending a prime symbol ( ′ ). Half turns are indicated by appending a 2. The letters L, R, F, B, U, and D indicate a clockwise quarter turn of the left, right, front, back, up, and down face respectively. To denote a sequence of moves on the 3×3×3 Rubik's Cube, this article uses "Singmaster notation", which was developed by David Singmaster. Main article: Rubik's Cube § Move notation An algorithm that solves a cube in the minimum number of moves is known as God's algorithm. There are many algorithms to solve scrambled Rubik's Cubes. In STM (slice turn metric), the minimal number of turns is unknown. These numbers are also the diameters of the corresponding Cayley graphs of the Rubik's Cube group. The maximal number of face turns needed to solve any instance of the Rubik's Cube is 20, and the maximal number of quarter turns is 26. A move to turn an outer layer two quarter (90°) turns in the same direction would be counted as two moves in the quarter turn metric (QTM), but as one turn in the face metric (FTM, or HTM "Half Turn Metric", or OBTM "Outer Block Turn Metric"). The second is to count the number of outer-layer twists, called "face turns". The first is to count the number of quarter turns. There are two common ways to measure the length of a solution. Optimal solutions for Rubik's Cube refer to solutions that are the shortest.
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