1991:. First all cubes are generated that are the result of applying 1 move to them. That is C * F, C * U, … Next, from this list, all cubes are generated that are the result of applying two moves. Then three moves and so on. If at any point a cube is found that needs too many moves based on the lower bounds to still be optimal it can be eliminated from the list.
2028:
positions. In the quarter-turn metric, only a single position (and its two rotations) is known that requires the maximum of 26 moves. Despite significant effort, no additional quarter-turn distance-26 positions have been found. Even at distance 25, only two positions (and their rotations) are known to exist. At distance 24, perhaps 150,000 positions exist.
30:
1274:
Initially, Thistlethwaite showed that any configuration could be solved in at most 85 moves. In
January 1980 he improved his strategy to yield a maximum of 80 moves. Later that same year, he reduced the number to 63, and then again to 52. By exhaustively searching the coset spaces it was later found
43:
are solutions that are the shortest in some sense. There are two common ways to measure the length of a solution. The first is to count the number of quarter turns. The second is to count the number of outer-layer twists, called "face turns". A move to turn an outer layer two quarter (90°) turns in
2011:
to show that all unsolved cubes can be solved in no more than 26 moves (in face-turn metric). Instead of attempting to solve each of the billions of variations explicitly, the computer was programmed to bring the cube to one of 15,752 states, each of which could be solved within a few extra moves.
2023:
that all cube positions could be solved with a maximum of 20 face turns. In 2009, Tomas
Rokicki proved that 29 moves in the quarter-turn metric is enough to solve any scrambled cube. And in 2014, Tomas Rokicki and Morley Davidson proved that the maximum number of quarter-turns needed to solve the
341:
It can be proven by counting arguments that there exist positions needing at least 18 moves to solve. 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 turns out that the
2027:
The face-turn and quarter-turn metrics differ in the nature of their antipodes. An antipode is a scrambled cube that is maximally far from solved, one that requires the maximum number of moves to solve. In the half-turn metric with a maximum number of 20, there are hundreds of millions of such
427:
and on extensive computer searches. Thistlethwaite's idea was to divide the problem into subproblems. 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. In
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1948:
Using these group solutions combined with computer searches will generally quickly give very short solutions. But these solutions do not always come with a guarantee of their minimality. To search specifically for minimal solutions a new approach was needed.
357:
would be a position that is very difficult. 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. In 1992, a solution for the superflip with 20 face turns was found by
1998:
will always find optimal solutions, there is no worst case analysis. It is not known in general how many iterations this algorithm will need to reach an optimal solution. An implementation of this algorithm can be found here.
1202:
1912:
is available, the search becomes virtually instantaneous: one need only generate 18 cube states for each of the 12 moves and choose the one with the lowest heuristic each time. This allows the second heuristic, that for
2015:
Tomas
Rokicki reported in a 2008 computational proof that all unsolved cubes could be solved in 25 moves or fewer. This was later reduced to 23 moves. In August 2008, Rokicki announced that he had a proof for 22 moves.
796:
1961:
IDA* is a depth-first search that looks for increasingly longer solutions in a series of iterations, using a lower-bound heuristic to prune branches once a lower bound on their length exceeds the current iterations
1952:
In 1997 Richard Korf announced an algorithm with which he had optimally solved random instances of the cube. Of the ten random cubes he did, none required more than 18 face turns. The method he used is called
1455:
684:
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that the worst possible number of moves for each stage was 7, 10, 13, and 15 giving a total of 45 moves at most. There have been implementations of
Thistlewaite's algorithm in various computer languages.
586:
2012:
All were proved solvable in 29 moves, with most solvable in 26. Those that could not initially be solved in 26 moves were then solved explicitly, and shown that they too could be solved in 26 moves.
1858:, which may narrow it down considerably, searching through that many states is likely not practical. To solve this problem, Kociemba devised a lookup table that provides an exact heuristic for
2237:
1357:
502:
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In 1995 Michael Reid proved that using these two groups every position can be solved in at most 29 face turns, or in 42 quarter turns. This result was improved by Silviu Radu in 2005 to 40.
2007:
In 2006, Silviu Radu further improved his methods to prove that every position can be solved in at most 27 face turns or 35 quarter turns. Daniel Kunkle and Gene
Cooperman in 2007 used a
44:
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").
378:
The first upper bounds were based on the 'human' algorithms. By combining the worst-case scenarios for each part of these algorithms, the typical upper bound was found to be around 100.
124:
indicate a clockwise quarter turn of the left, right, front, back, up, and down face respectively. A half turn (i.e. 2 quarter turns in the same direction) are indicated by appending a
1790:, much shorter overall solutions are usually obtained. Using this algorithm solutions are typically found of fewer than 21 moves, though there is no proof that it will always do so.
366:, providing a new lower bound for the diameter of the cube group. Also in 1995, a solution for superflip in 24 quarter turns was found by Michael Reid, with its minimality proven by
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1542:
1269:
960:
891:. For each element he found a sequence of moves that took it to the next smaller group. After these preparations he worked as follows. A random cube is in the general cube group
370:. In 1998, a new position requiring more than 24 quarter turns to solve was found. The position, which was called a 'superflip composed with four spot' needs 26 quarter turns.
2540:, a Wikibooks article that gives an overview over several algorithms that are simple enough to be memorizable by humans. However, such algorithms will usually not give an
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had come up with a different algorithm that took at most 160 moves. Soon after, Conway’s
Cambridge Cubists reported that the cube could be restored in at most 94 moves.
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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. These numbers are also the
333:
signifies a half turn and a prime (') indicates a turn counterclockwise. Note that these spacial rotations are usually represented with lowercase letters.
135:
However, because these notations are human-oriented, we use clockwise as positive, and not mathematically-oriented, which is counterclockwise as positive.
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is the largest and contains only 1082565 elements. The number of moves required by this algorithm is the sum of the largest process in each step.
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1980:
Clearly the number of moves required to solve any of these subproblems is a lower bound for the number of moves needed to solve the entire cube.
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690:
3185:
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1957:
and is described in his paper "Finding
Optimal Solutions to Rubik's Cube Using Pattern Databases". Korf describes this method as follows
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in early 1979. He simply counted the maximum number of moves required by his cube-solving algorithm. Later, Singmaster reported that
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1966:
It works roughly as follows. First he identified a number of subproblems that are small enough to be solved optimally. He used:
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88:
To denote a sequence of moves on the 3×3×3 Rubik's Cube, this article uses "Singmaster notation", which was developed by
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1851:
329:
direction. These cube rotations are often used in algorithms to make them smoother and faster. As with regular turns, a
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at most 18 moves, as
Michael Reid showed in 1995. By also generating suboptimal solutions that take the cube to group
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2020:
1940:, to be less precise and still allow for a solution to be computed in reasonable time on a modern computer.
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48:
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Non-standard moves are usually represented with lowercase letters in contrast to the standard moves above.
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56:
17:
2019:
Finally, in 2010, Tomas
Rokicki, Herbert Kociemba, Morley Davidson, and John Dethridge gave the final
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1988:
424:
415:
359:
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faces 1 quarter turn clockwise (left to right), as seen from the D face. As with regular turns, a
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The following are standard moves, which do not move centre cubies of any face to another location:
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802:
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367:
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71:
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2701:
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Proceedings of the
International Symposium on Symbolic and Algebraic Computation (ISSAC '07)
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contains 18 possible moves (each move, its prime, and its 180-degree rotation), that leaves
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1207:
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992:
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285:. Using this notation for a three-layer cube is more consistent with multiple-layer cubes.
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2217:
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83:
67:
39:
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2130:
Adventures in group theory: Rubik's Cube, Merlin's machine, and Other Mathematical Toys
1973:
The cube restricted to only 6 edges, not looking at the corners nor at the other edges.
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129:
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3106:
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2008:
423:. The approaches to the cube that led to algorithms with very few moves are based on
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245:
In multiple-layer cubes, numbers may precede face names to indicate rotation of the
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2814:
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381:
Perhaps the first concrete value for an upper bound was the 277 moves mentioned by
52:
2330:
Implementation of Thistlewaite's Algorithm for Rubik's Cube Solution in Javascript
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Intermediate state of the Rubik's Cube in Kociemba's algorithm. Any state from G
842:
2405:
2796:
2786:
2158:
1197:{\displaystyle G_{1}\setminus G_{0},G_{2}\setminus G_{1},G_{3}\setminus G_{2}}
962:. He applied the corresponding process to the cube. This took it to a cube in
350:
349:: it does not exhibit a concrete position that needs this many moves. It was
70:. An algorithm that solves a cube in the minimum number of moves is known as
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1995:
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The breakthrough, known as "descent through nested sub-groups" was found by
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faces 1 quarter turn clockwise (top to bottom), as seen facing the F face.
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faces 1 quarter turn clockwise (front to back), as seen facing the L face.
1796:
At first glance, this algorithm appears to be practically inefficient: if
3131:
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2771:
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791:{\displaystyle G_{3}=\langle L^{2},R^{2},F^{2},B^{2},U^{2},D^{2}\rangle }
203:
signifies a half turn and a prime (') indicates a turn counterclockwise.
2547:
2192:
345:
This argument was not improved upon for many years. Also, it is not a
179:(short for "Standing" layer) represents turning the layer between the
59:. In STM (slice turn metric), the minimal number of turns is unknown.
2901:
1984:
191:(short for "Equator" layer) represents turning the layer between the
167:(short for "Middle" layer) represents turning the layer between the
29:
2367:"Finding Optimal Solutions to Rubik's Cube Using Pattern Databases"
1287:
in 1992. He reduced the number of intermediate groups to only two:
2855:
2729:
2445:
2439:
Tom Rokicki (2008). "Twenty-Five Moves Suffice for Rubik's Cube".
919:
2329:
1970:
The cube restricted to only the corners, not looking at the edges
1450:{\displaystyle G_{1}=\langle U,D,L^{2},R^{2},F^{2},B^{2}\rangle }
679:{\displaystyle G_{2}=\langle L,R,F^{2},B^{2},U^{2},D^{2}\rangle }
1954:
2551:
2341:
2318:
2544:
solution which only uses the minimum possible number of moves.
1850:(over 1 quadrillion) cube states to be searched. Even with a
2504:
2103:
2072:
581:{\displaystyle G_{1}=\langle L,R,F,B,U^{2},D^{2}\rangle }
2207:
Michael Reid's Rubik's Cube page M-symmetric positions
2133:. Baltimore: Johns Hopkins University Press. pp.
128:. A counterclockwise turn is indicated by appending a
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1829:
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1604:
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is very large (~4.3×10), the right coset spaces
1091:
1049:
1022:
995:
989:. Next he looked up a process that takes the cube to
968:
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897:
851:
805:
693:
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511:
441:
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3229:
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3168:
3152:
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2795:
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2715:
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2229:
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1932:
1904:
1885:. When the exact number of moves needed to reach
1877:
1842:
1815:
1782:
1755:
1728:
1701:
1657:
1630:
1590:
1571:. Next he searched the optimal solution for group
1563:
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1449:
1351:
1263:
1223:
1196:
1104:
1062:
1035:
1008:
981:
954:
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883:
830:
790:
678:
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496:
163:are used to denote the turning of a middle layer.
2406:Press Release on Proof that 26 Face Turns Suffice
2202:
2200:
1352:{\displaystyle G_{0}=\langle U,D,L,R,F,B\rangle }
497:{\displaystyle G_{0}=\langle L,R,F,B,U,D\rangle }
147:Moving centre cubies of faces to other locations:
1504:, he would search through the right coset space
2505:"God's Number is 26 in the Quarter Turn Metric"
2104:"God's Number is 26 in the Quarter Turn Metric"
362:, of which the minimality was shown in 1995 by
261:are then used to denote turning layers next to
218:are also used to denote turning layers next to
2383:Michael Reid's Optimal Solver for Rubik's Cube
2319:Progressive Improvements in Solving Algorithms
2003:Further improvements, and finding God's Number
2563:
242:. This is more consistent with expectations.
8:
2098:
2096:
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2092:
2067:
2065:
1484:
1478:
1444:
1380:
1346:
1310:
825:
819:
785:
707:
673:
609:
575:
525:
491:
455:
2419:"Twenty-Six Moves Suffice for Rubik's Cube"
1665:were both done with a method equivalent to
1283:Thistlethwaite's algorithm was improved by
1081:will have the "+" and "–" symbols as shown.
3149:
2759:
2570:
2556:
2548:
2285:
2273:
2261:
918:. Next he found this element in the right
325:signifies the rotation of the cube on the
317:signifies the rotation of the cube in the
2444:
1976:The cube restricted to the other 6 edges.
1924:
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1709:needs at most 12 moves and the search in
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1021:
1000:
994:
973:
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628:
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594:
569:
556:
516:
510:
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1072:
841:Next he prepared tables for each of the
28:
2342:"Solve Rubik's Cube with Cube Explorer"
2037:
1686:
1615:
1521:
1248:
1181:
1155:
1129:
939:
868:
432:into the following chain of subgroups:
84:Rubik's Cube § Singmaster notation
18:Optimal solutions for Rubik's Cube
884:{\displaystyle G_{i+1}\setminus G_{i}}
273:respectively in the same direction as
230:respectively in the same direction as
7:
3274:1982 World Rubik's Cube Championship
1702:{\displaystyle G_{1}\setminus G_{0}}
1631:{\displaystyle G_{1}\setminus G_{0}}
1537:{\displaystyle G_{1}\setminus G_{0}}
1264:{\displaystyle G_{2}\setminus G_{1}}
955:{\displaystyle G_{1}\setminus G_{0}}
305:are used to signify cube rotations.
139:The following are non-standard moves
3268:The Simple Solution to Rubik's Cube
2218:Posted to Cube lovers on 2 Aug 1998
1763:and looking for short solutions in
309:signifies rotating the cube in the
2417:Kunkle, D.; Cooperman, C. (2007).
1231:are much smaller. The coset space
25:
2639:Rubik's family cubes of all sizes
2385:(requires a compiler such as gcc)
3253:Rubik's Cube in popular culture
2485:"Twenty-Nine QTM Moves Suffice"
2300:"The Subgroup H and its cosets"
2240:. Math Horizons. Archived from
2193:How to solve a 4x4 Rubik's Cube
1854:-based computer algorithm like
249:th layer from the named face.
2236:Rik van Grol (November 2010).
1085:Although the whole cube group
1:
2538:How to solve the Rubik's Cube
2238:"The Quest For God's Number"
2050:www.worldcubeassociation.org
2524:Notes on Rubik's Magic Cube
2182:How to solve the 3x3x4 Cube
1490:{\displaystyle G_{2}=\{1\}}
831:{\displaystyle G_{4}=\{1\}}
342:latter number is smaller.
206:Instead, lowercase letters
3316:
3216:Thistlethwaite's algorithm
2522:Singmaster, David (1981).
2462:— Domain of the Cube Forum
2460:Twenty-Three Moves Suffice
2395:Rubik can be solved in 27f
1544:to take the cube to group
1502:Thistlethwaite's algorithm
428:particular he divided the
411:Thistlethwaite's algorithm
401:Thistlethwaite's algorithm
81:
38:Optimal solutions for the
1943:
1278:
400:
3300:Computer-assisted proofs
2659:5×5×5 (Professor's Cube)
2472:twenty-two moves suffice
2046:"World Cube Association"
1987:cube C, it is solved as
289:Rotating the whole cube:
33:A scrambled Rubik's Cube
3259:Rubik, the Amazing Cube
2654:4×4×4 (Rubik's Revenge)
2159:"Rubik's Cube Notation"
2021:computer-assisted proof
1843:{\displaystyle 18^{12}}
3237:World Cube Association
3112:Anthony Michael Brooks
3072:Krishnam Raju Gadiraju
2127:Joyner, David (2002).
2002:
1934:
1906:
1879:
1844:
1817:
1784:
1757:
1730:
1703:
1669:(IDA*). The search in
1667:iterative deepening A*
1659:
1632:
1592:
1565:
1538:
1491:
1451:
1353:
1265:
1225:
1198:
1106:
1082:
1064:
1037:
1010:
983:
956:
912:
885:
832:
792:
680:
582:
498:
51:of the corresponding
34:
3230:Official organization
2884:Truncated icosahedron
2365:Richard Korf (1997).
1935:
1933:{\displaystyle G_{1}}
1907:
1905:{\displaystyle G_{1}}
1880:
1878:{\displaystyle G_{0}}
1845:
1818:
1816:{\displaystyle G_{0}}
1785:
1783:{\displaystyle G_{1}}
1758:
1756:{\displaystyle G_{1}}
1731:
1729:{\displaystyle G_{1}}
1704:
1660:
1658:{\displaystyle G_{1}}
1633:
1593:
1591:{\displaystyle G_{1}}
1566:
1564:{\displaystyle G_{1}}
1539:
1492:
1452:
1354:
1266:
1226:
1224:{\displaystyle G_{3}}
1199:
1107:
1105:{\displaystyle G_{0}}
1076:
1065:
1063:{\displaystyle G_{4}}
1038:
1036:{\displaystyle G_{3}}
1011:
1009:{\displaystyle G_{2}}
984:
982:{\displaystyle G_{1}}
957:
913:
911:{\displaystyle G_{0}}
886:
833:
793:
681:
583:
499:
407:Morwen Thistlethwaite
32:
2649:3×3×3 (Rubik's Cube)
2526:. Enslow Publishers.
2073:"God's Number is 20"
1917:
1889:
1862:
1827:
1800:
1767:
1740:
1713:
1673:
1642:
1602:
1575:
1548:
1508:
1462:
1364:
1294:
1279:Kociemba's algorithm
1235:
1208:
1116:
1089:
1047:
1020:
993:
966:
926:
895:
849:
803:
691:
593:
509:
439:
2924:Virtual combination
2756:combination puzzles
2718:combination puzzles
2644:2×2×2 (Pocket Cube)
1989:iterative deepening
416:Scientific American
353:that the so-called
66:to solve scrambled
3221:Rubik's Cube group
3067:Prithveesh K. Bhat
2991:Rubik's Revolution
2866:Great dodecahedron
2618:Oskar van Deventer
2298:Herbert Kociemba.
1930:
1902:
1875:
1840:
1813:
1780:
1753:
1726:
1699:
1655:
1628:
1598:. The searches in
1588:
1561:
1534:
1487:
1447:
1349:
1261:
1221:
1194:
1102:
1083:
1060:
1033:
1006:
979:
952:
908:
881:
828:
788:
676:
578:
494:
421:Douglas Hofstadter
413:were published in
347:constructive proof
57:Rubik's Cube group
35:
3282:
3281:
3194:
3193:
2919:
2918:
2683:Variations of the
2613:Panagiotis Verdes
132:( ′ ).
16:(Redirected from
3307:
3246:Related articles
3150:
3097:David Singmaster
3057:Shotaro Makisumi
3032:Jessica Fridrich
3010:Renowned solvers
2926:puzzles (>3D)
2874:Alexander's Star
2828:Pyraminx Crystal
2760:
2702:Nine-Colour Cube
2674:8×8×8 (V-Cube 8)
2669:7×7×7 (V-Cube 7)
2664:6×6×6 (V-Cube 6)
2586:Puzzle inventors
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2352:
2338:
2332:
2327:
2321:
2316:
2310:
2309:
2307:
2306:
2295:
2289:
2283:
2277:
2271:
2265:
2259:
2253:
2252:
2250:
2249:
2233:
2220:
2215:
2209:
2204:
2195:
2190:
2184:
2179:
2173:
2172:
2170:
2169:
2155:
2149:
2148:
2124:
2118:
2117:
2115:
2114:
2100:
2087:
2086:
2084:
2083:
2069:
2060:
2059:
2057:
2056:
2042:
1944:Korf's algorithm
1939:
1937:
1936:
1931:
1929:
1928:
1911:
1909:
1908:
1903:
1901:
1900:
1884:
1882:
1881:
1876:
1874:
1873:
1849:
1847:
1846:
1841:
1839:
1838:
1822:
1820:
1819:
1814:
1812:
1811:
1789:
1787:
1786:
1781:
1779:
1778:
1762:
1760:
1759:
1754:
1752:
1751:
1735:
1733:
1732:
1727:
1725:
1724:
1708:
1706:
1705:
1700:
1698:
1697:
1685:
1684:
1664:
1662:
1661:
1656:
1654:
1653:
1637:
1635:
1634:
1629:
1627:
1626:
1614:
1613:
1597:
1595:
1594:
1589:
1587:
1586:
1570:
1568:
1567:
1562:
1560:
1559:
1543:
1541:
1540:
1535:
1533:
1532:
1520:
1519:
1496:
1494:
1493:
1488:
1474:
1473:
1456:
1454:
1453:
1448:
1443:
1442:
1430:
1429:
1417:
1416:
1404:
1403:
1376:
1375:
1358:
1356:
1355:
1350:
1306:
1305:
1285:Herbert Kociemba
1270:
1268:
1267:
1262:
1260:
1259:
1247:
1246:
1230:
1228:
1227:
1222:
1220:
1219:
1203:
1201:
1200:
1195:
1193:
1192:
1180:
1179:
1167:
1166:
1154:
1153:
1141:
1140:
1128:
1127:
1111:
1109:
1108:
1103:
1101:
1100:
1069:
1067:
1066:
1061:
1059:
1058:
1042:
1040:
1039:
1034:
1032:
1031:
1015:
1013:
1012:
1007:
1005:
1004:
988:
986:
985:
980:
978:
977:
961:
959:
958:
953:
951:
950:
938:
937:
917:
915:
914:
909:
907:
906:
890:
888:
887:
882:
880:
879:
867:
866:
837:
835:
834:
829:
815:
814:
797:
795:
794:
789:
784:
783:
771:
770:
758:
757:
745:
744:
732:
731:
719:
718:
703:
702:
685:
683:
682:
677:
672:
671:
659:
658:
646:
645:
633:
632:
605:
604:
587:
585:
584:
579:
574:
573:
561:
560:
521:
520:
503:
501:
500:
495:
451:
450:
383:David Singmaster
90:David Singmaster
21:
3315:
3314:
3310:
3309:
3308:
3306:
3305:
3304:
3285:
3284:
3283:
3278:
3241:
3225:
3206:God's algorithm
3190:
3164:
3141:
3102:Ron van Bruchem
3027:Bob Burton, Jr.
3022:Édouard Chambon
3005:
3001:Rubik's Triamid
2952:
2925:
2915:
2896:
2878:
2860:
2837:
2809:
2791:
2755:
2749:
2725:Helicopter Cube
2717:
2711:
2684:
2678:
2622:
2581:
2576:
2534:
2521:
2518:
2516:Further reading
2513:
2512:
2503:
2502:
2498:
2489:
2487:
2482:
2481:
2477:
2470:
2466:
2458:
2454:
2438:
2437:
2433:
2421:
2416:
2415:
2411:
2404:
2400:
2393:
2389:
2381:
2377:
2369:
2364:
2363:
2359:
2350:
2348:
2340:
2339:
2335:
2328:
2324:
2317:
2313:
2304:
2302:
2297:
2296:
2292:
2286:Singmaster 1981
2284:
2280:
2274:Singmaster 1981
2272:
2268:
2262:Singmaster 1981
2260:
2256:
2247:
2245:
2235:
2234:
2223:
2216:
2212:
2205:
2198:
2191:
2187:
2180:
2176:
2167:
2165:
2157:
2156:
2152:
2145:
2126:
2125:
2121:
2112:
2110:
2102:
2101:
2090:
2081:
2079:
2071:
2070:
2063:
2054:
2052:
2044:
2043:
2039:
2034:
2005:
1946:
1920:
1915:
1914:
1892:
1887:
1886:
1865:
1860:
1859:
1830:
1825:
1824:
1803:
1798:
1797:
1770:
1765:
1764:
1743:
1738:
1737:
1716:
1711:
1710:
1689:
1676:
1671:
1670:
1645:
1640:
1639:
1618:
1605:
1600:
1599:
1578:
1573:
1572:
1551:
1546:
1545:
1524:
1511:
1506:
1505:
1465:
1460:
1459:
1434:
1421:
1408:
1395:
1367:
1362:
1361:
1297:
1292:
1291:
1281:
1251:
1238:
1233:
1232:
1211:
1206:
1205:
1184:
1171:
1158:
1145:
1132:
1119:
1114:
1113:
1092:
1087:
1086:
1080:
1050:
1045:
1044:
1043:and finally to
1023:
1018:
1017:
996:
991:
990:
969:
964:
963:
942:
929:
924:
923:
898:
893:
892:
871:
852:
847:
846:
806:
801:
800:
775:
762:
749:
736:
723:
710:
694:
689:
688:
663:
650:
637:
624:
596:
591:
590:
565:
552:
512:
507:
506:
442:
437:
436:
403:
387:Elwyn Berlekamp
376:
339:
86:
80:
72:God's algorithm
62:There are many
23:
22:
15:
12:
11:
5:
3313:
3311:
3303:
3302:
3297:
3287:
3286:
3280:
3279:
3277:
3276:
3271:
3264:
3263:
3262:
3249:
3247:
3243:
3242:
3240:
3239:
3233:
3231:
3227:
3226:
3224:
3223:
3218:
3213:
3208:
3202:
3200:
3196:
3195:
3192:
3191:
3189:
3188:
3183:
3178:
3176:Layer by Layer
3172:
3170:
3166:
3165:
3163:
3162:
3156:
3154:
3147:
3143:
3142:
3140:
3139:
3134:
3129:
3124:
3122:Feliks Zemdegs
3119:
3114:
3109:
3104:
3099:
3094:
3089:
3084:
3079:
3074:
3069:
3064:
3059:
3054:
3049:
3044:
3039:
3037:Chris Hardwick
3034:
3029:
3024:
3019:
3013:
3011:
3007:
3006:
3004:
3003:
2998:
2993:
2988:
2987:
2986:
2984:Master Edition
2976:
2971:
2966:
2960:
2958:
2954:
2953:
2951:
2950:
2948:Magic 120-cell
2945:
2940:
2935:
2929:
2927:
2921:
2920:
2917:
2916:
2914:
2913:
2910:Rubik's Domino
2906:
2904:
2898:
2897:
2895:
2894:
2888:
2886:
2880:
2879:
2877:
2876:
2870:
2868:
2862:
2861:
2859:
2858:
2853:
2847:
2845:
2839:
2838:
2836:
2835:
2833:Skewb Ultimate
2830:
2825:
2819:
2817:
2811:
2810:
2808:
2807:
2801:
2799:
2793:
2792:
2790:
2789:
2784:
2779:
2774:
2768:
2766:
2757:
2751:
2750:
2748:
2747:
2742:
2737:
2732:
2727:
2721:
2719:
2713:
2712:
2710:
2709:
2704:
2699:
2694:
2688:
2686:
2680:
2679:
2677:
2676:
2671:
2666:
2661:
2656:
2651:
2646:
2641:
2636:
2630:
2628:
2624:
2623:
2621:
2620:
2615:
2610:
2605:
2600:
2595:
2589:
2587:
2583:
2582:
2577:
2575:
2574:
2567:
2560:
2552:
2546:
2545:
2533:
2532:External links
2530:
2529:
2528:
2517:
2514:
2511:
2510:
2496:
2475:
2464:
2452:
2431:
2409:
2398:
2387:
2375:
2357:
2333:
2322:
2311:
2290:
2278:
2266:
2254:
2221:
2210:
2196:
2185:
2174:
2150:
2143:
2119:
2088:
2061:
2036:
2035:
2033:
2030:
2004:
2001:
1994:Although this
1978:
1977:
1974:
1971:
1964:
1963:
1945:
1942:
1927:
1923:
1899:
1895:
1872:
1868:
1837:
1833:
1810:
1806:
1777:
1773:
1750:
1746:
1723:
1719:
1696:
1692:
1688:
1683:
1679:
1652:
1648:
1625:
1621:
1617:
1612:
1608:
1585:
1581:
1558:
1554:
1531:
1527:
1523:
1518:
1514:
1498:
1497:
1486:
1483:
1480:
1477:
1472:
1468:
1457:
1446:
1441:
1437:
1433:
1428:
1424:
1420:
1415:
1411:
1407:
1402:
1398:
1394:
1391:
1388:
1385:
1382:
1379:
1374:
1370:
1359:
1348:
1345:
1342:
1339:
1336:
1333:
1330:
1327:
1324:
1321:
1318:
1315:
1312:
1309:
1304:
1300:
1280:
1277:
1258:
1254:
1250:
1245:
1241:
1218:
1214:
1191:
1187:
1183:
1178:
1174:
1170:
1165:
1161:
1157:
1152:
1148:
1144:
1139:
1135:
1131:
1126:
1122:
1099:
1095:
1078:
1057:
1053:
1030:
1026:
1003:
999:
976:
972:
949:
945:
941:
936:
932:
905:
901:
878:
874:
870:
865:
862:
859:
855:
839:
838:
827:
824:
821:
818:
813:
809:
798:
787:
782:
778:
774:
769:
765:
761:
756:
752:
748:
743:
739:
735:
730:
726:
722:
717:
713:
709:
706:
701:
697:
686:
675:
670:
666:
662:
657:
653:
649:
644:
640:
636:
631:
627:
623:
620:
617:
614:
611:
608:
603:
599:
588:
577:
572:
568:
564:
559:
555:
551:
548:
545:
542:
539:
536:
533:
530:
527:
524:
519:
515:
504:
493:
490:
487:
484:
481:
478:
475:
472:
469:
466:
463:
460:
457:
454:
449:
445:
402:
399:
395:Richard K. Guy
375:
372:
338:
335:
82:Main article:
79:
76:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3312:
3301:
3298:
3296:
3293:
3292:
3290:
3275:
3272:
3270:
3269:
3265:
3261:
3260:
3256:
3255:
3254:
3251:
3250:
3248:
3244:
3238:
3235:
3234:
3232:
3228:
3222:
3219:
3217:
3214:
3212:
3209:
3207:
3204:
3203:
3201:
3197:
3187:
3184:
3182:
3179:
3177:
3174:
3173:
3171:
3167:
3161:
3158:
3157:
3155:
3151:
3148:
3144:
3138:
3135:
3133:
3130:
3128:
3125:
3123:
3120:
3118:
3115:
3113:
3110:
3108:
3107:Eric Limeback
3105:
3103:
3100:
3098:
3095:
3093:
3090:
3088:
3085:
3083:
3080:
3078:
3075:
3073:
3070:
3068:
3065:
3063:
3060:
3058:
3055:
3053:
3050:
3048:
3045:
3043:
3040:
3038:
3035:
3033:
3030:
3028:
3025:
3023:
3020:
3018:
3015:
3014:
3012:
3008:
3002:
2999:
2997:
2996:Rubik's Snake
2994:
2992:
2989:
2985:
2982:
2981:
2980:
2979:Rubik's Magic
2977:
2975:
2974:Rubik's Clock
2972:
2970:
2967:
2965:
2962:
2961:
2959:
2955:
2949:
2946:
2944:
2941:
2939:
2936:
2934:
2931:
2930:
2928:
2922:
2911:
2908:
2907:
2905:
2903:
2899:
2893:
2890:
2889:
2887:
2885:
2881:
2875:
2872:
2871:
2869:
2867:
2863:
2857:
2854:
2852:
2849:
2848:
2846:
2844:
2840:
2834:
2831:
2829:
2826:
2824:
2821:
2820:
2818:
2816:
2812:
2806:
2805:Skewb Diamond
2803:
2802:
2800:
2798:
2794:
2788:
2785:
2783:
2780:
2778:
2775:
2773:
2770:
2769:
2767:
2765:
2761:
2758:
2752:
2746:
2743:
2741:
2738:
2736:
2733:
2731:
2728:
2726:
2723:
2722:
2720:
2714:
2708:
2705:
2703:
2700:
2698:
2695:
2693:
2690:
2689:
2687:
2681:
2675:
2672:
2670:
2667:
2665:
2662:
2660:
2657:
2655:
2652:
2650:
2647:
2645:
2642:
2640:
2637:
2635:
2632:
2631:
2629:
2627:Rubik's Cubes
2625:
2619:
2616:
2614:
2611:
2609:
2606:
2604:
2601:
2599:
2598:Larry Nichols
2596:
2594:
2591:
2590:
2588:
2584:
2580:
2573:
2568:
2566:
2561:
2559:
2554:
2553:
2550:
2543:
2539:
2536:
2535:
2531:
2525:
2520:
2519:
2515:
2506:
2500:
2497:
2486:
2483:Tom Rokicki.
2479:
2476:
2473:
2468:
2465:
2461:
2456:
2453:
2447:
2442:
2435:
2432:
2427:
2420:
2413:
2410:
2407:
2402:
2399:
2396:
2391:
2388:
2384:
2379:
2376:
2368:
2361:
2358:
2347:
2343:
2337:
2334:
2331:
2326:
2323:
2320:
2315:
2312:
2301:
2294:
2291:
2288:, p. 30.
2287:
2282:
2279:
2276:, p. 26.
2275:
2270:
2267:
2264:, p. 16.
2263:
2258:
2255:
2244:on 2014-11-09
2243:
2239:
2232:
2230:
2228:
2226:
2222:
2219:
2214:
2211:
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2203:
2201:
2197:
2194:
2189:
2186:
2183:
2178:
2175:
2164:
2160:
2154:
2151:
2146:
2144:0-8018-6947-1
2140:
2136:
2132:
2131:
2123:
2120:
2109:
2105:
2099:
2097:
2095:
2093:
2089:
2078:
2074:
2068:
2066:
2062:
2051:
2047:
2041:
2038:
2031:
2029:
2025:
2022:
2017:
2013:
2010:
2009:supercomputer
2000:
1997:
1992:
1990:
1986:
1981:
1975:
1972:
1969:
1968:
1967:
1960:
1959:
1958:
1956:
1950:
1941:
1925:
1921:
1897:
1893:
1870:
1866:
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1137:
1133:
1124:
1120:
1097:
1093:
1075:
1071:
1055:
1051:
1028:
1024:
1001:
997:
974:
970:
947:
943:
934:
930:
921:
903:
899:
876:
872:
863:
860:
857:
853:
844:
822:
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776:
772:
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750:
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733:
728:
724:
720:
715:
711:
704:
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629:
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78:Move notation
77:
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69:
68:Rubik's Cubes
65:
60:
58:
54:
53:Cayley graphs
50:
45:
42:
41:
31:
27:
19:
3295:Rubik's Cube
3266:
3257:
3153:Speedsolving
3127:Collin Burns
3082:Frank Morris
3047:Rowe Hessler
2964:Missing Link
2815:Dodecahedron
2777:Pyraminx Duo
2685:Rubik's Cube
2579:Rubik's Cube
2541:
2523:
2499:
2488:. Retrieved
2478:
2467:
2455:
2434:
2428:. ACM Press.
2425:
2412:
2401:
2390:
2378:
2360:
2349:. Retrieved
2346:kociemba.org
2345:
2336:
2325:
2314:
2303:. Retrieved
2293:
2281:
2269:
2257:
2246:. Retrieved
2242:the original
2213:
2188:
2177:
2166:. Retrieved
2162:
2153:
2129:
2122:
2111:. Retrieved
2107:
2080:. Retrieved
2076:
2053:. Retrieved
2049:
2040:
2026:
2024:cube is 26.
2018:
2014:
2006:
1993:
1982:
1979:
1965:
1951:
1947:
1795:
1792:
1499:
1282:
1273:
1084:
840:
425:group theory
414:
404:
380:
377:
374:Upper bounds
364:Michael Reid
344:
340:
337:Lower bounds
330:
326:
322:
318:
314:
310:
306:
302:
298:
294:
293:The letters
292:
288:
287:
282:
278:
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270:
266:
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151:The letters
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146:
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130:prime symbol
125:
121:
117:
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109:
105:
101:
100:The letters
99:
95:
94:
87:
61:
46:
40:Rubik's Cube
37:
36:
26:
3199:Mathematics
3181:CFOP method
3160:Speedcubing
3137:Mátyás Kuti
3092:Gilles Roux
3087:Lars Petrus
3017:Yu Nakajima
2969:Rubik's 360
2957:Derivatives
2943:MagicCube7D
2938:MagicCube5D
2933:MagicCube4D
2851:Impossiball
2843:Icosahedron
2782:Pyramorphix
2764:Tetrahedron
2716:Other cubic
2707:Sudoku Cube
2608:Tony Fisher
2603:Uwe Mèffert
843:right coset
419:in 1981 by
391:John Conway
368:Jerry Bryan
351:conjectured
321:direction.
313:direction.
3289:Categories
3042:Kevin Hays
2797:Octahedron
2787:BrainTwist
2593:Ernő Rubik
2490:2010-02-19
2351:2018-11-27
2305:2013-07-28
2248:2013-07-26
2168:2017-03-19
2113:2017-02-20
2108:cube20.org
2082:2017-05-23
2077:cube20.org
2055:2017-02-20
2032:References
1016:, next to
430:cube group
64:algorithms
3211:Superflip
3146:Solutions
3117:Mats Valk
3077:Tyson Mao
2754:Non-cubic
2745:Gear Cube
2735:Dino Cube
2697:Bump Cube
2692:Void Cube
2446:0803.3435
1996:algorithm
1852:heuristic
1687:∖
1616:∖
1522:∖
1445:⟩
1381:⟨
1347:⟩
1311:⟨
1249:∖
1182:∖
1156:∖
1130:∖
940:∖
869:∖
786:⟩
708:⟨
674:⟩
610:⟨
576:⟩
526:⟨
492:⟩
456:⟨
355:superflip
49:diameters
3132:Max Park
3062:Toby Mao
3052:Leyan Lo
2892:Tuttminx
2823:Megaminx
2772:Pyraminx
2740:Square 1
2634:Overview
1983:Given a
1500:As with
3186:Optimal
3169:Methods
2912:(2x3x3)
2542:optimal
845:spaces
55:of the
2902:Cuboid
2141:
1985:random
1962:bound.
922:space
393:, and
120:, and
2856:Dogic
2730:Skewb
2441:arXiv
2422:(PDF)
2370:(PDF)
2163:Ruwix
920:coset
2139:ISBN
1955:IDA*
1856:IDA*
1638:and
1204:and
301:and
281:and
269:and
257:and
238:and
226:and
214:and
195:and
183:and
171:and
159:and
92:.
3291::
2424:.
2344:.
2224:^
2199:^
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2137:.
2106:.
2091:^
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2064:^
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1832:18
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1479:{
1476:=
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742:2
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721:,
716:2
712:L
705:=
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696:G
669:2
665:D
661:,
656:2
652:U
648:,
643:2
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635:,
630:2
626:F
622:,
619:R
616:,
613:L
607:=
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571:2
567:D
563:,
558:2
554:U
550:,
547:B
544:,
541:F
538:,
535:R
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523:=
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514:G
489:D
486:,
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477:B
474:,
471:F
468:,
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462:,
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331:2
327:F
323:z
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315:y
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126:2
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110:F
106:R
102:L
20:)
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