# Virtual Cubes

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### Rubik's Cube

The Rubik's Cube is a 3-D mechanical puzzle invented in 1974 by Hungarian sculptor and professor of architecture Ernö Rubik. Each of the six faces of a Rubik's Cube is covered by nine stickers, among six solid colors (traditionally blue, green, red, orange, yellow, and white). A pivot mechanism enables each face to turn independently, thus mixing up the colors. For the puzzle to be solved, each face must be a solid color.

A Rubik's Cube consists of 26 unique miniature cubes, also called 'cubies': 8 corners, 12 edges, and 6 centers. The six centers are affixed to the core mechanism. The 3x3x3 cube has a total of 54 stickers.

#### The number of possible positions

The number of possible positions of the cube is 8! × 37 × 12! / 2 × 211 = 43'252'003'274'489'856'000 ≈ 43.252 × 1018 which is about 43.3 quintillion on the short scale or 43.3 trillion on the long scale.

The Rubik's Cube has eight corners and twelve edges. There are 8! (40'320) ways to arrange the corners. Seven can be oriented independently, and the orientation of the eighth depends on the preceding seven, giving 37 (2'187) possibilities. There are 12! / 2 (239'500'800) 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 (2'048) possibilities.

A Rubik's Cube usually has no orientation markings on the centers and therefore solving it does not require any attention orienting those parts correctly. Marking the Rubik's Cube's centers increases its difficulty because this expands the set of distinguishable possible positions. There are 46 / 2 (2'048) ways to orient the centers, since an even permutation of the corners implies an even number of quarter turns of the centres as well. An alternative explanation is: There are 45 × 2 = 211 (2'048) ways to orient the centers, since five of the six centers can be twisted independently of one another into each of four different orientations, and for the sixth center there still exist two possibilities. Thus orientations of centers increases the total number of possible positions from 43'252'003'274'489'856'000 ≈ 43.252 × 1018 to 88'580'102'706'155'225'088'000 ≈ 88.58 × 1021.

#### The diameter of the Rubik's Cube

In July 2010, Tomas Rokicki, Herbert Kociemba, Morley Davidson, and John Dethridge proved the so-called God's Number to be 20 in face-turn metric (ftm). This means that every position of Rubik's Cube can be solved in twenty moves or less. Distance-20 positions or antipodes (positions that are maximally far from solved) are both rare and plentiful; they are rarer than one in a billion positions, yet there are probably more than one hundred million such positions. The exact number of antipodes is not yet known.

In August 2014, Tomas Rokicki and Morley Davidson proved that God's Number in the quarter-turn metric (qtm) is 26. In qtm, only a single position (Superflip composed with four spots) plus its two rotations is known that requires the maximum of 26 moves.

The slice turn metric allows any quarter- or half-turn of a middle 'slice' as one move. (There seems to be less interest in counting only slice quarter turns.) Since these allowed moves include the HTM moves, the upper bound here is 20. The conjecture is that God's number in this STM is 18. See also What is God's Number if Slice Moves are Allowed? for further informations.

In the Super Group, where the rotations of the centers are regarded as well, the number of possible positions is 2'048 times higher then in the regular Cube Group (46/2 = 2'048). Walter Randelshofer discovered in July 19, 2014 that the 'Pure Superflip' (Superflip with untouched centers) can be solved optimally in 24 moves in face-turn metric (ftm). In July 22, 2014 he also found that the '180° Superflip' (Superflip with all centers rotated around 180°) requires 20 moves in slice-turn metric (stm). Bruce Norskog assumed that there must exist positions that require at least 28 moves in quarter-turn metric (qtm). This was confirmed in September 9, 2014 by Herbert Kociemba for a specific 'Superflip composed with four spots' position, where all centers except the upper one are rotated around 180°. See also the article Lower bounds for the 3x3x3 Super Group for further informations.

More generally, it has been shown that an N × N × N Rubik's Cube can be solved optimally in Θ(n2 / log(n)) moves. See also Algorithms for Solving Rubik's Cubes by Erik D. Demaine, Martin L. Demaine, Sarah Eisenstat, Anna Lubiw and Andrew Winslow.

#### Possible orders

In July 27, 1981 Jesper C. Gerved and Torben Maack Bisgaard proved that there are 73 different possible orders in the group of the Rubik's Cube. The orders are: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 18, 20, 21, 22, 24, 28, 30, 33, 35, 36, 40, 42, 44, 45, 48, 55, 56, 60, 63, 66, 70, 72, 77, 80, 84, 90, 99, 105, 110, 112, 120, 126, 132, 140, 144, 154, 165, 168, 180, 198, 210, 231, 240, 252, 280, 315, 330, 336, 360, 420, 462, 495, 504, 630, 720, 840, 990, 1'260.

The most common order is 60 with a total of 4'199'961'633'799'421'952 elements (9.710% of all cube positions), followed by the orders 36 (8.712%), 24 (7.615%), 12 (5.387%) and 180 (5.364%). Th most uncommon order is 11 with a total of 44'590'694'400 elements (0.000'000'103% of all cube positions), followed by the orders 2 (0.000'000'395%) and 3 (0.000'078%). The maximal order is 1'260 and counts 51'490'480'088'678'400 elements (0.119% of all cube positions).

See the document Number of elements of each order in the group of Rubik's Cube for further informations.

In June 2022 Ben Whitmore messaged Tomas Rokicki that the numbers he had published on SpeedSolving.com in 2021 disagreed with the numbers found by Jesper C. Gerved and Torben Maack Bisgaard. Both, Lucas Garron and Herbert Kociemba confirmed, that the numbers computed in 1981 had been wrong. See also this post from TwistyPuzzles.com.

Order Algorithm Order Algorithm Order Algorithm Order Algorithm
1   24 U R2 D' (3 ftm) 80 U R B (3 ftm) 240 U R' D2 F' (4 ftm)
2 U2 (1 ftm) 28 U R D R' (4 ftm) 84 U R F (3 ftm) 252 U R D F (4 ftm)
3 (U2 R2)2 (4 ftm) 30 U R2 (2 ftm) 90 U R D (3 ftm) 280 U R U L F' (5 ftm)
4 U (1 ftm) 33 U R F' D' (4 ftm) 99 U2 R D2 F (4 ftm) 315 U R D L (4 ftm)
5 U R U R' (4 ftm) 35 U R2 D' R2 (4 ftm) 105 U R (2 ftm) 330 U' R' F' D L2 (5 ftm)
6 U2 R2 (2 ftm) 36 U R B2 (3 ftm) 110 U R D2 B R D L' (7 ftm) 336 U R U F D2 (5 ftm)
7 U R U' F (4 ftm) 40 U R D R2 (4 ftm) 112 U R D2 B L' B2 (6 ftm) 360 U R F' (3 ftm)
8 U R2 D (3 ftm) 42 U R' U2 R2 (4 ftm) 120 U R D' F (4 ftm) 420 U R D L' (4 ftm)
9 U R F2 (3 ftm) 44 U' R F' D (4 ftm) 126 U' R' F' L (4 ftm) 462 U' R F' D2 L' (5 ftm)
10 U' R' F' R (4 ftm) 45 U R D R (4 ftm) 132 U R D F' (4 ftm) 495 U R D' R2 B' (5 ftm)
11 (U R' L F D2)2 (10 ftm) 48 U2 R U F (4 ftm) 140 U R' U F' (4 ftm) 504 U' R D2 F (4 ftm)
12 U2 R' F2 (3 ftm) 55 U R D F' R' B' (6 ftm) 144 U R' F' D2 (4 ftm) 630 U R' F' D L2 (5 ftm)
14 U R U B R' F (6 ftm) 56 U' R F' D2 (4 ftm) 154 U R U F L D' (6 ftm) 720 U' R' U' F' D2 (5 ftm)
15 (U R2)2 (4 ftm) 60 U R' F' (3 ftm) 165 U R U2 L' F (5 ftm) 840 U' R F D2 (4 ftm)
16 U R U' F D (5 ftm) 63 U R' (2 ftm) 168 U R D2 (3 ftm) 990 U R U B L' D' (6 ftm)
18 U' R F' D2 L (5 ftm) 66 U R2 F' D' L' (5 ftm) 180 U R D' (3 ftm) 1'260 U R' U F' D2 (5 ftm)
20 U' R D2 R' (4 ftm) 70 U R B2 D B2 (5 ftm) 198 U' R F2 D2 (4 ftm)
21 U' R2 D' F2 (4 ftm) 72 U R' F' D (4 ftm) 210 U R' D L' (4 ftm)
22 U R' L F D2 (5 ftm) 77 U R D B' (4 ftm) 231 U R F' D (4 ftm)

#### The maximum permutation order

The maximum permutation order of a regular Rubik's Cube is 1'260. It can be achieved by twisted corner 3-cycles and 5-cycles: 3 × (3 × 5) = (9 × 5); twisted edge 2-cycles and 7-cycles: 2 × (2 × 7) = (4 × 7). This results in a Least Common Multiple (LCM) of (9 × 5) × (4 × 7) = 1'260. The order can be reached by the following algorithm found by David Singmaster: B2 R' F D' F (5 ltm). Note, that the maxium permutation order for a Super Cube version is higher.

#### The difficulty of a Permutation Puzzle

TwistyPuzzle.com member kastellorizo proposed in 2010 a formula which allowes to classify the difficulty of Permutation Puzzles:

cc × Σ [ n_i × ( L_i × l_i ) / ( H_i × h_i ) ]

First let us assume that a puzzle has i different generators g_i, and each such generator is repeated n_i times (Generators are considered to be repeated, if their action results in a similar change). For example the 3x3x3 has one type of generator, but it can be found six times. The 2x2x2 has one type, repeated three times. The Pyraminx has two types, the trivial tip generator (repeated four times) and another deeper cut generator (also repeated four times).

Each generator is defined it by its cutting movement. It divides the puzzle into two parts, and each part has a number of pieces presented as stickers or tiles (for most puzzles that is). So we get two numbers, L_i is the low and H_i is the high number (when compared). Finally, we check how many 'different type' of pieces each part has, and we define them as l_i and h_i respectively for the parts with low number and high number. For example, the 3x3x3 has a total of three different type of pieces, the 2x2x2 has one, the Pyraminx has three, and so on.

The shape of a puzzle plays a huge role as well. Therefore, the entire previous formula has to be multiplyed by the concentration coefficient 'cc' of the puzzle. It is the maximum number of pieces which are adjacent (with edges) to one piece. For example, the 'cc' for the Dogic, the Pyraminx, is equal to 3, for the 3x3x3 it is equal to 4, and for the Megaminx, it is 5! For the Boob Cube it is 1, and for the 2x2x1 it is 2.

1x1x1: 0.0000 (no generators)
1x1x2 (Boob Cube): 1*1*(5*2)/(5*2) = 1.0000
1x2x2: 2*2*(8*1)/(8*1) = 4.0000
Floppy Cube 1x3x3: 4*(4*(11*1)/(19*2)) = 4.6316
Chronos: 4*(6*(1*1)/(5*1)) = 4.8000
Braintwist: 2*(8*(6*1)/(18*1)) = 5.3333
Rubik's Magic (4-tiles): 2*(2*(8*2)/(8*2)+1*(8*2)/(8*2)) = 6.0000
Orbik: 2*(12*(2*1)/(12*1)) = 6.0000
Pyraminx: 3*[4*(3*1)/(33*3) + 4*(12*3)/(24*3)] = 6.3636
15-Puzzle: 4*(24*(1*1)/(14*1)) = 6.8571
Dino Cube: 3*(8*(6*1)/(21*1)) = 6.8571
Rubik's Rings: 4*(2*(16*1)/(18*1)) = 7.1111
Magic Octahedron: 3*[8*(4*1)/(68*3) + 8*(16*3)/(56*3)] = 7.3278
Helicopter Cube: 3*12*(10*2)/(48*2) = 7.5000 (jumbling is not counted!)
Octo: 4*(1*(16*1)/(16*1)+8*(4*1)/(28*1)) = 8.5714
Orbit: 2*(1*(28*1)/(28*1)+2*(8*1)/(48*1)+2*(20*1)/(36*1)+2*(28*1)/(28*1)) = 8.8889
2x2x2: 3*3*(12*1)/(12*1) = 9.0000
2x2x3: 4*(2*(12*1)/(20*2)+2*(16*2)/(16*2)) = 10.4000
Impossiball: 3*(12*(5*1)/(15*1) = 12.0000
Puck: 2*(6*(6*1)/(6*1)) = 12.0000
Cubedron: 4*12*(5*2)/(20*2) = 12.0000
2x2x4: 4*(2*(12*1)/(28*2)+2*(20*2)/(20*2)+1*(20*2)/(20*2)) = 12.8571
Dogic: 3*[12*(5*1)/(75*2) + 12*(20*2)/(60*2)] = 13.2000
Columbus Egg: 2*(10*(20*1)/(30*1)) = 13.3333
Rubik's Clock: 1*(8*(5*4/13*6)+8*(11*4)/(7*6)+4*(9*4)/(9*6)+2*(5*4)/(13*6)) = 13.6117
Rubik's Magic (8-tiles): 2*(2*(16*2)/(16*2)+1*(16*2)/(16*2)+4*(16*2)/(16*2)) = 14.0000
Megaminx: 5*12*(26*3)/(106*3) = 14.7170
Pyraminx Crystal: 3*12*(35*2)/(85*2) = 14.8236
3x3x3: 4*6*(21*3)/(33*3) = 15.2728
Skewb = 4*4*(15*2)/(15*2) = 16.0000
Astrolabacus: 2*(6*(18*1)/(18*1)+6*(6*1)/(18*1)) = 16.0000
Void Cube: 4*6*(20*2)/(28*2) = 17.1428
24 Cube: 3*6*(12*1)/(12*1) = 18.0000
FT Octahedron: 4*8*(27*3)/(45*3) = 19.2000
Masterball: 4*(2*(8*1)/(24*2)+1*(16*1)/(16*1)+4*(16*2)/(16*2)) = 21.3333
4x4x4: 4*[6*(32*4)/(64*4) + 3*(48*4)/(48*4)] = 24.0000
5x5x5: 4*[6*(45*7)/(105*7) + 6*(65*7)/(85*7)] = 28.6388
7x7x7: 4*[6*(77*12)/(217*12) + 6*(105*12)/(189*12) + 6*(133*12)/(161*12)] = 41.3760

See this TwistyPuzzles.com thread for further informations.