# Virtual Cubes

### Revenge Cube

The Revenge Cube (also known als the 'Rubik's Revenge' or the 'Master Cube') is the 4x4x4 version of the Rubik's Cube. Invented by Péter Sebestény, the Rubik's Revenge was nearly called the 'Sebestény Cube' until a somewhat last-minute decision changed the puzzle's name to attract fans of the original Rubik's Cube. Unlike the original puzzle, it has no fixed facets: the centre facets (four per face) are free to move to different positions.

A Revenge Cube consists of 56 unique miniature cubes, also called 'cubies': 8 corners, 24 edges, and 24 centers. The 4x4x4 cube has a total of 96 stickers.

#### The number of possible positions

The number of possible positions of the cube is 8! × 37 × 24!2 / 4!6 × 24 ≈ 7.401 × 1045 which is about 7.4 septillion or 7.4 septilliard on the long scale or 7.4 quattuordecillion on the short scale.

The number of possible centre arrangements is 24! / 4!6 = 3'246'670'537'110'000 ≈ 3.246 × 1015 which is about 3.2 billiard on the long scale or 3.2 quadrillion on the short scale. In September 9, 2005 Jaap Scherphuis showed that the centers can always be solved in at most 22 moves. See also The 4x4x4 centres can be solved in 22 moves for further informations.

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

The first upper bound for the Rubik's Revenge was definded in July 7, 2006 by Bruce Norskog. In God's algorithm calculations for the 4x4x4 'squares set', posted in the Domain of the Cube Forum, he claims that the 4x4x4 can be solved for any legally scrambled configuration in no more than 79 slice turns (11 + 18 + 14 + 17 + 19 = 79)! A new lower bound was defined in the same thread as well: «At most 3629 + 3628 + 3627 + ... (approx. 1/6 of the number of positions of a 4x4) positions can be reached in 29 slice turns or less, so there are positions that require at least 30 slice turns.»

In February 14, 2007, Bruce Norskog completed a new five-stage analysis which shows that the 4x4x4 can always be solved using at most 68 block turns (10 + 15 + 12 + 15 + 16 = 68). See Solving the 4x4x4 in 68 turns for further informations.

In August 13, 2007, Bruce Norskog completed a new analyses for 'twist turns' and 'block turns' for stage 2 of his five-stage-solution. The total number of moves that could be required to perform all five stages is now 11 + 17 + 18 + 16 + 20 = 82 for twist turns, and 10 + 14 + 12 + 15 + 16 = 67 for block turns. See New upper bounds: 82 twist turns, 67 block turns for further informations.

In September 30, 2013, Shuang Chen lowered the upper bound to 57 moves in OBTM (outer-block-turn metric). His solving algorithm is based on Charles Tsai's 8-Step-Method. Shuang achieved better results by merging step 3 and 4 into one step. See also the article Solving the 4x4x4 in 57 moves (OBTM) for further informations. This drops the upper bound for BTM (block-turn metric) to 57 moves as well. Actually the value should even be lower, because of the larger set of basic moves in BTM.

In March 3, 2015, Tomas Rokicki confirmed the upper bound of 57 moves in OBTM. With the same code he also calculated new upper bounds of 56 moves in SSTM (single-slice-turn metric) and 53 moves in BTM (block-turn metric). In March 7 and 9, 2015, Shunag Chen further improved these upper bounds to 55 OBTM and 53 SSTM respectively. See 4x4x4 upper bounds: 57 OBTM confirmed; 56 SST and 53 BT calculated for further informations.

In July 18, 2011, Tomas Rokicki computed lower bounds for the N × N × N Rubik's Cubes using six different metrics. According to this, the 4x4x4 has a lower bound of 29 block turns. See also the article Lower Bounds for n x n x n Rubik's Cubes (through n=20) in Six Metrics for further informations.

#### The maximum permutation order

The maximum permutation order of a regular Revenge Cube is 765'765. It can be achieved by twisted corner 3-cycles and 5-cycles: 3 × (3 × 5) = (9 × 5); edge 7-cycle and 17-cycle: (7 × 17); center 11-cycle and 13-cycle: (11 × 13). This results in a Least Common Multiple (LCM) of (9 × 5) × (7 × 17) × (11 × 13) = 765'765. The order can be reached by the following algorithm found by Tony Forbes: NF NU2 B' TD2 TU (5 btm). Note, that the maxium permutation order for a Super Cube version is higher.

In January 12, 2008 Max Daniel, Benjamin Lipp and Jakob von Raumer proved for the 'Jugend Forscht' project, that the maximum permutation order of the Revenge Cube is 765'765. Their algorithm to reach it was: U TL NB' · CR' (3 btm). They discovered, that possible orders can be computed as the LCM from three values, where one value has to be from the list S8 (corner orders) and two values from the list S24 (edge and center orders).

S8: 2, 3, 4, 5, 6, 7, 8, 10, 12, 15.

S24: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 28, 30, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 48, 51, 52, 55, 56, 57, 60, 63, 65, 66, 68, 70, 72, 76, 77, 78, 80, 84, 85, 88, 90, 91, 95, 99, 102, 104, 105, 110, 112, 114, 117, 119, 120, 126, 130, 132, 140, 143, 154, 156, 165, 168, 170, 180, 182, 195, 198, 204, 210, 220, 231, 234, 240, 252, 260, 264, 273, 280, 308, 312, 315, 330, 360, 364, 385, 390, 396, 420, 440, 462, 504, 630, 660, 840.

This will deliver 360'360 and 255'255 as the highest values: LCM (5, 504, 143) = 360'360; LCM (15, 119, 143) = 255'255. However, this isn't already the maximal order, because flips have been ignored. From the total order that also regards flips, the relation to the orders above is either the factor 1 or 3. Since for all permutation orders of 360'360 the factor is 1 and for 255'255 the factor 3 exists, the maximum order is therefore 255'255 × 3 = 765'765. See also the German document Gruppentheorie am 4-Rubikwürfel for further informations.