# Virtual Cubes

### Professor Cube

The Professor's Cube invented by Udo Krell is the 5x5x5 version of the Rubik's Cube. It has qualities in common with both the original 3x3x3 Rubik's Cube and the 4x4x4 Rubik's Revenge, and knowing how to solve either can help when working on the 5x5x5 cube.

A Professor's Cube consists of 98 unique miniature cubes, also called 'cubies': 8 corners, 36 edges, and 54 centers. The six true centers are affixed to the core mechanism. The 5x5x5 cube has a total of 150 stickers.

#### The number of possible positions

The number of possible positions of the cube is 8! × 37 × 12! × 210 × 24!3 / 4!12 ≈ 2.828 × 1074 which is about 282.8 duodecillion on the long scale or 282.8 tresvigintillion on the short scale.

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

In December 30, 2010, Jaap Scherphuis roughly defines the lower bound to be at least 50 moves. He also states, that currently no good upper bounds for the 5x5x5 exist at all. See also the article Bigger Rubik Cube bound 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 5x5x5 has a lower bound of 42 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 Professor's Cube is 281'801'520. It can be achieved by twisted corner 3-cycles and 5-cycles: 3 × (3 × 5) = (9 × 5); twisted edge 4-cycle and 8-cycle: 2 × (8); edge 7-cycle and 17-cycle: (7 × 17); center 11-cycle, 13-cycle and 23-cycle: (11 × 13 × 23). This results in a Least Common Multiple (LCM) of (9 × 5) × (2 × 8) × (7 × 17) × (11 × 13 × 23) = 281'801'520. The order can be reached by the following algorithm found by Tony Forbes: NF NR' L2 F2 B U2 ND NL R F U (11 btm). Note, that the maximum permutation order for a Super Cube version is higher.