Pocket Cube
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The pocket cube is the 2×2×2 equivalent of a Rubik's cube. The cube consists of 8 corner cubelets, and no other types of cubelets.
Any permutation of the 8 corner cubelets is possible (8! positions), and 7 of the cubelets can be independently rotated (37 positions). There is nothing identifying the orientation of the cube in space, reducing the positions by a factor of 24. The number of possible positions of the cube is
- <math>\frac{8!\,3^7}{24}=7!\,3^6=3674160<math>
which factors as <math>2^4 3^8 5^1 7^1<math>.
The maximum number of twists required to solve the cube is up to 11 full twists, or up to 14 quarter twists. An optimal (least number of twists) solution from any position can be found by a computer with a brute force algorithm.
The number f of positions that require n full twists and number q of positions that require n quarter turn twists are
| n | f | q |
|---|---|---|
| 0 | 1 | 1 |
| 1 | 9 | 6 |
| 2 | 54 | 27 |
| 3 | 321 | 120 |
| 4 | 1847 | 534 |
| 5 | 9992 | 2256 |
| 6 | 50136 | 8969 |
| 7 | 227536 | 33058 |
| 8 | 870072 | 114149 |
| 9 | 1887748 | 360508 |
| 10 | 623800 | 930588 |
| 11 | 2644 | 1350852 |
| 12 | 782536 | |
| 13 | 90280 | |
| 14 | 276 |