Pocket Cube

From left to right: original Pocket Cube, Eastsheen cube, V-Cube 2, V-Cube 2b.

The Pocket Cube (also known as the Mini Cube or the Ice Cube) is the 2×2×2 equivalent of a Rubik's Cube. The cube consists of 8 pieces, all corners.

History

In March 1970, Larry D. Nichols invented a 2×2×2 "Puzzle with Pieces Rotatable in Groups" and filed a Canadian patent application for it. Nichols's cube was held together with magnets. Nichols was granted U.S. Patent 3,655,201 on April 11, 1972, two years before Rubik invented his Cube.

Nichols assigned his patent to his employer Moleculon Research Corp., which sued Ideal in 1982. In 1984, Ideal lost the patent infringement suit and appealed. In 1986, the appeals court affirmed the judgment that Rubik's 2×2×2 Pocket Cube infringed Nichols's patent, but overturned the judgment on Rubik's 3×3×3 Cube.[1]

Permutations

Pocket Cube in different forms. From top (to bottom):
i. Solved pocket cube.
ii. Scrambled pocket cube.
iii.Pocket cube with one side tilted.

Any permutation of the eight corners is possible (8! positions), and seven of them 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. This is because all 24 possible positions and orientations of the first corner are equivalent due to the lack of fixed centers. This factor does not appear when calculating the permutations of N×N×N cubes where N is odd, since those puzzles have fixed centers which identify the cube's spatial orientation. The number of possible positions of the cube is

The maximum number of turns required to solve the cube is up to 11 half or quarter turns, or up to 14 quarter turns only.[2]

The number a of positions that require n any (half or quarter) turns and number q of positions that require n quarter turns only are:

n a q a(%) q(%)
0 1 1 0.000027% 0.000027%
1 9 6 0.00024% 0.00016%
2 54 27 0.0015% 0.00073%
3 321 120 0.0087% 0,0033%
4 1847 534 0.050% 0.015%
5 9992 2256 0.27% 0.061%
6 50136 8969 1.36% 0.24%
7 227536 33058 6.19% 0.90%
8 870072 114149 23.68% 3.11%
9 1887748 360508 51.38% 9.81%
10 623800 930588 16.98% 25.33%
11 2644 1350852 0.072% 36.77%
12 0 782536 0% 21.3%
13 0 90280 0% 2.46%
14 0 276 0% 0.0075%

For the miniature (2 × 2 × 2) Rubik’s cube, the two-generator subgroup (the number of positions generated just by rotations of two adjacent faces) is of order 29,160. [3]

Two simple algorithms that can be combined to solve the second layer:

1.To fix the positions of the corners: R U' L' U R' U' L

2.To adjust the corners: D R' D' R

World Records

Vicente Albíter of Mexico solving it in 1.55 seconds at the Mexican Open 2008

The world record solve is 0.49 seconds, set by Maciej Czapiewski of Poland on 20 March 2016 at Grudziądz Open 2016. [4]

The world record average of 5 solves (excluding fastest and slowest) is 1.51 seconds, set by Lucas Etter of the United States on 12 September 2015 at Music City Speedsolving 2015, with the times (1.24), 1.69, (2.21), 1.45, and 1.39 seconds. [5]

Top 5 solvers by single solve

Name Fastest solve Competition
Maciej Czapiewski 0.49s Grudziądz Open 2016
Michał Rzewuski 0.52s Grudziądz Open 2016
Mats Valk 0.56s Kaohsiung Open 2016
Rami Sbahi 0.58s Canadian Open 2015
Kim Roger Høyland Larsen 0.58s Sandnes Open 2016

Top 5 solvers by average of 5 solves

Name Fastest average Competition
Lucas Etter 1.51s Music City Speedsolving 2015
Martin Vædele Egdal 1.53s Lucia Open 2016
Rami Sbahi 1.55s Canadian Open 2015
Jayden McNeill 1.55s Australian Nationals 2015
Antonie Paterakis 1.55s Belgium Summer Open 2017

References

  1. ↑ "Moleculon Research Corporation v. CBS, Inc". Digital-law-online.info. Retrieved 2012-06-20.
  2. ↑ Jaapsch.net: Pocket Cube
  3. ↑ http://sporadic.stanford.edu/bump/match/morepolished.pdf
  4. ↑
  5. ↑ World Cube Association Official Results - 2×2×2 Cube.
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