Virtual Cubes

Pocket Cube

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Pocket Cube

The Pocket Cube (also known as the 'Mini Cube'), invented by Ernö Rubik, is the 2x2x2 equivalent of the Rubik's Cube.

A Pocket Cube consists of 8 unique miniature cubes, also called 'cubies': 8 corners, and no other types of cubies. The 2x2x2 cube has a total of 24 stickers.

The number of possible positions

The number of possible positions of the cube is 8! × 37 / 24 = 7! × 36 = 3'674'160 ≈ 3.674 × 106 which is about 3.7 million.

The diameter of the Pocket Cube

God's Number for the Pocket Cube is known since 1981. It requires at least 11 moves in face-turn metric (ftm) or 14 moves in quarter-turn metric (qtm) to solve any position of the Pocket Cube. A total of 2'644 antipodes (positions that are maximally far from solved) exist for FTM and 276 for QTM. A total of 108 positions require 11 face-turns and 14 quarter-turns, making them antipodes in both, FTM and QTM. Reduced by symmetry there are 77 symmetry distinct antipodes in FTM and and 10 in QTM. For further informations see also the Cube Lovers post God's Algorithm for the 2x2x2 Pocket Cube and the Mini Cube page by Jaap Scherphuis.

In May 17, 2014 Bruce D. MacKenzie discovered that, reduced in anti-symmetry, there exist 8 symmetric distinct antipodes in quarter-turn-metric. See also the article 2x2x2 Cube for further informations. In May 5, 2016, Walter Randelshofer stated that from these 8 symmetric distinct antipodes in QTM, 4 symmetric distinct positions are antipodes in both, QTM and FTM:
1. L F2 L' D' L U2 B' D' R D' R2 (11* ftm, 14* qtm)
2. R B R2 U2 B R' B U' R2 U L' (11* ftm, 14* qtm)
3. B2 R F' D R2 B' R D B' L B2 (11* ftm, 14* qtm)
4. L2 U B2 L' U' L B' L B' U2 B (11* ftm, 14* qtm)

Possible orders

In November 11, 2015 Walter Randelshofer stated that there are 17 different possible orders in the group of the Pocket Cube. The orders are: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 15, 18, 21, 30, 36, 45.

The maximum permutation order

The maximum permutation order of the Pocket Cube is 45. It can be achieved by twisted corner 3-cycles and 5-cycles: 3 × (3 × 5) = (9 × 5). This results in a Least Common Multiple (LCM) of (9 × 5) = 45. The order can be reached by the following algorithm found by David Singmaster: B2 R' F D' F (5 ftm).

© Walter Randelshofer. All rights reserved.