Pocket Cube
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The Pocket Cube (also known as the Mini Cube and Twizzle) is a 2×2×2
combination puzzle In mathematics, a combination is a selection of items from a set (mathematics), set that has distinct members, such that the order of selection does not matter (unlike permutations). For example, given three fruits, say an apple, an orange and a ...
invented in 1970 by American puzzle designer Larry D. Nichols. The
cube A cube or regular hexahedron is a three-dimensional space, three-dimensional solid object in geometry, which is bounded by six congruent square (geometry), square faces, a type of polyhedron. It has twelve congruent edges and eight vertices. It i ...
consists of 8 pieces, which are all corners.


History

In February 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 on April 11, 1972, two years before Rubik invented the 3×3×3 cube. Nichols assigned his
patent A patent is a type of intellectual property that gives its owner the legal right to exclude others from making, using, or selling an invention for a limited period of time in exchange for publishing an sufficiency of disclosure, enabling discl ...
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.


Group Theory

The
group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ( ...
of the 3×3×3 cube can be transferred to the 2×2×2 cube. The elements of the group are typically the moves of that can be executed on the cube (both individual rotations of layers and composite moves from several rotations) and the group operator is a concatenation of the moves. To analyse the group of the 2×2×2 cube, the cube configuration has to be determined. This can be represented as a 2-tuple, which is made up of the following parameters: * Position of the corner pieces as a
bijective In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equival ...
function (
permutation In mathematics, a permutation of a set can mean one of two different things: * an arrangement of its members in a sequence or linear order, or * the act or process of changing the linear order of an ordered set. An example of the first mean ...
) * Orientation of the corner pieces as
vector Vector most often refers to: * Euclidean vector, a quantity with a magnitude and a direction * Disease vector, an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematics a ...
x Two moves M_1and M_2 from the set A_Mof all moves are considered equal if they produce the same configuration with the same initial configuration of the cube. With the 2×2×2 cube, it must also be considered that there is no fixed orientation or top side of the cube, because the 2×2×2 cube has no fixed center pieces. Therefore, the
equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric, and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. A simpler example is equ ...
\sim is introduced with M_1 \sim M_2 := M_1 and M_2 result in the same cube configuration (with optional rotation of the cube). This relation is reflexive, as two identical moves transform the cube into the same final configuration with the same initial configuration. In addition, the relation is symmetrical and transitive, as it is similar to the mathematical relation of equality. With this equivalence relation,
equivalence class In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements ...
es can be formed that are defined with M := \ \subseteq A_M on the set of all moves A_M. Accordingly, each equivalence class /math> contains all moves of the set A_M that are equivalent to the move with the equivalence relation. /math> is a subset of A_M. All equivalent elements of an equivalence class /math> are the representatives of its equivalence class. The
quotient set In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements ...
A_M / \sim can be formed using these equivalence classes. It contains the equivalence classes of all cube moves without containing the same moves twice. The elements of A_M / \sim are all equivalence classes with regard to the equivalence relation \sim . The following therefore applies: A_M / \sim := \. This quotient set is the set of the group of the cube. The 2×2×2 Rubik's cube, has eight permutation objects (corner pieces), three possible orientations of the eight corner pieces and 24 possible rotations of the cube, as there is no unique top side. Any permutation of the eight corners is possible (8 ! positions), and seven of them can be independently rotated with three possible orientations (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 (similar to what happens in circular permutations). 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 :\frac=7! \times 3^6=3,674,160. This is the order of the group as well. The largest order of an element in this group is 45. For example, one such element of order 45 is :(UR^2L'). Any cube configuration can be solved in up to 14 turns (when making only quarter turns) or in up to 11 turns (when making half turns in addition to quarter turns).Jaapsch.net: Pocket Cube
/ref> 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: The two-generator subgroup (the number of positions generated just by rotations of two adjacent faces) is of order 29,160. Code that generates these results can be found here.


Methods

A pocket cube can be solved with the same methods as a 3x3x3 Rubik's cube, simply by treating it as a 3x3x3 with solved (invisible) centers and edges. More advanced methods combine multiple steps and require more algorithms. These algorithms designed for solving a 2×2×2 cube are often significantly shorter and faster than the algorithms one would use for solving a 3×3×3 cube. The Ortega method, also called the Varasano method, is an intermediate method. First a face is built (but the pieces may be permuted incorrectly), then the last layer is oriented (OLL) and lastly both layers are permuted (PBL). The Ortega method requires a total of 12 algorithms. The CLL method first builds a layer (with correct permutation) and then solves the second layer in one step by using one of 42 algorithms. A more advanced version of CLL is the TCLL Method also known as Twisty CLL. One layer is built with correct permutation similarly to normal CLL, however one corner piece can be incorrectly oriented. The rest of the cube is solved, and the incorrect corner orientated in one step. There are 83 cases for TCLL. One of the more advanced methods is the EG method. It starts by building a face like in the Ortega method, but then solves the rest of the puzzle in one step. It requires knowing 128 algorithms, 42 of which are the CLL algorithms. Top-level speedcubers may also 1-look the puzzle, which involves inspecting the entire cube and planning out the entire solution in the 15 seconds of inspection allotted to the solver before the solve, with the best solvers being able to plan more than one solution, considering movecount and ergonomics of each.


Notation

Notation is based on 3×3×3 notation but some moves are redundant (All moves are 90°, moves ending with ‘2’ are 180° turns): *R represents a clockwise turn of the right face of the cube *U represents a clockwise turn of the top face of the cube *F represents a clockwise turn of the front face of the cube *R' represents an anti-clockwise turn of the right face of the cube *U' represents an anti-clockwise turn of the top face of the cube *F' represents an anti-clockwise turn of the front face of the cube


World records

The world record for the fastest single solve time is 0.43 seconds, set by Teodor Zajder of
Poland Poland, officially the Republic of Poland, is a country in Central Europe. It extends from the Baltic Sea in the north to the Sudetes and Carpathian Mountains in the south, bordered by Lithuania and Russia to the northeast, Belarus and Ukrai ...
at Warsaw Cube Masters 2023. The world record average of 5 solves (excluding fastest and slowest) is 0.88 seconds set by Yiheng Wang (王艺衡) of
China China, officially the People's Republic of China (PRC), is a country in East Asia. With population of China, a population exceeding 1.4 billion, it is the list of countries by population (United Nations), second-most populous country after ...
at Hangzhou Open 2024 with the times of (1.26), (0.84), 0.91 0.89, and 0.85 seconds. World Cube Associationbr>Official Results – 2×2×2 Cube
An average of 0.78 seconds was set by Wang previously with times of 0.74, (0.70), (0.97), 0.78, and 0.81 seconds, but frame-by-frame analysis revealed his use of 'sliding,' a technique breaking several of the World Cubing Association's (WCA) regulations. Yiheng Wang also set a record of 0.86 seconds but got penalized for the same reason After deliberation between the WCA's Board of Directors and WCA Regulations Committee, Wang was retroactively penalized with additional seconds added to four of his solves.


Top 5 solvers by single solve


Top 5 solvers by Olympic average of 5 solves


See also

* Rubik's Cube (3×3×3) * Rubik's Revenge (4×4×4) * Megaminx * Professor's Cube (5×5×5) * V-Cube 6 (6×6×6) * V-Cube 7 (7×7×7) * V-Cube 8 (8×8×8) *
Combination puzzle In mathematics, a combination is a selection of items from a set (mathematics), set that has distinct members, such that the order of selection does not matter (unlike permutations). For example, given three fruits, say an apple, an orange and a ...


References


External links


Methods for speedsolving the 2×2×2code for enumerating all permutations of a Rubik's cube
{{Rubik's Cube Rubik's Cube