Rubik's Cube in the context of Abstract group

Trillion is a number with two distinct definitions:

• 1,000,000,000,000, i.e. one million million, or 10 (ten to the twelfth power), as defined on the short scale. This is now the meaning in both American and British English.
• 1,000,000,000,000,000,000, i.e. 10 (ten to the eighteenth power), as defined on the long scale. This is one million times larger than the short scale trillion. This is the historical meaning in English and the current use in many non-English-speaking countries where trillion and billion 10 (ten to the twelfth power) maintain their long scale definitions.

A combination puzzle, also known as a sequential move puzzle, is a puzzle which consists of a set of pieces which can be manipulated into different combinations by a group of operations. Many such puzzles are mechanical puzzles of polyhedral shape, consisting of multiple layers of pieces along each axis which can rotate independently of each other. Collectively known as twisty puzzles, the archetype of this kind of puzzle is the Rubik's Cube. Each rotating side is usually marked with different colours, intended to be scrambled, then solved by a sequence of moves that sort the facets by colour. Generally, combination puzzles also include mathematically defined examples that have not been, or are impossible to, physically construct.

Ernő Rubik (Hungarian: [ˈrubik ˈɛrnøː]; born 13 July 1944) is a Hungarian architect and inventor, widely known for creating the Rubik's Cube (1974), Rubik's Magic, and Rubik's Snake.

While Rubik became famous for inventing the Rubik's Cube and his other puzzles, much of his recent work involves the promotion of science in education. Rubik is involved with several organizations such as Beyond Rubik's Cube, the Rubik Learning Initiative and the Judit Polgar Foundation, all of which aim to engage students in science, mathematics, and problem solving at a young age.

Pentangle, later Pentangle Puzzles, was a British manufacturer and distributor of burr puzzles and other mechanical puzzles. It operated in the UK from 1971 until 2018. It was best known as the first company to distribute what became called "Rubik's Cube" outside Hungary.

Ideal Toy Company was an American toy company founded by Morris Michtom and his wife, Rose. During the post–World War II baby boom era, Ideal became the largest doll-making company in the United States. Their most popular dolls included Betsy Wetsy, Toni, Saucy Walker, Shirley Temple, Miss Revlon, Patti Playpal, Tammy, Thumbelina, Tiny Thumbelina, and Crissy. The company is also known for selling the Rubik's Cube.

Thomas Kremer (26 May 1930 – 24 June 2017) was a game inventor and marketer who acquired the rights to market the Rubik's Cube.

Tom Kremer was a games designer, entrepreneur and publisher, best known for his discovery and popularisation of the Rubik's Cube. As an octogenarian he founded the publishing house Notting Hill Editions, with the aim of reinvigorating the lost art of the essay.

In mathematics, a group is a set with an operation that combines any two elements of the set to produce a third element within the same set and the following conditions must hold: the operation is associative, it has an identity element, and every element of the set has an inverse element. For example, the integers with the addition operation form a group.

The concept of a group was elaborated for handling, in a unified way, many mathematical structures such as numbers, geometric shapes and polynomial roots. Because the concept of groups is ubiquitous in numerous areas both within and outside mathematics, some authors consider it as a central organizing principle of contemporary mathematics.

In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are sets with specific operations acting on their elements. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term abstract algebra was coined in the early 20th century to distinguish it from older parts of algebra, and more specifically from elementary algebra, the use of variables to represent numbers in computation and reasoning. The abstract perspective on algebra has become so fundamental to advanced mathematics that it is simply called "algebra", while the term "abstract algebra" is seldom used except in pedagogy.

Algebraic structures, with their associated homomorphisms, form mathematical categories. Category theory gives a unified framework to study properties and constructions that are similar for various structures.

In abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as groups endowed with additional operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right.

Various physical systems, such as crystals and the hydrogen atom, and three of the four known fundamental forces in the universe, may be modelled by symmetry groups. Thus group theory and the closely related representation theory have many important applications in physics, chemistry, and materials science. Group theory is also central to public key cryptography.

A mechanical puzzle is a puzzle presented as a set of mechanically interlinked pieces in which the solution is to manipulate the whole object or parts of it. While puzzles of this type have been in use by humanity as early as the 3rd century BC, one of the most well-known mechanical puzzles of modern day is the Rubik's Cube, invented by the Hungarian architect Ernő Rubik in 1974. The puzzles are typically designed for a single player, where the goal is for the player to discover the principle of the object, rather than accidentally coming up with the right solution through trial and error. With this in mind, they are often used as an intelligence test or in problem solving training.

Rubik's Magic, like the Rubik's Cube, is a mechanical puzzle invented by Ernő Rubik and first manufactured by Matchbox in the mid-1980s.

The puzzle consists of eight black square tiles (changed to red squares with goldish rings in 1997) arranged in a 2 × 4 rectangle; diagonal grooves on the tiles hold wires that connect them, allowing them to be folded onto each other and unfolded again in two perpendicular directions (assuming that no other connections restrict the movement) in a manner similar to a Jacob's ladder toy. The front side of the puzzle shows, in the initial state, three separate, rainbow-colored rings; the back side consists of a scrambled picture of three interconnected rings. The goal of the game is to fold the puzzle into a heart-like shape and unscramble the picture on the back side, thus interconnecting the rings.

A Rubik's Snake (also Rubik's Twist, Rubik's Transformable Snake, Rubik's Snake Puzzle) is a toy with 24 wedges that are right isosceles triangular prisms. The wedges are connected by spring bolts, so that they can be twisted, but not separated. By being twisted, the Rubik's Snake can be made to resemble a wide variety of objects, animals, or geometric shapes. Its "ball" shape in its packaging is a non-uniform concave rhombicuboctahedron.

The snake was invented by Ernő Rubik, better known as the inventor of the Rubik's Cube.

The Rubik's Cube group ${\displaystyle (G,\cdot )}$ represents the mathematical structure of the Rubik's Cube mechanical puzzle. Each element of the set ${\displaystyle G}$ corresponds to a cube move, which is the effect of any sequence of rotations of the cube's faces. With this representation, not only can any cube move be represented, but any position of the cube as well, by detailing the cube moves required to rotate the solved cube into that position. Indeed with the solved position as a starting point, there is a one-to-one correspondence between each of the legal positions of the Rubik's Cube and the elements of ${\displaystyle G}$. The group operation ${\displaystyle \cdot }$ is the composition of cube moves, corresponding to the result of performing one cube move after another.

The Rubik's Cube is constructed by labeling each of the 48 non-center facets with the integers 1 to 48. Each configuration of the cube can be represented as a permutation of the labels 1 to 48, depending on the position of each facet. Using this representation, the solved cube is the identity permutation which leaves the cube unchanged, while the twelve cube moves that rotate a layer of the cube 90 degrees are represented by their respective permutations. The Rubik's Cube group is the subgroup of the symmetric group ${\displaystyle S_{48}}$ generated by the six permutations corresponding to the six clockwise cube moves. With this construction, any configuration of the cube reachable through a sequence of cube moves is within the group. Its operation ${\displaystyle \cdot }$ refers to the composition of two permutations; within the cube, this refers to combining two sequences of cube moves together, doing one after the other. The Rubik's Cube group is non-abelian as composition of cube moves is not commutative; doing a sequence of cube moves in a different order can result in a different configuration.