The original Rubik's Cube was a mechanical 3×3×3 cube puzzle invented in 1974 by the Hungarian sculptor and professor of architecture Ernő Rubik. Extensions of the Rubik's Cube have been around for a long time and come in both hardware and software forms. The major extension have been the availability of cubes of larger size and the availability of the more complex cubes with marked centres. The properties of Rubik’s family cubes of any size together with some special attention to software cubes is the main focus of this article. Many properties are mathematical in nature and are functions of the cube size variable.
Definitions In the main, the terminology used here is in agreement with what is in general use. Elsewhere, some terms are used with different meanings. To avoid misconceptions, the meaning of most terms in use in this article is defined below.
Cube size The standard Rubik's cube is often referred to as a 3×3×3 cube. That cube will be referred to as a size 3 cube and in general an cube will be referred to as a size cube. Rubik Cube family Cubes that have similar rotational properties to the standard Rubik's size 3 cube and obey generalized rules for a size cube are considered to be members of the Rubik Cube family. Cubes of size 2 and above that meet this condition are available. Cubie Individual cube elements will be referred to as cubies (others sometimes refer to them as "cubelets"). There are three types of cubies: corner cubies (three coloured surfaces), edge cubies (two coloured surfaces), and centre cubies (one coloured surface). The absolute centre cubies for odd size cubes sit on the central axes of the six faces, and their relative positions never change. Cubicle A cubicle is the compartment in which a cubie resides. For a permutation (defined below), cubicles are considered to occupy fixed positions in the space occupied by the cube object, but their contents (cubies) may shift position. Facelet A facelet is a visible coloured surface of a cubie (corner cubies have three facelets, edge cubies have two, and centre cubies have one). Cube style Two cube styles are referred to herein: firstly a standard cube with unmarked centres and secondly a cube with marked centres. Cube state A particular arrangement of the cubies will be referred to as a cube state. What looks the same is considered to be the same (unless specific mention to the contrary is made). Each state has equal probability of being produced after a genuine random scrambling sequence. A rotation of the whole cube does not change the state considered herein. In other texts, the various states are often referred to as permutations or arrangements. Cube layer A cube layer is a one-cubie-thick slice of the cube perpendicular to its axis of rotation. Outer layers (faces) contain more cubies than inner layers. For a cube of size , there will be layers along any given axis. Cube face The meaning of a cube face depends on the context in which it is used. It usually means one of the six three-dimensional outer layers but can also refer to just the outside layer's surface which is perpendicular to its axis of rotation. The faces are usually designated as up (U), down (D), front (F), back (B), left (L), and right (R). Set state The set (or solved) state of the cube is one for which a uniform colour appears on each of the six faces. For cubes with marked centres, the set state is characterised by a unique arrangement of all centre cubies, and the labelling of those cubies must reflect that. Scrambled state The scrambled state is the starting point for unscrambling the cube. It arises when a cube in the set or any other state is subject to a large number of randomly chosen layer rotations. "Fixed-in-space" axes of rotation There are three mutually perpendicular axes of rotation for the cube. One set of axes defined in terms of D, U, B, F, L, and R terms can be considered to have a fixed orientation in space. Think of these axes as belonging to a cube-shaped container in which the cube object can be positioned in any of 24 orientations. One axis can be drawn through the centres of the D and U faces (the DU axis). The others are the BF and LR axes. "Cube object" axes of rotation Another set of axes can be defined for the cube object itself. These axes relate to the face colours, the most common being white, red, orange, yellow, green, and blue. The axes are usually yellow-white, red-orange, and blue-green. For odd size cubes, these axes are always fixed relative to the internal frame of the cube object. For even size cubes, these axes remain fixed relative to the internal frame of the cube object after initial selections. The origin for the axes is the centre of the cube object. Layer rotation The only way that cube state can be changed is by rotations of cube layers about their axes of rotation. All changes of state involve rotation steps that can be considered as a sequence of single layer quarter turns. Orbit For a basic quarter turn of a cube layer for cubes of all sizes, sets-of-four cubies move in separate four-cubicle trajectories. When all the possible trajectories for a given cubie type are considered for the whole cube, we will refer to all the possible movement positions as being in a given orbit. We consider that the size 3 cube has two orbits, one in which the eight corner cubies are constrained to move and one in which the 12 edge cubies are constrained to move. Transfer of cubies between these orbits is impossible.
For cubes of size 4 and above, we will also define an edge cubie orbit as comprising 12 cubies, but will use the term complementary orbit to describe a pair of orbits between which edge cubies can move. A pair of complementary edge cubie orbits contains a total of 24 cubies. Cubes of size 4 and above include centre cubie orbits that contain 24 cubies. Transfer of cubies between one such orbit and another is not possible (applies to cubes of size 5 and above). Move A move is a quarter turn rotation of a layer or a sequence of such quarter turns that a person would apply as a single step. Move notation A clockwise quarter turn of an outer layer is usually expressed as U, D, F, B, L, or R. In other respects the notation used varies among authors. For example, a counterclockwise quarter turn of an outer layer could be expressed as U', D', F', B', L', or R'. Algorithm An algorithm defines a sequence of layer rotations to transform a given state to another (usually less scrambled) state. Usually an algorithm is expressed as a printable character sequence according to some move notation. An algorithm can be considered to be a "smart" move. All algorithms are moves, but few moves are considered to be algorithms. Permutation A permutation of the cube as used herein means the act of permuting (i.e. rearranging) the positions of cubies. A permutation is an all-inclusive term which includes a sequence of quarter turns of any length. Even the solving of the cube from a scrambled state represents a permutation. The term "permutation" is used extensively by mathematicians who use Group Theory to quantify the process involved in a rearrangement of cubies.
The term "permutation" is also often used to mean the state of the cube that results after it is rearranged, but that meaning will not be used herein. In such cases, the term "cube state" will be used. That allows the term "permutation" to be used when the permutation results in no change of state - an area of special interest for Rubik’s family cube permutations. Parity A cube permutation can be represented by a number of swaps of two cubies. If that number is even, the permutation has even parity, and if the number is odd, the permutation has odd parity.
Cube types
Hardware cubes Hardware (physical) cubes are based on the original size 3 cube invented by Ernő Rubik in 1974. These cubes usually use coloured stickers on the facelets for cubie identification. The size 3 standard Rubik’s Cube gained peak interest in the 1980s and was closely followed by the size 4 (Rubik's Revenge) cube. Other, usually more recently available, hardware forms of the cube come in size 2 (Pocket Cube), size 5 (Professor's Cube), size 6 (V-Cube 6), and size 7 (V-Cube 7). Lesser known hardware cubes of larger sizes have also been produced. Currently, the largest hardware cube made is size 49, and the largest mass-produced is size 21.
Software cubes In parallel with the hardware form of the cube there are many software forms available that obey the same rules as the hardware forms. Software cube emulators are not subject to the physical restraints that impose a size limit on the hardware forms. Hence the only really big cubes available are those in software form. Also, unlike the hardware forms, a range of cube sizes can easily be accommodated by the one program. The design characteristics of programs that allow users to unscramble cubes vary considerably with such features as the capability to allow the user to save a partially unscrambled state being often available.
Software cubes were in use in the 1980s when monochrome monitors were in common use. The lack of colour meant a different way of face identification was required. A program that retained 1980s monochrome capability (using numerals 1 to 6 to identify facelets) for cubes in the size 2 to 11 range was produced in 1991 (together with colour capability in the size 2 to 15 range). More recently developed software cubes use coloured facelets as for hardware cubes.
The most common, but by no means universal, approach is to emulate the cube by providing a "three-dimensional" display of the cube that makes it look like a real hardware cube. A drawback of the "three-dimensional" display is that, without some extra enhancements, the state of parts of the cube for any given view is hidden.
Other interactive software approaches that do not emulate a three-dimensional cube are also used by some programmers. Generally, an aim of such approaches is to allow the state of all cubies to be in view all the time but has the disadvantage (for some viewers) that the display does not appear like a real-world cube. A conventional two-dimensional (unfolded) display with all cube elements appearing of equal size is one approach. Another form of display where all cube elements do not appear of equal size is also in use. The upper cube size limit for software cubes is limited by the monitor’s available pixels and what viewers find as acceptable which, in turn, is a function of their visual acuity. For cubes of large size it can be advantageous to allow a portion of the cube to be scrolled out of view.
All emulators provide a means for the user to change the cube state step-by-step and unscramble the cube. Most emulators use mouse movements to control the rotation of the cube elements, others use keyboard commands, and some use a combination of both. Some also include the functionality for the computer to automatically find the fastest series of movements to solve the virtual cube and execute them.
Software cubes present some major capabilities that are not possible with hardware cubes. Instant return to the set state is always available. If the program allows a partially unscrambled state to be saved then, by regularly updating the saved state, users need not despair if they do something that leaves their cube in a mess. They can return to their previously recorded state and continue from there. The bigger the cube the more helpful such a capability becomes.
Some Freeware large cube (size greater than 10) implementations are available.
Cube design variants While there are multiple variants in use only two will be considered here: Standard cubes with unmarked centres. Cubes with marked centres.
Standard cubes with unmarked centres A 2-layer (size 2) cube has corner cubies only.