Rubik's family cubes of all sizes

Hardware (physical) cubes are based on the original size 3 cube invented by Erno 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 the size 33, and the largest mass-produced is size 19.[1]

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.

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.

A 2-layer (size 2) cube has corner cubies only.

Cubes of size 2 and size 3 have single solutions meaning that all the cube elements can have only one correct location for a solved cube.

Centre cubies differ from the corner and edge cubies in that their orientation or position has multiple possibilities. For cubes of odd size, there will be a centre cubie that is centrally located on the cube face and that cubie has only one correct location for a solved cube. However, multiple locations of all other centre cubies apply for a solved cube. The centre cubies (other than the single central one for cubes of odd size) form sets-of-four on each face and sets-of-24 for the whole cube for the various orbits. These centre cubies have four possible final positions (their orientation changes with position but cannot be changed independently) that would satisfy the solved state.

Typically, hardware cubes with marked centres use images or logos on the faces to designate what centre cubie(s) orientation is required for a solved cube. Such cubes are also referred to as "supercubes" and the application of markings of this type is generally restricted to just cubes of very small size.

Solving a cube with marked centres is considerably more complex than that for standard cubes. The use of jig-saw style image marking on cubes of large size would make a difficult task even more difficult. Two possibilities in current use on software cubes are the use of a numerical graphic in the "1" to "4" range and the use of a corner marking graphic.

There is a direct correspondence between numerical and corner marking. A top left corner quadrant marking is equivalent to a numerical marking 1, second quadrant to 2, third quadrant to 3, and fourth quadrant to 4. The following image illustrates these different forms of marking.

Because transfer of cubies between orbits is impossible, the same 1-2-3-4 markings can be used for each orbit. With the exception of the absolute centre cubies for cubes of odd size, there are 24 centre cubies (4 per face) in each orbit. If ${\displaystyle n}$ is cube size, there will be ${\displaystyle {\frac {\left(n-2\right)^{2}-a}{4}}}$ orbits where ${\displaystyle a}$ is zero if ${\displaystyle n}$ is even or ${\displaystyle a}$ is one if ${\displaystyle n}$ is odd.

Numerical marking is typically applicable for cubes up to about size 32. Corner marking while less user-friendly can allow the marked centres range to be extended beyond the numerical marking limit.

Except for absolute centre marking for cubes of odd size, numerical marking would provide the best means of centre cubie marking for hardware cubes as their size range is limited. Rotation of the numbers would mean a small inconvenience relative to non-rotated numbers that can be used for software cubes. The big advantage of numbers is that they reduce the complexity of solving the last cube face when markings are in use (e.g. if the set-of-four sequence is 1-3-4-2 (even parity, needs two swaps to become the required 1-2-3-4) then the algorithm requirement is clear. Algorithms have been defined in[2] and are, of course, equally applicable to hardware cubes.

Cube solving involves starting with a scrambled cube and applying step-by-step layer rotations to eventually end up with a solved cube. For cubes with unmarked centres that means all faces would need to appear in uniform colour. For cubes with marked centres a unique arrangement of all centre cubies in addition to the uniform colour requirement would need to apply. Since the starting point is always different, there can never be a unique set of rotations that can be applied to solve a cube. Usually, people work through the solution with the possible use of algorithms, mainly in the latter stage of the unscrambling. In theory, it is possible for a person to write a computer program that "thinks" like a human and solves the cube without human intervention (refer to the Solving by computer program section).

The aim of most software cube emulators is to provide a means for the user to interact with the program to solve (unscramble) the cube in a similar manner to the way they would unscramble a hardware cube.

Efficient rotation sequences (algorithms) can be developed using Group Theory permutation mathematics. However, there are many references to the relevant rotation sequences required to solve cubes of small size (refer to some for size 3, 4, and 5 cubes[11][12][13][14]) and there are multiple approaches to what steps that can be used. There is no such thing as a wrong way to solve a cube. The steps involved in solving any cube of size greater than 4 are fairly straightforward extensions of those required to solve size 3 and size 4 cubes. However, there is limited availability of generalized instructions that can be applied for solving cubes of any size (particularly the large ones). Generalized guidance on one way of solving standard cubes[15] and cubes with marked centres[2] of all sizes is available.

Anybody who can solve a size 4 cube should be able to solve cubes of larger size provided they accept an increased time-to-solve penalty. Software design features, unavailable in hardware cubes, can simplify the cube solving process. For a given set of cube design features the complexity (difficulty) of solving a Rubik’s family cube increases if the number of reachable states increases. Three main properties affect that number:

1. Cube size: The number of cubies to be placed is a quadratic (second order polynomial) function of cube size and therefore has a major influence on cube solving complexity.
2. Odd or even size: Even size cubes have an additional effect to just cube size that adds complexity relative to odd size cubes. This effect is relatively small and is independent of cube size (the added contribution when cube size changes from ${\displaystyle n}$ to ${\displaystyle \left(n+1\right)}$ for ${\displaystyle n}$ odd is constant). This effect will be expanded upon when the number of reachable states is considered later.
3. Unmarked or marked centre cubies: Centre cubie marking adds complexity to cube solving.

Additional algorithms to assist users to solve the size 3[16] and to solve any size[2] cube with marked centres have been defined.

Large cube emulators claimed to cater for cubes up to and beyond size 100 are available. Irrespective of what upper size limit is claimed, available pixels (that vary according to the monitor in use) and the user's visual acuity are going to impose practical limits on the maximum cube size that a person can handle.

As indicated in the Rules for Rubik’s family cubes section, total number of cubies is ${\displaystyle \left\{6\left(n-1\right)^{2}+2\right\}}$, and the number of centre cubies is ${\displaystyle 6\left(n-2\right)^{2}}$, where ${\displaystyle n}$ is cube size. For large size cubes, the number of centre cubies becomes very dominant as indicated below.

It follows that placement of centre cubies will become increasingly more significant than the placement of other cubies as cube size is increased. The time to solve a cube will rise dramatically with cube size. For example, there are about 24 times as many cubies to place in a size 16 cube than there are in a size 4 cube. If the average time to place a cubie were the same in both cases, that factor of 24 in time would also apply. The 24 factor is likely to be an under-estimation, because the presence of a large number of cubies makes it more difficult (and time-consuming) to identify what belongs where.

Providing a software program that allows the state of cubes of big size to be changed is not much more difficult than doing the same thing for cubes of small size. However, solving big cubes is a much more demanding and time-consuming task than doing the same for small cubes. Therefore, it is likely that most really big software cubes that are available have never been solved.

Identifying the exact locations to look for cubies (mainly the quadruple centre cubie sets) is a major issue for big cubes. Use of a secondary marker grid[10] can ease identification. For example, a marker grid to form 4×4 segments for a size 16 cube (16 such segments per face) could be used.

A common set of six cubie colours adopted for both hardware cubes and software cubes comprises white, red, orange, yellow, green, and blue. This colour set may be non-optimum for software cubes of large size where the number of pixels per cubie is small. For instance, the differentiation between white and yellow may be problematic. Reducing the number of colors in the red to blue range from five to four and adding violet (the colour at the extreme of the visible spectrum) produces a colour set that may be considered more suitable for cubes of large size. Some software cube implementations allow users to change the default colour set, if desired. This is a useful addition for users whose colour perception is at variance with the norm.

Cube solving by computer program[17] (as distinct from the normal way people solve a cube) for small size (e.g. size 3) cubes has been developed, and it is equally easy to solve large size cubes by computer.

A "final layer problem" is defined here to mean a need for a rearrangement of final layer edge cubies that cannot be achieved using standard size 3 cube moves. These are often referred to as parity problems or errors, but such terminology may be misleading. If moves were limited to those available to the size 3 cube, such states would be unreachable (break parity rules). There are numerous variations in the way the final layer problems are presented and the algorithms to resolve them, but the correction requirement will be similar to that described below. The problems considered here apply equally to standard cubes and to those with marked centres but in the latter case additional final layer issues arise for aligning centre cubies. The problems for larger cubes can be considered as straightforward extensions of those that apply to the size 4 cube. Basically, two types of problem can arise:

• There is a need to flip a complementary pair or a complete set of edge cubies in the final edge set. This condition will be referred to as an OLL (orientation of last layer) requirement.
• There is a need to swap the positions of two edge cubie sets in the final layer. This condition will be referred to as a PLL (permutation of last layer) requirement.

OLL and PLL as used here can be considered to be sub-sets of the usual definitions of these terms. There are many references to moves that can be used to resolve these problems. Fewer references[5][18] demonstrate how these moves satisfy parity rules. From a parity perspective, there is a need to consider the rearrangement of centre cubies which is not readily observable in cubes with unmarked centres. Only OLL parity compliance will be illustrated here.

A typical OLL correction for a size 9 cube is shown. The cubies shown in colour are the only ones in the cube that change positions.

OLL before correction for size 9 cube

OLL after correction for size 9 cube

For the OLL correction there are ${\displaystyle n-2}$ centre cubie swaps and overall there are ${\displaystyle \left(n-1\right)}$ swaps when the edge pair is included. For odd size cubes ${\displaystyle \left(n-1\right)}$ is always even (and conforms with the universal even parity requirement for odd size cubes). For even size cubes ${\displaystyle \left(n-1\right)}$ is always odd which means in this case a parity reversal always occurs, an allowable parity condition for even size cubes.

For the complete edge set flip (a requirement that can arise only for cubes of even size), the number of swaps will be ${\displaystyle \left(n-1\right)\left({\frac {n}{2}}-1\right)}$. The overall number of swaps will be even if ${\displaystyle \left({\frac {n}{2}}-1\right)}$ is even (i.e. ${\displaystyle {\frac {n}{2}}}$ is odd). The overall number of swaps will be odd if ${\displaystyle {\frac {n}{2}}}$ is even. Hence overall parity will be even if ${\displaystyle {\frac {n}{2}}}$ is odd and odd if ${\displaystyle {\frac {n}{2}}}$ is even.

The parity of a given algorithm can, of course, also be deduced from its content using the rules detailed in the Rules for Rubik’s family cubes section.

For standard cubes the rearrangement of centre cubies to resolve the OLL and PLL problems is unimportant. For cubes with marked centre cubies the effect of this rearrangement of these cubies is a serious drawback. For cubes with marked centres it is not possible (except for the size 4 cube) to align all final layer centre cubies until all edge cubies have been placed in their final positions.

Instructions for people on how to solve Rubik's type cubes are normally conveyed either in purely graphical form or as sequences defined using a printable character notation. A character sequence that can be translated and applied to perform a sequence of layer rotations to transform a given state to another (usually less scrambled) state is often referred to as an algorithm. Algorithms are most commonly used when unscrambling the latter portion of the cube but can be applied more extensively if desired. Algorithms can be written down as instructions that can be memorized or looked up in a document. The printable characters used (e.g. to indicate an anticlockwise quarter turn, a single layer quarter turn, or a multiple layer quarter turn) in algorithm instructions vary among authors, as does their positions in the instructions. Where people interpret instructions the way they are presented is insignificant. The only time the form of presentation has significance is when computer keyboard entry is used to change the state of software cubes, and automatic updating of the screen image occurs whenever a valid instruction is received. For example, if F′ is used to represent an anticlockwise quarter turn of the front face, then, as the user types in F, a clockwise quarter turn will occur, and a correction will be needed when the user types the ′ character. The end result will still be correct, but use of −F rather than F′ would eliminate the superfluous rotation. Any text enhancements, such as superscripts or subscripts, must be avoided in the method of presenting cube rotation sequences when users communicate with software cubes via keyboard commands. When computer keyboard entry of instructions is used, macros (which map a short input text string to a longer string) can be used[10][15][19] as algorithm shortcuts.

Speedcubing (or speedsolving) is the practice of solving a cube in the Rubik's cube family in the shortest time possible (which usually implies reducing the number of quarter turn moves required). It is most commonly applied to cubes of small size, and there are numerous solving methods that have been documented. An international team of researchers using computer power from Google has found every way the standard size 3 Rubik's cube can be solved and have shown that it is possible to complete the solution in 20 moves or fewer[20] for any initial scrambled state (where a move here is defined as a quarter or a half turn of a face). Generally, speed solving methods apply more to specialist cubists than typical cubists and are more complex than simple layer-by-layer type methods used by most other people.

If, for a cube of any size, m represents the number of reachable states, u represents the number of unreachable states, and t equals their sum:

${\displaystyle t=u+m}$

${\displaystyle t=km}$ where ${\displaystyle k}$ is a positive integer

${\displaystyle u=\left(k-1\right)m}$

Both m and k are functions of cube size ${\displaystyle n}$. Values for m and k will be considered in the following sections. In other texts, "reachable states" are often referred to as "permutations".

The number of reachable states is based on:

• Standard permutations and combinations mathematics.[21]
• Reduction factors that must be applied to above to reflect movement restrictions specific to Rubik’s family cubes.

The number of different states that are reachable for cubes of any size can be simply related to the numbers that are applicable to the size 3 and size 4 cubes. Hofstadter in his 1981 paper[22] provided a full derivation of the number of states for the standard size 3 Rubik's cube. More recent information sources that adequately justify the figures for the size 3[3][4][23] and size 4[24] cubes are also available. References that indicate the number of possible states for a size ${\displaystyle n}$ cube are available.[24][25][26] The brief material provided below presents the results in the form used in one of these references[24] which covers the topic in far more detail.

For cubes with unmarked centre cubies the following positive integer constants (represented by P, Q, R, and S) apply. These constants are in agreement with figures frequently quoted for the size 3 and size 4 cubes.

The value of S may warrant a word of explanation as it is commonly inferred that the number of possible states for centre cubies with identifying markings for a size 4 cube is 24!. Use of that value is guaranteed to yield the wrong answer if cubes with marked centres are under consideration. The first 20 cubies can be arbitrarily placed giving rise to factor 24!/4!. However, for each possible arrangement of edge cubies, only half the 4! hypothetical arrangements for the last four are reachable.[2][24] Hence the correct value for the cube with marked centres is 24!/2. If the markings are removed, then a "permutation with some objects identical"[21] applies. For the standard cube the marked cube value needs to be divided by (4!)⁶/2 (the 2 divisor must also be applied here). That gives an overall S value for the size 4 cube of 24!/(4!)⁶. All states for 24-centre-cubie orbits for standard Rubik’s family cubes are reachable (if required, even parity is always achievable by swapping the positions of a couple of centre cubies of the same colour).

${\displaystyle m={\textrm {P}}{\textrm {Q}}^{a}{\textrm {R}}^{b}{\textrm {S}}^{c}}$

where ${\displaystyle a}$, ${\displaystyle b}$, and ${\displaystyle c}$ are positive integer variables (functions of cube size ${\displaystyle n}$) as given below.

${\displaystyle a=n{\textrm {mod}}2}$ (i.e. 0 if ${\displaystyle n}$ is even or 1 if ${\displaystyle n}$ is odd)

${\displaystyle b={\frac {n-2-a}{2}}}$

${\displaystyle c={\frac {\left(n-2\right)^{2}-a}{4}}}$

For even size cubes ${\displaystyle {\textrm {Q}}^{a}=1}$ (see exponentiation).

For further simplification, parameter ${\displaystyle m}$ may also be expressed as ${\displaystyle m=10^{y}}$ where ${\displaystyle y={\textrm {log}}_{10}m}$. Parameter ${\displaystyle y}$ can be related to ${\displaystyle n}$ by a continuous quadratic function subject to the restriction that ${\displaystyle n}$ must be an integer greater than 1 when referring to possible states for cubes:

${\displaystyle y={\textrm {A}}n^{2}+{\textrm {B}}n+{\textrm {C}}}$

where A, B, and C are constants. Constants A and B are the same for ${\displaystyle n}$ even and for ${\displaystyle n}$ odd, but the value of C is different.

In graphical terms, when ${\displaystyle y}$ is plotted,[24] two parabolae of exactly the same shape are involved, with "even" cube values lying on one and "odd" cube values lying on the other. The difference is imperceptible except when plotted over a small range of ${\displaystyle n}$, as indicated in the graphs reproduced below. Only Rubik’s family values for ${\displaystyle n}$ equal to 2 and 3 are included in the second graph.

Use of the log function y provides the only practical means of plotting numbers that vary over such a huge range as that for the Rubik's cube family. The difference between the curves translates as a factor of 505.08471690483 (equal to ${\displaystyle {\frac {{\textrm {R}}^{0.5}{\textrm {S}}^{0.25}}{\textrm {Q}}}}$). This is the factor that defines the effect of even size, relative to odd size, on the number of reachable states for cubes with unmarked centres.

Hence, with the logarithmic presentation the number of cube states can be expressed using just four[27] numbers (A, B, and the two C values). Furthermore, the number of cube states form a restricted set of values for a more general continuous quadratic (parabolic) function for which ${\displaystyle n}$ can have non-integer and negative values. Calculating the value of m from the corresponding value of y is a straightforward process.

Centre cubies are different from corner or edge cubies in that, unless they have indicative markings, there are multiple possibilities for their final orientation and/or locations. The number of different ways centre cubies can be arranged to yield a solved cube with unmarked centre cubies may be of interest. To calculate that, the impact of centre cubie marking needs to be assessed. Define ${\displaystyle m_{\textrm {M}}}$, ${\displaystyle {\textrm {Q}}_{\textrm {M}}}$, and ${\displaystyle {\textrm {S}}_{\textrm {M}}}$ to be the changed parameters for marked centre cubies (P and R remain unchanged).

${\displaystyle {\textrm {Q}}_{\textrm {M}}={\textrm {TQ}}}$ where ${\displaystyle {\textrm {T}}={\frac {4^{6}}{2}}=2048}$

${\displaystyle {\textrm {S}}_{\textrm {M}}={\textrm {VS}}}$ where ${\displaystyle {\textrm {V}}={\frac {(4!)^{6}}{2}}=95551488}$

${\displaystyle m_{\textrm {M}}={\textrm {P}}\left({\textrm {Q}}_{\textrm {M}}\right)^{a}{\textrm {R}}^{b}\left({\textrm {S}}_{\textrm {M}}\right)^{c}}$

${\displaystyle m_{\textrm {M}}=m_{\textrm {D}}m}$

${\displaystyle m_{\textrm {M}}={\textrm {T}}^{a}{\textrm {V}}^{c}}$

Parameter ${\displaystyle m_{\textrm {M}}}$ defines the number of reachable states for cubes with marked centres. Factor ${\displaystyle m_{\textrm {D}}}$ gives the number of different arrangements of unmarked centre cubies that will provide a solved size ${\displaystyle n}$ cube. It is also the factor by which the number of different states for a standard cube needs to be multiplied by when marked centres apply.

The number of unreachable states far exceeds the number of reachable states. There are many references to the number of unreachable states for the size 3 cube but very few for larger size cubes.

The unreachable arrangements for corner and edge cubies are the same for cubes with or without marked centres.

If a corner cubie for cubes of any size is considered, then a 1/3 twist clockwise leaving everything else unchanged will represent an unreachable state, and similarly for a 1/3 twist counter-clockwise. Hence only 1/3 of the twist possibilities are reachable.

For the central edge cubie for odd size cubes the behaviour is the same as that for the size 3 cube. Only half the conceivable positions are reachable and only half the conceivable orientations are reachable. Hence only 1/4 of the central edge cubie movement possibilities are reachable.

Edge cubies that comprise 12 complementary pairs (24 cubies total) behave as if the complementary cubies did not look the same. Any given edge cubie can move to any position in the 24-cubie orbit but for any given position there is one reachable and one unreachable orientation for that cubie. The reverse applies for the complementary edge cubie. For a given cubie (1-2) the reachable and unreachable orientations for a given face for a given orbit for a size 8 cube is illustrated below. One of the 24 reachable possibilities for a given edge cubie matches that of the set cube.

The number of unreachable states for a 24-edge-cubie set is the same as the number of reachable states (24! in each case).

In the case of the marked centre cubies only half the conceivable arrangements for each set of 24 cubies for any given orbit are reachable.[2] The same parity rules that apply for marked centre cubies also apply for the unmarked centre cubies. A quarter turn of a set-of-four centre cubies cannot be achieved without changing the arrangement elsewhere to meet the parity requirement. Because there are 95551488 ways of arranging the individual centre cubies so that the resulting arrangement appears exactly the same, parity rules can be met without any observable indication of how the parity compliance is achieved. Hence, for the normal case (24 cubies comprising four of each of six colours) there is no restriction on the achievable states for the centre cubies.

The following table uses the values noted above to represent the k component factors for the size ${\displaystyle n}$ cube. Exponents a, b and c are functions of cube size ${\displaystyle n}$ as defined above.

Reduction components for factor k (for standard cube with unmarked centres) and for ${\displaystyle k_{\textrm {M}}}$ (for cube with marked centres)	Unmarked centres' cube type	Marked centres' cube type
Corner cubie factor	3	3
Central edge cubie factor (such cubies exist only for cubes of odd size)	${\displaystyle 2^{2a}}$	${\displaystyle 2^{2a}}$
Complementary edge cubie factor for all 12-pair sets combined	${\displaystyle 2^{b}}$	${\displaystyle 2^{b}}$
Absolute centre cubie factor (such cubies exist only for cubes of odd size)	1	${\displaystyle 2^{a}}$
Centre cubie factor for all 24-cubie sets combined	1	${\displaystyle 2^{c}}$

Taking the product of these factors:

For the standard size ${\displaystyle n}$ cube	${\displaystyle k=\left(3\right)2^{2a+b}}$
For the marked centres' size ${\displaystyle n}$ cube	${\displaystyle k_{\textrm {M}}=\left(3\right)2^{3a+b+c}}$

Some values for cubes of small size are given below.

Cube size	2	3	4	5	6	7	8
Value of ${\displaystyle k}$	3	12	6	24	12	48	24
Value of ${\displaystyle k_{\textrm {M}}}$	3	24	12	192	192	6144	12288

The number of unreachable states is given by ${\displaystyle (k-1)m}$ for standard cubes and by ${\displaystyle (k_{\textrm {M}}-1)m_{\textrm {M}}}$ for cubes with marked centre cubies.