How Many Combinations Does a Twisty Puzzle Have?

A standard 3×3 Rubik's Cube has exactly 43,252,003,274,489,856,000 possible positions — around 43 quintillion. That number is already enormous, but increasing the number of layers causes the number of possible combinations to grow extremely quickly.

A 4×4 has around 7.40 × 10⁴⁵ positions, while larger cubes quickly reach numbers with hundreds, thousands or eventually millions of digits.

The same idea applies to other twisty puzzles. A Pyraminx, Master Pyraminx, Megaminx, Gigaminx and Kilominx all have their own piece structures and therefore need different formulas.

The calculator above uses separate mathematical formulas for three major families:

• NxNxN cubes

• Pyraminx-style tetrahedral puzzles

• Minx-style dodecahedral puzzles

But where do these enormous numbers actually come from?

The basic idea: permutations, orientations and restrictions

Most twisty-puzzle combination calculations work in broadly the same way.

First, we ask how many ways the pieces can be permuted — in other words, moved between different locations.

If there are eight completely interchangeable positions for eight different pieces, there are:

8! = 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 40,320

possible permutations.

We then account for orientation. A cube corner, for example, can usually be twisted into three orientations.

Finally, we need to remove arrangements that cannot actually be produced by legal puzzle moves.

This last part is important. You cannot simply dismantle a Rubik's Cube, put every piece back however you want and assume that position could have been reached by turning the puzzle. Twisty puzzles have restrictions involving things such as corner orientation, edge orientation and permutation parity. Jaap Scherphuis describes this general approach as counting permutations and orientations and then applying the puzzle's constraints.

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Cubes

The 3×3

The easiest familiar example is the ordinary 3×3.

There are:

• 8 corners

• 12 edges

The eight corners can be arranged in:

8!

ways.

Each corner has three possible orientations, but once seven corners have been oriented, the orientation of the final corner is forced. This gives:

3⁷

possible corner orientations.

The 12 edges can be arranged in:

12!

ways.

Edges have two possible orientations, but their orientation is also constrained.

There is also a parity relationship between the edge and corner permutations.

Combining all of these restrictions gives:

8! × 3⁷ × 12! × 2¹⁰

which equals:

43,252,003,274,489,856,000

possible positions.

This is roughly:

4.33 × 10¹⁹

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What changes on a 4×4?

A 4×4 still has eight corners, but the remaining pieces are very different.

Instead of 12 ordinary edge pieces, it has 24 wing pieces.

It also has 24 movable centre pieces.

For a normal six-colour 4×4, those 24 centres consist of four pieces of each colour.

The formula can be written as:

8! × 3⁷ × 24! × 24! / 24⁶

This produces:

7,401,196,841,564,901,869,874,093,974,498,574,336,000,000,000

or approximately:

7.40 × 10⁴⁵

positions.

That is an extraordinary increase from the 3×3.

Why do we divide the centres by 24⁶?

Imagine temporarily giving every centre piece its own number.

There would be:

24!

ways of arranging those 24 numbered centres.

But that isn't what we see on an ordinary cube.

There are four white centres, four yellow centres, four red centres and so on. Swapping two white centre pieces with each other does not produce a visibly different position.

For every colour there are:

4! = 24

ways to rearrange those four indistinguishable pieces without changing what the cube looks like.

There are six colours, so we divide by:

(4!)⁶ = 24⁶

This is why an ordinary 4×4 centre orbit contributes:

24! / 24⁶

visibly different arrangements.

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What about picture cubes?

This creates an interesting difference for picture cubes.

Consider a 4×4 world-map cube where every section of the picture printed on the 24 centres is different.

Those centres are no longer visually interchangeable.

Moving one blue-looking piece of ocean into the position occupied by another blue-looking piece may still change the picture because the printed sections themselves are different.

In that case, we no longer divide the centre arrangements by 24⁶.

The centre contribution becomes the complete:

24!

This makes the number of visibly distinguishable positions:

24⁶ = 191,102,976

times greater than on the ordinary six-colour 4×4.

Instead of around:

7.40 × 10⁴⁵

positions, the distinct-centre picture version has approximately:

1.41 × 10⁵⁴

visibly different positions.

This is why the calculator includes a Distinct Picture Centres option.

There is one important qualification: the calculator's picture option distinguishes the movable centre pieces. On odd-layer cubes such as the 3×3, 5×5 and 7×7, the very middle centre also has a visible orientation on some picture-cube designs. That introduces another orientation problem and is not currently included in the calculator's picture-centre option. A conventional 3×3 picture cube, for example, can have more states because its fixed centres can be visibly rotated.

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General NxNxN cubes

As the cube gets larger, the pieces form groups known as orbits.

Pieces within the same orbit can move into one another's positions. Pieces in different orbits cannot.

This is the key to generalising the calculation.

The calculator uses:

3⁷ × 8! ÷ 24^((n+1) mod 2)

× (24! / 24⁶)^⌊((n−2)/2)²⌋

× (24!)^⌊(n−2)/2⌋

× (2¹⁰ × 12!)^(n mod 2)

where n is the number of layers.

It looks intimidating, but each part represents something recognisable.

3⁷ × 8!

This is the corner contribution.

Every ordinary NxNxN cube has eight corners, regardless of whether it is a 2×2 or a 100×100.

(24!)^⌊(n−2)/2⌋

This represents the different wing orbits.

As a cube grows, extra types of edge pieces appear.

A 4×4 has one 24-piece wing orbit.

A 5×5 also has one.

A 6×6 has two.

A 7×7 has two.

And so on.

(24! / 24⁶)^⌊((n−2)/2)²⌋

This accounts for the movable centre orbits on a normal six-colour cube.

The number of these centre orbits grows approximately with the square of the number of layers.

That is one of the main reasons the combination count becomes so enormous on very large cubes.

With the calculator's picture-centre option, each:

24! / 24⁶

is replaced by:

24!

because every centre in that orbit is treated as distinguishable.

The odd/even terms

Odd cubes have fixed face centres and a set of ordinary middle edges that even cubes do not have.

Even cubes, meanwhile, have no fixed centre defining their absolute orientation, so whole-cube rotational symmetry has to be handled differently.

Those differences produce the remaining factors in the general formula.

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Pyraminx combinations

The Pyraminx family looks quite different from a cube, so it needs a completely different formula.

The standard 3-layer Pyraminx has:

933,120

meaningful positions when the four trivial tips are ignored.

If the tips are included, each of the four tips can independently occupy three orientations.

That multiplies the result by:

3⁴ = 81

so the total becomes:

75,582,720

positions.

OEIS lists the general position sequence for corner-turning tetrahedral puzzles and separately gives the version including tip rotations, which is exactly 81 times the no-tip count.

Larger Pyraminxes

The same family includes puzzles such as:

• 3 layers — Pyraminx

• 4 layers — Master Pyraminx

• 5 layers — Professor Pyraminx

• larger generalized Pyraminxes

The general formula is slightly unusual because it depends on whether the number of layers is divisible by three.

If 3 divides n, the number of positions, ignoring tips, is:

5 × 2^(2n²−3n−1) × 3^(n²/3+3n−6) × 1925^(n²/3−n)

Otherwise it is:

5 × 2^(2n²−3n−1) × 3^(n²/3+3n−16/3) × 1925^(n²/3−n−1/3)

Although some of those exponents appear to contain fractions, they evaluate to integers for the relevant values of n.

For example:

3-layer Pyraminx: 933,120

while the 4-layer Master Pyraminx already jumps to:

2,681,795,837,952,000

possible positions before the trivial tips are included.

So just one additional layer increases the state space by almost three billion times.

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Minx puzzles

The Minx family is even more interesting because it contains both odd- and even-layer puzzles.

These include:

• 2 layers — Kilominx

• 3 layers — Megaminx

• 4 layers — Master Kilominx

• 5 layers — Gigaminx

• 6 layers — Elite Kilominx

• 7 layers — Teraminx

• 8 layers — Super Elite Kilominx

• 9 layers — Petaminx

• 11 layers — Examinx

• 13 layers — Zettaminx

• 15 layers — Yottaminx

Unlike a cube, these puzzles have 12 pentagonal faces, 20 corners and 30 physical edges.

This changes the size of their piece orbits dramatically.

A full edge or centre orbit on a higher-order Minx contains 60 pieces, compared with the 24-piece orbits found on cubes.

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Odd-layer Minxes

For odd-layer puzzles such as the Megaminx, Gigaminx and Teraminx, we can write the number of layers as:

L = 2m + 1

The generalized formula is:

30! × 20! × (60!)^(m²−1) × 2^(28−m) × 3¹⁹ / (5!)^(12m²−12m)

OEIS gives this formula for generalized odd-size Megaminx puzzles.

For a normal 3-layer Megaminx:

m = 1

and the result is approximately:

1.0067 × 10⁶⁸

possible positions.

That already makes the Megaminx state space enormously larger than that of a 3×3.

Gigaminx

A Gigaminx has five layers, so:

m = 2

Putting that into the formula gives approximately:

3.65 × 10²⁶³

positions.

That means the full number contains 264 digits. OEIS also records that the corresponding sequence entry has 264 digits.

Teraminx

At seven layers:

m = 3

and the number grows to approximately:

1.15 × 10⁵⁷³

positions.

At this point, simply writing out the full number is already becoming rather impractical.

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Even-layer Minxes

Even-layer Minxes have a different structure.

The simplest is the Kilominx, which contains only the 20 corner pieces.

Its position count can be written as:

(19! / 2) × 3¹⁸

giving approximately:

2.36 × 10²⁵

positions.

The 4-layer Master Kilominx adds one 60-piece wing orbit and one 60-piece centre orbit.

Its structure gives:

((19! / 2) × 3¹⁸)

× (60! / 2)

× (60! / (5!)¹²)

which produces approximately:

9.15 × 10¹⁶³

positions.

The calculator extends this same orbit pattern to larger even-layer Minxes.

For an even-layer Minx where:

L = 2m

there are:

m − 1

60-piece wing orbits, and:

(m − 1)²

60-piece centre orbits.

This gives the general formula:

((19! / 2) × 3¹⁸)

× (60! / 2)^(m−1)

× (60! / (5!)¹²)^((m−1)²)

For the 6-layer Elite Kilominx, for example:

m = 3

so there are:

• 2 wing orbits = 120 wing pieces

• 4 centre orbits = 240 movable centre pieces

• 20 corners

The resulting state space is approximately:

3.09 × 10⁴¹⁶

positions.

The general even-layer expression used here is an extension of the orbit structure that reproduces the established Kilominx and Master Kilominx calculations; unlike the odd-layer Megaminx formula, it is not being presented here as a separately published OEIS formula.

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Why do these numbers grow so quickly?

The important thing to notice is that adding layers does not simply add a fixed number of combinations.

It adds new groups of movable pieces, each of which can often be permuted in an enormous number of ways.

A new 60-piece Minx orbit, for example, can contribute a factor involving:

60!

and:

60! ≈ 8.32 × 10⁸¹

That is an 82-digit number from a single factorial.

Even more importantly, the number of centre orbits grows roughly with the square of the puzzle's layer count.

This is why the exponent in scientific notation itself starts growing very rapidly.

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Why doesn't the calculator always show the exact number?

For ordinary puzzles, the calculator uses JavaScript's BigInt system to calculate the complete integer exactly.

That means that if you ask for the number of combinations of a 4×4, Megaminx or even a much larger puzzle, the calculator can genuinely calculate every digit rather than using a floating-point approximation.

However, eventually that stops being useful.

Suppose an answer contains 10,000 digits. It is still possible to calculate and display it, although hardly anyone is going to read the complete number.

Now imagine an answer containing a million digits.

Or a billion.

Or several trillion.

For extremely large layer counts, constructing the entire integer would consume huge amounts of memory and processing time simply to produce a number that is impossible to meaningfully display.

The calculator therefore changes methods automatically.

For results containing up to 10,000 digits, it uses an exact integer calculation.

Above that threshold, it instead calculates:

log₁₀(N)

where N is the number of positions.

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How logarithms let us calculate absurdly large puzzles

Scientific notation writes a number in the form:

a × 10ᵇ

where a is between 1 and 10.

If:

log₁₀(N) = 573.061...

then:

N ≈ 1.15 × 10⁵⁷³

We also immediately know that the full number contains:

574 digits

because:

digits = floor(log₁₀(N)) + 1

This means the calculator does not actually have to construct an integer containing trillions of digits.

Instead of calculating something like:

60! × 60! × 60! × ...

millions or billions of times conceptually, it can use the logarithmic identity:

log(a × b) = log(a) + log(b)

and:

log(aᵇ) = b × log(a)

A vast multiplication therefore becomes a comparatively tiny set of additions and multiplications.

This is why the calculator can give a scientific-notation answer for a hypothetical million-layer puzzle almost immediately even though explicitly writing the complete answer would be impossible in practice.

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Exact numbers versus meaningful numbers

Once numbers become this large, their exact decimal expansion is not usually the interesting part.

Knowing that one puzzle has:

1.15 × 10⁵⁷³

positions tells us far more than seeing a block of 574 digits.

The exponent gives us an immediate idea of scale and makes different puzzles much easier to compare.

A 3×3 has an exponent of about 19.

A 4×4 reaches about 45.

A Megaminx reaches about 68.

A Master Kilominx reaches about 163.

A Gigaminx reaches about 263.

An Elite Kilominx reaches about 416.

A Teraminx reaches about 573.

And from there, the numbers continue accelerating.

The interesting question very quickly stops being:

"Can we write down every digit?"

and becomes:

"Just how quickly does the number of possible positions grow?"

With the calculator above, you can experiment with anything from a 2×2 or Kilominx to hypothetical puzzles with thousands or even millions of layers — without asking your browser to store a number containing trillions of digits.