How Many Combinations Does a 4×4 Rubik’s Cube Have?

A standard 3×3 Rubik’s Cube has 43,252,003,274,489,856,000 possible combinations — around 43 quintillion. This famous number is already unimaginably large, but adding just one more layer to create a 4×4 makes the number explode.

A standard 4×4 cube has:

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

That's approximately 7.4 × 10⁴⁵, or around 7.4 quattuordecillion positions.

But where does that number come from? And what happens when we replace the six solid colours with a detailed picture, as on the Ziina Star Flag World Map 4×4, where all 24 centre pieces are visibly different?

The answer gets even bigger.

How is the number of 4×4 combinations calculated?

A 4×4 has three main types of movable pieces:

• 8 corners

• 24 wing pieces, which combine to form 12 edges when the cube is solved

• 24 centre pieces, with four on each face

The number of positions of a normal 4×4 can be calculated using:

8! × 3⁷ × 24! × 24! ÷ 24⁶

Let's break down what each part means.

The 8 corner pieces

The corners work in much the same way as those on a 3×3.

There are eight corners, which can be arranged in:

8! = 40,320 ways

Each corner can also have three different orientations.

At first, you might expect this to give 3⁸ possible corner orientations. However, the orientation of the final corner is determined by the orientations of the other seven. You cannot legally twist just one corner of a cube.

That leaves:

3⁷ = 2,187 possible corner orientations

So the corner contribution is:

8! × 3⁷

or:

88,179,840 possible corner arrangements and orientations.

The 24 wing pieces

This is where the 4×4 starts becoming dramatically more complicated than the 3×3.

A 3×3 has 12 edge pieces. A 4×4 instead has 24 individual wing pieces.

For example, instead of having one white-red edge piece, a 4×4 has two separate white-red wings which pair together during a reduction solve.

These 24 wings can be permuted in:

24! ways

And 24! is already:

620,448,401,733,239,439,360,000

Unlike the edges of a 3×3, we don't simply multiply this by 2²⁴ for wing orientation. In the usual mathematical representation of a 4×4, a wing's position determines its orientation.

This is also connected to the famous 4×4 parity cases. What appears to be a single flipped edge after reduction isn't literally an ordinary 3×3 edge piece that has somehow been flipped on its own. Instead, it results from the permutation of the individual wing pieces.

The 24 centre pieces

A 4×4 has no fixed centres.

Instead, there are four centre pieces on each of its six faces:

6 × 4 = 24 centres

If every one of these pieces were unique, there would be:

24!

ways to arrange them.

However, that's not what we see on an ordinary six-colour 4×4.

Imagine a solved white face. It contains four white centre pieces. If you removed two of those white centres and swapped them with each other, the cube would look exactly the same.

As far as the visible state of a normal cube is concerned, the four white centres are therefore interchangeable.

The same applies to red, blue, orange, green and yellow.

For each group of four identical centres, we have overcounted the arrangements by:

4! = 24

Since there are six colours, we therefore divide the centre permutations by:

(4!)⁶ = 24⁶

So the number of visibly distinct centre arrangements on an ordinary 4×4 is:

24! ÷ 24⁶

Putting everything together

We can now combine the corners, wings and centres:

8! × 3⁷ × 24! × (24! ÷ 24⁶)

or more neatly:

(8! × 3⁷ × (24!)²) ÷ 24⁶

This gives:

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

possible positions.

That's approximately:

7.4 × 10⁴⁵

For comparison, the 3×3's approximately 4.33 × 10¹⁹ positions suddenly looks rather small!

What happens with a 4×4 picture cube?

This is where things get particularly interesting.

Picture cubes such as the Ziina Star Flag World Map 4×4 and its terrain-map counterpart have 24 distinct centre pieces.

Instead of four centres simply being the same colour, each centre contains a different section of the world map.

This changes something surprisingly important about the mathematics of the puzzle.

On a normal 4×4, swapping two blue centres doesn't visibly change anything. They're both just blue.

On the World Map 4×4, swapping two centres moves two different parts of the world into the wrong places. The two positions can now be distinguished from one another.

Therefore, all 24 centres effectively become unique pieces.

So how many combinations does the Ziina World Map 4×4 have?

Because all 24 centre pieces are distinct, we no longer divide the centre permutations by 24⁶.

Instead of:

24! ÷ 24⁶

we now have the full:

24!

The number of distinguishable positions is therefore:

8! × 3⁷ × 24! × 24!

This is exactly 24⁶ times greater than the conventional figure for a normal 4×4.

And:

24⁶ = 191,102,976

So making the 24 centres distinguishable increases the number of visibly different positions by a factor of more than 191 million.

The result is approximately:

1.414 × 10⁵⁴ distinguishable positions.

That's roughly 1.4 septendecillion positions.

Why doesn't the physical cube have more positions?

There is an important distinction here.

The picture cube doesn't actually have 191 million times as many physical configurations as a normal 4×4. The mechanisms can reach the same underlying states.

The difference is that a conventional colour scheme hides information from us.

Suppose the four white centres are labelled internally:

A, B, C and D

There are:

4! = 24

ways those four pieces can be arranged within a completed white centre.

On an ordinary cube, every one of those arrangements looks exactly the same:

White – White
White – White

So we count them as one visible position.

If those same four pieces contain different parts of a world map, all 24 arrangements can potentially look different.

Now repeat this across all six faces:

24 × 24 × 24 × 24 × 24 × 24 = 24⁶

That's where the enormous 191,102,976× multiplier comes from.

Why picture 4×4s are harder to solve

This isn't just a mathematical technicality. You actually encounter the difference while solving the puzzle.

When solving a normal 4×4, building a centre is relatively straightforward. If you have four white centre pieces together, you've solved the white centre.

You don't care which white piece is in the top-left, top-right, bottom-left or bottom-right position.

With a world map cube, you do.

You might successfully get the four pieces belonging to one face together while still having the geography completely wrong. Individual sections of countries, oceans and flags need to line up with their neighbours.

In other words, there are two challenges:

Which face does this centre belong to?

and:

Where within that face does it belong?

The second question simply doesn't exist on an ordinary 4×4.

The artwork reveals complexity that was already there

This is perhaps the most interesting part of the calculation.

A normal 4×4 and a picture 4×4 can use essentially the same type of mechanism. Adding a world map doesn't magically create hundreds of millions of new mechanical states.

Instead, the artwork allows us to distinguish states that were already mechanically different but visually identical.

For a conventional six-colour 4×4:

≈ 7.40 × 10⁴⁵ visibly distinguishable positions

For a 4×4 with 24 individually distinguishable centres, such as the Ziina World Map cubes:

≈ 1.414 × 10⁵⁴ visibly distinguishable positions

That's 191,102,976 times as many.

So the next time somebody asks how many combinations a 4×4 has, 7.4 quattuordecillion isn't necessarily the whole story.

Give every centre piece its own identity, as on the Ziina Star Flag World Map 4×4, and the number of positions you can actually tell apart jumps to around 1.4 septendecillion.

Not bad for simply changing what's printed on the pieces.