How Many Combinations Does the Cube Have?
A standard 3x3 cube can be mixed into 43,252,003,274,489,856,000 different positions. That is about 43 quintillion, or 4.3 × 1019. Here is where the number comes from.
The working out
The centres don’t move compared with each other, so we only count the 8 corners and 12 edges.
| Part | Sum | Ways |
|---|---|---|
| Arrange the 8 corners | 8! = 8 × 7 × 6 × … × 1 | 40,320 |
| Twist the corners | 37 (the last corner’s twist is decided by the others) | 2,187 |
| Arrange the 12 edges | 12! ÷ 2 (half are impossible, see below) | 239,500,800 |
| Flip the edges | 211 (the last edge’s flip is decided by the others) | 2,048 |
| Total | 8! × 37 × 12!/2 × 211 | 43,252,003,274,489,856,000 |
Why some positions are impossible
If you take a cube apart and put the pieces back at random, you get 12 times as many positions: 519,024,039,293,878,272,000. Only one in twelve of those can be solved by turning, because turning follows three hidden rules:
- Corner twists add up. The total twist of all the corners is always a whole number of turns, so you can’t twist just one corner (that’s the × 3).
- Edge flips come in pairs. You can’t flip just one edge (that’s the × 2).
- Swaps come in pairs. Every face turn swaps pieces in a way that keeps corners and edges “in step”, so you can’t swap just two pieces (the other × 2).
Our solver checks these rules when you enter a cube, and tells you if your cube has a twisted corner, a flipped edge or two swapped pieces.
How big is 43 quintillion?
- If you looked at one position every second, it would take about 1.4 trillion years to see them all. That is around 100 times the age of the universe.
- A stack of 43 quintillion cubes (5.7 cm each) would reach about 260 light years into space.
- Laid side by side, they would cover the whole surface of the Earth about 275 layers deep.
Even so, every one of those positions can be solved in 20 moves or fewer.
Questions people ask
How many combinations does a 3x3 cube have?
43,252,003,274,489,856,000, which is about 43 quintillion.
Why is it divided by 12?
Because of three rules turning can’t break: corner twists (÷3), edge flips (÷2) and piece swaps (÷2). Put together, only 1 in 12 random reassemblies can be solved.