The Rubik's Cube: 43 Quintillion and God's Number

Quick answer: a Rubik's Cube has 43,252,003,274,489,856,000 possible arrangements, and every one of them can be solved in 20 moves or fewer. That upper limit is called God's number, and it was proved in July 2010.

Best for: anyone who wants to understand the cube rather than memorise it.

Two numbers define this puzzle, and the gap between them is the whole story. Forty-three quintillion positions. Twenty moves. Those facts sit very uncomfortably together, and the discomfort is what makes the cube mathematically interesting rather than merely popular.

What Makes the Rubik's Cube Different?

It never comes apart, and almost everything else in this field does.

Ernő Rubik built it in 1974. The pieces are captive on an internal mechanism, so the cube stays a cube throughout and you are rearranging colours on a surface rather than assembling an object. Nothing is hidden, nothing is discovered, and there is no moment where you see how it works.

That puts it in the combination family, also called sequential movement, which is one of the seven types in the guide to mechanical puzzles. It is the opposite end of the field from the interlocking and packing puzzles we make, and people who love one often dislike the other.

How Many Combinations Does a Rubik's Cube Have?

43,252,003,274,489,856,000. Roughly forty-three quintillion.

The number comes from counting what can be rearranged and then dividing out what cannot. There are eight corner pieces that can be permuted and twisted, and twelve edge pieces that can be permuted and flipped. Not every combination of those is reachable, because the mechanism enforces constraints: you cannot twist a single corner in isolation, or flip a single edge, or swap just two pieces.

Those constraints divide the raw count by twelve, which is why the final figure is what it is. It is the same kind of reasoning as the parity argument that makes half of all 15 puzzle positions unreachable, and the position classes that make the European peg solitaire board unsolvable.

What Is God's Number?

The largest number of moves ever needed from any position, assuming perfect play.

For the 3 by 3 cube it is 20. Every one of those forty-three quintillion positions can be solved in at most twenty face turns, counting a half turn as one move, and some positions genuinely require all twenty.

The proof came in July 2010 and needed serious computing: the positions were sorted into symmetry classes, reducing the problem enormously, and the remainder was checked exhaustively. It is a proof rather than a strong conjecture.

The striking part is the ratio. A space of forty-three quintillion states with a diameter of twenty is extraordinarily well connected, which is a fact about group structure rather than about toys.

Why Can People Solve It in Seconds Then?

Because speedsolvers are not finding twenty-move solutions.

Competitive methods use algorithms: memorised sequences that make one predictable change while leaving other parts of the cube undisturbed. A typical solve runs to fifty or sixty moves, far from optimal, but every move is executed without thought.

That is the defining property of the combination family, and it is why the cube is the odd one out among classic puzzles. You can get extremely good at it through recall and practice. Nothing like that exists for a burr or a tangram, where knowing one answer does not generalise to the next.

Finding the actual shortest solution for a given position is a hard computational problem, which is the gap between solving fast and solving well.

How Long Did Rubik Take to Solve His Own Cube?

About a month, which is the detail worth keeping from the whole story.

He built the thing, understood its mechanism completely, having designed it, and then could not undo what he had done to it. There was nobody to ask and no method to look up, because neither existed yet.

That is the purest version of the experience every puzzle sells. Not being beaten by someone else's cleverness, but being beaten by an object you fully understand, which is the same thing people describe when a two-piece puzzle defeats them. Knowing how something works and knowing how to solve it turn out to be different kinds of knowledge.

Is It Really a Mechanical Puzzle?

Yes, and it is the most successful one ever made. The quibble people raise is worth stating anyway.

Every other category here involves taking something apart or putting it together. The cube does neither, which is why some collectors treat twisty puzzles as a separate hobby with its own competitions, vocabulary and shops. The overlap between people who collect burrs and people who collect twisty puzzles is smaller than you would expect.

What it undeniably did was make mechanical puzzles mainstream. Everything in this field benefits from the fact that most people have held one puzzle and that puzzle was a cube.

What Should You Try if You Liked It?

We do not make twisty puzzles, and we would be bad at it. If the cube is what you want, the speedcubing world is a well-served place and we are not part of it.

If what you enjoyed was the sequence rather than the colours, the closest things we make are Snake Cube, where 27 blocks on a cord can only be wrong in their order, and EcstaTIC, a two-piece cube that needs a rotation almost nobody tries. The turning interlocking cube family is explained in turning interlocking cubes, and the other great sequence puzzle is the Tower of Hanoi.

The full range is in mechanical puzzles and cube puzzles.

Rubik's other famous design, the Rubik's Snake, took the same instinct for constrained rotation and removed the goal entirely.

For a much smaller puzzle with a genuinely elegant mathematical solution, Instant Insanity is solved with a page of graph theory.

The other Hungarian puzzle of that era, the Hungarian Rings, turned out to be a re-release of an 1893 American patent.

Ideal followed the Cube with the Missing Link, a hybrid of sliding and rotation with 327 billion arrangements.

Frequently Asked Questions

How many combinations does a Rubik's Cube have?

43,252,003,274,489,856,000, about forty-three quintillion. The raw count of piece arrangements is twelve times larger, but the mechanism makes those positions unreachable: you cannot twist one corner, flip one edge, or swap only two pieces.

What is God's number for the Rubik's Cube?

Twenty. Every possible position can be solved in at most twenty face turns, and some require exactly twenty. It was proved in July 2010 by reducing the positions to symmetry classes and checking the rest exhaustively.

Why do speedsolvers use more than 20 moves?

Because they use memorised algorithms rather than searching for the optimal route. A fast solve typically runs fifty or sixty moves, every one executed without hesitation. Finding the true shortest solution for a given scramble is computationally hard.

Who invented the Rubik's Cube?

Ernő Rubik, in 1974. It is the best-selling puzzle ever made and the reason most people have handled a mechanical puzzle at all.

Is a Rubik's Cube the same as other cube puzzles?

No. A Rubik's Cube never comes apart: the pieces are captive on an internal mechanism and you rearrange colours on its surface. Most other cube puzzles separate completely into pieces that must be reassembled, which is a different problem entirely.