How many possible combinations does a Rubik’s Cube have?

How to get close to this number

This is calculable rather than memorable. Eight corner pieces can be arranged and each twisted three ways; twelve edge pieces can be arranged and each flipped two ways; then divide by the arrangements that cannot be reached by legal turns. The result is around forty-three quintillion, and the derivation is the point rather than the digits.

The startling fact in the explanation is the one that should not be true: every one of those positions can be solved in twenty moves or fewer. A space of forty-three quintillion states with a maximum solving distance of twenty is extraordinarily shallow, and proving it required enumerating the entire space by computer in 2010.

That combination — astronomically many states, very short paths between them — is what makes the puzzle solvable by humans at all. If the diameter of the state space were large, no set of memorisable algorithms would work. The speedcubing records elsewhere on this site exist because of this mathematical property rather than despite the combination count.