How many ways are there to solve a Rubiks cube?

Question

Friends have been arguing that a rubiks cube could never be mastered cause there are too many different ways to solve it cause there are 43,252,003,274,489,856,000 positions the Cube could have, so how many possible solutions to solve all those faces? Also we where using this article http://www.cube20.org/

Answer 1

Asking the number of ways to solve a Rubik's Cube is like asking the number of ways to drive from New York to Washington DC. There's one shortest way, but you could also drive literally anywhere, detouring through Los Angeles, Salt Lake City, Minneapolis, back to Los Angles, then to Washington, DC. There's no limit on how many moves a solution can have.

Answer 2

Herbert Kociemba wrote a generating function (in Mathematica code) which gives how many "canonical sequences (where commutating moves and cancellations like U U' etc. are taken into account) in OBTM" for a given maneuver length. (OBTM is outer block turn metric. This the same move metric for cube20.org.)

See his post here.

By following his post (that is, inputting his generating function into Mathematica), we can see, for example, that there are 43946585901564160587264 3x3x3 "canonical sequences" of length 20 (which is 1016.06 times larger than the number of legal 3x3x3 positions which many know to be 43252003274489856000).

So we can can certainly use Herbert's generating function to give an upperbound on the number of solutions/generating move sequences of a given length for a given position since his generating function supposedly counts the number of all possible move sequences--which means that this number includes move sequences which generate multiple positions, not just one.

Clearly, if you do not limit the number of moves the sequences can be, the more possible maneuvers there are. Thus it is only reasonable for us to put a limit to how many moves a solution is to therefore get an upperbound for that maximum solution length by summing all coefficients of the Taylor Series expansion together.

If we sum all coefficients of the Taylor series expansion for the 3x3x3 up to depth 20, we get 47505455028489778073776, which is about 1098.34 times larger than the number of positions.

So we say that there can be at most 47505455028489778073776 different solutions to solve any given configuration because any given configuration can be solved in 20 moves or less. It's a weak upperbound, but, assuming Herbert is correct (which has always been the case in my experience), then there's your number if you were focused on cube20.org in regards to solutions 20 moves or less. :)