Most "cubers" solve a Rubik's cube in about 60 moves, using special algorithms and quick fingers. But any Rubik's cube can be solved in 20 moves or less (cube20.org). Is it possible that there is a way (without computers) to look at a cube and solve it in 20 moves or less, without trying to memorize several sextillion different positions?
Simple answer is that there is no known algorithm that can solve the Rubiks Cube in 20 moves maximum for an arbitrary starting position. Nevertheless, it is known that all positions are solvable in 20 moves, thus some algorithm must exist. Whether or not this algorithm requires less work than memorizing all the possible moves is currently somewhat open. (not comoletely.... We have used patterns and some group theory to reduce the number of solutions we needed to check to prove the number 20.) If we ever do find a simpler algorithm the proof will be much easier!