I'm late to notice this, but recently a team proved that a Rubik's Cube can always be solved in 20 moves or less, regardless of its initial configuration. It had already been proven, back in 1995, that at least one initial configuration requires 20 moves to solve. Thus, all positions can be solved in 20 moves, and some positions require the full 20.
What is particularly interesting is that the these researchers found the 20-move bound by having a computer solve all of the 4E19 initial positions exhaustively. The best proof that didn't do an exhaustive search only proved an upper bound of 22 moves.
I wonder if a human-digestible proof will be found for the 20-move upper bound, or if we'll be left with computers generating the stronger proof? At any rate, this is yet one more problem where a mathematical result depends crucially on some very heavy computation.