For a fun and mathematical card game online, based on modular math, visit my Calculation page.
Most of these pages are devoted to the standard 3×3×3 Rubik cube. There are many other related puzzles, including the 2×2×2 pocket cube, the 4×4×4 super cube, the Rubik tetrahedron, the Rubik Snake, and Rubik's magic. Since all of these puzzles are explorations into various permutation groups, they all lend themselves to the techniques presented here. Still, I concentrate on the standard Rubik cube, because it is by far the most popular, and in many ways the most interesting. It is difficult enough to pose a significant challenge, yet its complexity is not overwhelming.
The definition of a "move" has not (to my knowledge) been rigorously established. Clearly the 12 quarter turns are the primary moves, but are the 6 half turns considered moves as well? Some say yes, and some say no. A turn of any face through any angle is one continuous hand motion, so perhaps a 180 degree twist should be considered a single move. On the other hand, if all moves are quarter turns, the problem becomes a bit more tractable (mathematically). For instance, take any scrambled cube and evaluate the parity of its corners, as a permutation within the symmetric group S8. Since each primary move induces an odd permutation on the corners, The answer tells you whether you are an even or an odd number of moves away from start. We don't know how far the cube is from start, since the puzzle has not been completely analyzed, but the parity of the path to start is known. I find this aesthetically pleasing, so I will define a move as a quarter turn of any face.
If you are interested in the underlying mathematics, please read the corresponding chapter of my online book.