Laws of the cube
Only a 12th of all states reachable by assembling the cube piece-by-piece can actually be reached by twisting the 6 faces (from the solved state). Here's why:
- Only an even number of cubie swaps is possible (1/2 of all permutations) 
- Only an even number of edges can be in a flipped state (1/2 of all edge orientations) 
- The total number of corner twists must be divisible by 3 (1/3 of all corner orientations) 
So combined together, these laws mean that:
1/2 * 1/2 * 1/3 = 1/12 of all states are reachable.
-  Assuming permutations are hypothetically solved using a sequence cubie swaps.
-  An edge is defined as 'flipped' if cannot be solved using only <U, D, L R, F2, B2>.
-  If an oriented corner is defined as one which can be solved using only <U, D, R2, L2, F2, B2>, the number of corner twists can be counted as the number of counter clockwise 120° twists required to orient it.
Recognising an Unsolvable Cube
As a consequence of the laws above, if any of the following cases appear at the end of a solve, you can be certain the cube is not solvable with legal moves:
- Only two pieces to swap
- Only a single edge to flip
- Only a single corner to twist
This also holds true for combinations of the above. For example, everything solved apart from a single flipped edge and a single twisted corner.