If you take apart your Rubik's Cube and then reassemble it scrambled, there is only a 1/12 chance that it will be solvable.
Legal moves are moves that are done by twisting a layer of the cube or preforming an algorithm. Illegal moves are moves that are done by taking the cube apart and reassembling it.
It is impossible to
- flip just 1 edge without disturbing the positions of already solved pieces using any legal moves
- orient just 1 corner without disturbing the positions of already solved pieces using any legal moves
- swap 2 edges without something else being swapped as well
- swap 2 corners without something else being swapped as well
as the following proofs will show you.
If the last layer of the cube passes all of the test, the cube can be solved. If not, it's unsolvable.
Test #1: Corner Parity TestEdit
When you have solved the first two layers and the top cross, assign each correctly oriented corner a value of 0, each corner that needs to be rotated clockwise a value of +1 and each corner that needs to be rotated anticlockwise a value of -1. The sum of the values for all the corners must be divisible by 3 and be either -3, 0, or 3. If it is anything else, the cube is impossible.
Divisible means that a number can be divided without producing a .5 at the end of the resulting number. -3 is divisible by its positive counterpart 3; you get -1. 3 is divisible by itself; you get 1. 0 is divisible by 3, you get 0. However, if 1 corner is twisted clockwise, you get 1. 1 is not divisible by 3 because 1 divided by 3 is 0.33333333333. If 1 corner is rotated anticlockwise, you get -1, when divided by 3 you get -0.33333333333.
0.33333333333 can also be called the "Corner-Unsolvable Number", because it is impossible to twist any 1 corner in any direction using any legal moves without messing up the rest of the cube.
Each legal move of the cube always twists the corners in such a way that the sum of all of their orientations always remains the same, and so if 1 corner was twisted, it would be impossible to solve because its total corner orientation is 1, which isn't divisible by 3. So the reason you cannot orient just 1 corner on the cube is because that would change the total corner orientation number from 1 or -1 to 0. Because the total corner orientation number will ALWAYS stay the same with legal moves, if the initial and final corner orientation number do not match, the corner orientation state is unreachable, and therefore, unsolvable.
The corner parity test means that you cannot orient just 1 corner without changing the corner orientation number, which NEVER EVER changes with legal moves.