Unsolvable Rubik’s Cube
When you take apart a Rubik’s Cube and put it back together randomly you have only a 1/12 chance that your scrambled configuration is solvable. In case of other Rubik’s Cube variations where the orientation of the center piece counts the chances are even smaller.
If the online Rubik’s solver throws the ‘Invalid scramble’ error you must double check your puzzle and make sure you entered the color of every field correctly.
If you keep getting this error then you must to disassemble and rebuild it in solved position because probably it has an unsolvable state.
some impossible scrambles
Not every random scramble can be solved by legal moves because of the parity which refers to whether a permutation is even or odd (can that permutation be represented by an even or odd number of swaps):
- Corner orientation
Every corner piece has three possible orientations. It can be oriented correctly (0), clockwise (1) or counterclockwise (2). The sum of corner orientations always remain divisible by 3, no matter how many legal turns you make. See the first two cases on the image.
- Edge orientation
Every legal rotation on the Rubik’s Cube always flips an even number of edges so there can’t be only one piece oriented wrong. See the third example.
- Piece swaps
Considering the permutation of all the corners and edges, the overall parity must be even which means that each legal move always performs the equivalent of an even number of swaps (ignoring orientation). See the last example.
Combining all these laws we get 1/3 * 1/2 * 1/2 = 1/12
Experienced cubers realize that something’s wrong when they reach the last layer of the puzzle.
These scrambles can’t be solved, you have to take your cube apart to fix it.