Rubiks cube

Did you know that in a Rubiks cube, you cannot rotate the corner cubes arbitrarily? Starting with a solved puzzle, you can rotate a corner piece by one third or by two thirds (put differently, you an rotate it one third clockwise or one third anti-clockwise). But if you rotate it one third clockwise then either another piece has rotated one third anti-clockwise, or twho other pieces have rotated clockwise by one third. This means that the total amount of rotation for a corner piece is always an integral number: the law of integral rotation.

Similarly, you can flip the edge pieces only in an even number. The total amount of "flip" needs to be one, the law of integral flip.

This means that you can assemble the cube in six different ways. First, you can take it apart and reassemble it with one edge piece flipped. Now the cube cannot be solved, you will always end up with one piece flipped.
Also, you can reassemble it with one corner piece rotated clockwise, or anti-clockwise. This gives three different cubes, one that can be solved and two that can't. The total number of different cubes you can assemble is six.