Once again, to establish this, it is necessary to decide on a frame of reference for correct corner orientation. It turns out that each legal move on Rubik's Cube always twists the corners in such a way that the sum of all of their orientations is exactly divisible by 3, and so the above state would by impossible to reach since its total corner orientation is 1 (which is not divisible by 3). Let's say that a corner has orientation '0' if it is twisted the correct way, it has an orientation of '1' if it is twisted clockwise, and it has an orientation of '2' if it is twisted even one step more clockwise (this is the same as just twisting anti-clockwise). With corner orientations, things are slightly more difficult to account for, since each corner piece has three possible orientations, not two. Only one third of the corner orientations are reachable Therefore, it is never possible, using only legal moves, to flip an odd number of edges. In both of these cases, a 90 degree move will flip all 4 edges, which is again an even number of flips. The only remaining faces are the front and back faces. Using this frame of reference, it is easy to see that any move on the left, right, top and bottom faces will always flip zero edges, which is an even number. The most common frame of reference is to say that an edge in the wrong position has correct orientation if, when it is moved to its correct position using only the left, right, top and bottom faces, it would have correct orientation. To establish this, it is necessary to decide on a frame of reference for correct edge orientation, regardless of where an edge is positioned on the cube. It also turns out that each legal move on Rubik's Cube always flips an even number of edges, and so the above state would be impossible to reach via legal moves. Only half of the edge orientations are reachable Since exactly half of the conceivable permutations are even and the other half are odd, only half of the cube's permutations (ignoring orientation) are reachable by legal moves. Therefore, no matter how many moves you perform, always an even number of swaps will have been performed. All together, this is 6 swaps which is even. ![]() Similarly, the new edge state can be obtained via 3 swaps. The new corner state can be obtained via 3 swaps (swap C1/C4, swap C1/C3, swap C1/C2). No matter how many moves you perform, the number of accumulated swaps will therefore always remain even.įor example, consider the turning of one face by 90 degrees: C1 ![]() To understand why this is so, we need to realise that each legal move always performs the equivalent of an even number of swaps. ![]() Since the above cube has an odd number of swaps ("one" swap), this state cannot be reached. As it turns out, every cube state reachable by legal moves can always be represented by an even number of swaps, and at the same time cannot be represented by an odd number of swaps (the two are mutually exclusive).
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