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Rubik's Cube theory

Laws of the cube

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An interesting fact about Rubik's Cube is that if you take it apart and put it together randomly, there will be only a 1 in 12 chance of it actually being solvable by legal moves (that is, without taking the cube apart again).

Only half of the permutations are reachable

An impossible state

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). Since the above cube has an odd number of swaps ("one" swap), this state cannot be reached.

To understand why this is so, we need to realise that each legal move always performs the equivalent of an even number of swaps. No matter how many moves you perform, the number of accumulated swaps will therefore always remain even.

For example, consider the turning of one face by 90 degrees:

C1 E1 C2
E4 E2
C4 E3 C3
C4 E4 C1
E3 E1
C3 E2 C2

The new corner state can be obtained via 3 swaps (swap C1/C4, swap C1/C3, swap C1/C2). Similarly, the new edge state can be obtained via 3 swaps. All together, this is 6 swaps which is even. Therefore, no matter how many moves you perform, always an even number of swaps will have been performed.

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.

Only half of the edge orientations are reachable

An impossible state

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. 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. 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. 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 only remaining faces are the front and back faces. In both of these cases, a 90 degree move will flip all 4 edges, which is again an even number of flips. Therefore, it is never possible, using only legal moves, to flip an odd number of edges.

Only one third of the corner orientations are reachable

An impossible state

With corner orientations, things are slightly more difficult to account for, since each corner piece has three possible orientations, not two. 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). 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).

Once again, to establish this, it is necessary to decide on a frame of reference for correct corner orientation. Notice that every corner either belongs to the top or bottom and therefore each corner has one of its stickers with either the colour of the top face or the colour of the bottom face. The most common frame of reference for correct corner orientation is to say that a corner has correct orientation if its top/bottom sticker is facing up or down. If it is facing in any other direction, then this corner is not correctly oriented. Using this frame of reference, it is easy to see that any twist of the top and bottom faces will not change the orientation of the corners, and therefore the total orientation will remain exactly divisible by 3. For any of the 4 sides, a 90 degree turn will twist two corners by orientation 1 and the other two corners by orientation 2. The total change in orientation is 1+1+2+2 = 6, which is divisible by 3. Therefore, no matter how many legal moves you make in a row, the corner orientation will always remain divisible by 3.

Overall

Combining these laws, only 1/12 (1/2 * 1/2 * 1/3) of the conceivable cube states are reachable by legal moves.

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