Can the state of Rubiks cube be determined by viewing it standing on the table (three sides)?

Question

When the Rubiks Cube lies on the table you can see only three sides when you are not moving. Is it possible to calculate all pieces of the cube?

In this position, there are only 7 pieces completely not visible. 12 we can see partially. And 7 are completely visible.

The question came to mind when I was looking at the permutations of the cube. Every include at least one that is outside the set of invisible pieces. Does that mean that the cube can be recognized only by three sides?

Answer 1

The Rubik's cube cannot be determined from three faces. There are moves that allow, for instance, two adjacent edge pieces to be reversed (not swapped) without impacting any other pieces.

To see this, imagine that you are looking at the cube and want to do this reversal with the top edge nearest to you (call this E1) and the top edge on the right (E2). These edges are both in the top face. You can do this as follows:

1. Reverse E1 without moving anything else on the top face. This will obviously mess up the rest of the cube, but now the top face will have E1 reversed and be otherwise unchanged.
2. Turn the top face clockwise, so that E2 is where E1 was before.
3. Undo the moves that you did in E1.
4. Turn the top face anti-clockwise.

Note that if you had not done step 2 or step 4, step 3 would simply have undone step 1 and returned you to where you began.

The neat thing, then, is that step 3 fixes everything that was messed up in the rest of the cube in step 1, while reversing the edge that's on the top and closest to you. Because of step 2, this causes E2 to be reversed rather than E1.

The result is that both E1 and E2 are reversed, while nothing else is changed.