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I have a (nearly solved) Rubik's Cube:

Like this

However, as you can see, one edge only is flipped, making it unsolvable. I have scrambled this unsolvable Rubik's Cube and now it looks like this:

Here

And Here'

What are the signs that this Rubik's Cube is now unsolvable?

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Select any edge. In your head it should be fairly easy to imagine how to bring it to its home location using only moves of the U, D, L and R faces, and not the F or B faces (though if it makes it easier, you can use half turns F2 and B2). If it would arrive at its location flipped, then consider the edge flipped, and if it would arrive correctly oriented then it is not flipped.

Mentally do this for each edge, to determine which edges are flipped. If an odd number of edges are flipped, it is not solvable.

You can also recognise flipped edges by their colours, but I always get confused about the equator edges, so I prefer what I described above.

Corner orientation is easier to tell by their colours. Assuming you have white and yellow on the U/D centres, a corner has no twist if its yellow/white facelet is next to a yellow or white centre. It has +1 twist if it is twisted once clockwise from its no-twist orientation, and a -1 twist if it is twisted once anti-clockwise from its no-twist orientation.

Add up all the twists (i.e. count the clockwise twisted corners minus the number of anticlockwise twisted corners) and you should have a multiple of 3 (i.e. -6, -3, 0, 3, or 6). If not, the cube is not solvable.

There is also permutation parity, which is much harder to determine on a mixed cube.

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