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I am trying to solve the Rubik's cube in the picture using the layers' method. I used successfully the same algorithm on another cube, but somehow when I reach the last stage it just does not work. Does this configuration look legit (hence, I just need to try harder :) ) to you or is this cube corrupted?

enter image description here

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This position is not solvable without twisting a corner, I just tested it on a cube that I have handy.

If you are unable to twist a corner (as that cube doesn't seem like one that would be easy to do that on), you can pop off the caps of each center piece and loosen the screw underneath.
Then, either twist the corners once the cube is looser (though this could be some trial and error), or you could disassemble the whole cube and rebuild it, that way you can ensure that everything is how it belongs.

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You might need to twist the corner pieces. If you can't, its probably broken.

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The cube has a twisted cube which is the yellow-green-red. Twist it and after that do an easy algorithm:

R' D' R D

Then after the corner is solved, make an U' move and repeat the step 4 times.

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