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I bought a new Rubik's Cube and was able to solve it with some tutorials. However, I made a wrong movement, so that one piece came out. I think, I put it wrong together, because I'm not able to solve the cube anymore. When I try to solve my cube online with https://cube-solver.com/ I get the following error:

This configuration of the cube is not solvable, please check that you've entered all pieces correctly.

This is my cube. As you can see, I can't solve the last layer anymore:

enter image description here

Is it possible to find out, which piece is wrongly assembled?

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  • $\begingroup$ You might be interested in this post, which explains how to solve permutation puzzles using commutators and with a brief explanation of parities. $\endgroup$
    – user21820
    Jan 22, 2022 at 5:18

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There is never a particular piece that is the wrong one, but you can find out how to fix it.

There are three conditions that a solvable position must satisfy.

  1. The permutation parity must be even. This means that if you were to put the pieces in their correct locations by swapping pieces, the number of swaps you need must be even. If it is odd, then the position is not solvable using normal turns.

Your cube satisfies this condition. There is a 3-cycle of corners, and a 3-cycle of edges. Each 3-cycle would take 2 swaps to fix. This is 4 swaps all together, which is even.

  1. The corner orientations must be balanced. This means that if you were to fix the corner orientations by twisting them only clockwise, the total number of twist steps is a multiple of 3. If it is not a multiple of 3, then the position is not solvable using normal turns.

Your cube also satisfies this condition. Only two corners need to be twisted to make the last layer corners show yellow. One needs one twist step, the other two steps. This is 3 steps all together, which is a multiple of 3.

  1. The edge orientations must be balanced. This means that if you were to fix the edge orientations by flipping them one by one, the total number of flips is even. If it is not even, then the position is not solvable using normal turns.

Your cube fails this condition. There are three edges that need to be flipped to make the last layer edges show yellow, which is odd.

To fix your cube, you need to take out one edge piece (it does not matter which one, it may even be a currently solved one) and put it back in reversed. This will make your cube solvable again.

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  • $\begingroup$ For reference, what do you do for the other two cases? For the first case, I presume you swap two pieces, taking care to preserve or restore balance as you do. And for the second, I presume you rotate a corner either once or twice, depending on if the total corner twists is 2 or 1 mod 3. Is that correct? $\endgroup$
    – No Name
    Jan 22, 2022 at 2:04
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    $\begingroup$ @NoName: For (1), you don't need to care about orientation, because you only care about number of swaps of the pieces. For (2), you can just add mod 3 if the corners are in the right positions. For (3), you can just count the number of flips if the edges are in the right positions. If you're asking how to compute the parity without putting the pieces in the right positions, it's not hard, but not obvious and not that simple. Let me know if you want an answer to that. $\endgroup$
    – user21820
    Jan 22, 2022 at 5:24
  • $\begingroup$ No you answered my questions, thank you (although I'm sure there's a more avid cuber somewhere who would appreciate it) $\endgroup$
    – No Name
    Jan 22, 2022 at 14:32
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The accepted answer is very nice and will help you understand a lot about Rubik's cubes, but the way I have found most practical is to:

Remove as many pieces as necessary and put it back together in the fully solved state.

This is nice and simple to remember and execute, and for most people and most cubes it takes less time to just rearrange the whole cube if necessary vs. figuring out theoretically how to do the minimum number of rearrangements.

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  • $\begingroup$ It does depend on how well you understand the cube. For me, it was obvious in one second when looking at the question that there was at least an edge parity. If I saw the actual cube instead of a flat diagram, it would be equally obvious that there is no corner parity. But it's true that some parities are not obvious, such as the edge-corner joint-parity. Still, it's faster to solve the whole cube until the parity issue becomes obvious (< 30 seconds) than to disassemble the whole cube and put everything back. $\endgroup$
    – user21820
    Jan 22, 2022 at 5:31
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If you want a quick answer: just take any one of the last layer edges (two colored pieces) out, flip it and insert back in the same place. This is one of the most common issues that can happen and the WCA regulations (5b3 and 5b5) allows you to fix it this way in a competition.

If you want a more detailed explanation, check Jaap Scherphuis' answer, it's very good. Or try reading some simple guidelines to easily check if your cube is in an "invalid" (non solvable by usual movements) state, there's also this article on wikipedia about the Rubik's cube and also this other one more theory-oriented. There's quite a bit of nice reading material in the references if you're interested in learning the math behind such puzzles.

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