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Is it possible for a Rubik's cube to be Unsolvable?

Is it possible for a Rubik's cube to be Unsolvable?

If you just have an old cube lying around you may wonder whether it is actually possible to solve, I can assure you that every scramble that is possible to achieve using regular legal moves is solveable - and there are 43,252,003,274,489,856,000 of these possible scrambles.

However, there are some impossible cases and these can be the result of taking the cube apart and putting it back together incorrectly. Modern speedcubes I have available here should rarely have pieces come apart but it is designed for it to be possible if you intend to take it apart - just not too easy that it can happen by accident. Some older cubes are so sturdy and robust (but slow and stiff) that they will basically never come apart, while others are really loose and can come apart easily. If a few pieces come out and the cube is scrambled, you are basically guessing where to put them back, and if just 2 edges and 2 corners come out and you put them back in randomly then there is only a 1 in 12 chance that the cube is still possible to solve!

So what are these impossible cases?

Well, there are 3 of them, and you can only see these cases by almost solving the cube, so to find out if your cube is solveable you are going to have to get very close to solving it, if not actually solving it.

case 1:

This is a flipped edge case (assuming the other unseen sides are all solved), this is not possible to solve, only even numbers of flipped edges are possible when they are all placed in the correct position so if you are faced with this case then you need to push up the middle layer about half a turn and then twist the edge and carefully but firmly pull it out, you can then push and twist it back in the correct way round, if the cube is really tight then you may have to loosen the screws to do this successfully.

Case 2:

Assuming everything else is solved on the unseen sides, this case is also not possible as there are only two swapped edges (just two swapped corners is also an impossible case but that can be changed into two swapped edges), this case can never be reached with legal moves so the cube must have been taken apart at some point and reassembled incorrectly, you therefore have to take these two edges out, you can do this by pushing up the middle layer about half a turn and then twisting the edge and pulling it out, you can then more easily take out the other edge and put them back the correct way.

Case 3:

This is probably the most common impossible case because it can occur accidentally without the cube actually ever appearing to come apart, it is simply a twisted corner, if you randomly twist all the corners on a cube then there is a 1 in 3 chance the cube will be solveable, you can check if it is by asigning a 1 if the corner is twisted clockwise, 2 if it is twisted anticlockwise and 0 if it is not twisted, if the total number of twists is a multiple of 3 then the cube is solveable, if it is not then the cube is not solveable, however, you can always reduce it down to just one twisted corner, to fix this you simply twist the corner back, on most modern cubes this is very easy, if your cube is very tight then you might have to loosen it a bit to do this.

Recognising these cases during last layer:

If you know full OLL and PLL you should be able to spot all these cases very early on but it can be fun to give newer cubers cases like this and watch them get confused repeating F R U R' U' F' or R U R' U R U2 R' over and over again, often until you put them out of their misery and tell them what you did. But if you now know that having one flipped edge is impossible then you can spot that as soon as you solve the first two layers of your cube, if you know that one twisted corner is impossible then you can spot that during the second half of two look OLL or you can check by adding up the twists, case 2 is probably the hardest to spot early on, especially if you don't know full PLL but with experience you can spot it anyway and people won't be able to prank you too easily.

Now, of course, this article assumes that the individual corners and edges remained in one piece at all times...

 

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