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I wanted to start cubing, so I have a cube that allows for the center pieces to be swapped, so when it was dropped on the ground, they all fell. Now there's 6! combinations of center pieces. Is there a way for me to find the correct combination without checking each one of them to see if it is solvable? I can provide images of the sides.

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The corner and edge pieces can be used to determine which color sides are adjacent. Find all corner pieces for one color face, and from there you can determine the colors of the adjacent 4 faces and how they are connected. The final color goes on the opposite face.

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    $\begingroup$ Unfortunately there is a 50% chance that the cube is unsolvable due to permutation parity. You will have to do a check like this to be sure (or solve it until you have two pieces that need to be swapped), and if it is unsolvable, cyclically shift the four centres of one slice to fix that. $\endgroup$ Commented Feb 14 at 18:03
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    $\begingroup$ @JaapScherphuis As I understand it, the OP just wants to know how to arrange the 6 center squares - none of the other cubes have been rearranged. If only the center squares came out of a solvable cube, there is no possibility the cube is unsolvable when reconstructed. $\endgroup$ Commented Feb 14 at 18:11
  • $\begingroup$ If none of the other cubes are rearranged, it is trivial to put the centres back in correctly. They also offer to provide images of the sides, so I'm pretty sure the cube was scrambled before the centres fell out. $\endgroup$ Commented Feb 14 at 18:12
  • $\begingroup$ @JaapScherphuis The cube is still scrambled. By "not rearranged", I mean there haven't been any other illegal parity-breaking moves. If you start with any solvable cube configuration and take the centers out, you can of course put the centers back in a solvable configuration. The OP isn't starting from an unsolvable cube and wondering where to put the centers. $\endgroup$ Commented Feb 14 at 18:16
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    $\begingroup$ On a normal cube the only 4-spot pattern you can make is one where two pairs of opposite centres are swapped, as if the inner structure has been rotated 180 degrees. It is impossible to get the pattern where it is rotated 90 degrees because that would be an odd parity permutation. Therefore, if you put the centres in as you describe, you have a 50% chance of it being solvable. $\endgroup$ Commented Feb 14 at 19:15
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Step 1: choose one corner and set 3 center colors accordingly. Step 2: 3 remaining colors belong to opposite sides, orientation determined with opposite corner. Step3: correct one of 3 possible orientation can be reduced to 1 with any of edge pieces.

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If you put the yellow cap on top, and the white cap on the bottom, then place the blue in any place, put red to the right of the blue face and green to the right of the red face and at last place the orange on the last remaining side.

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