You own a
traditional 3x3x3 Rubik's cube, with nine stickers on each of its six faces.
When you're not looking, a mischievous sprite randomly selects two stickers, peels them off and swaps their places on the cube. What is the probability that the cube can still be solved?
If the two stickers have the same color (probability 8/53), the cube is still solvable.
I think that is the only way it will still be solvable.
When they are not the same color:
If the pieces are of different types (center, edge, corner) the color count of the types will be upset.
If both are center stickers, if they are opposite faces, the handedness of the cube will change but not the handedness based on the corner pieces. If adjacent faces the two corner pieces the faces share will also have a different handedness.
If both are corner pieces, the handedness will change (if along same edge) or the actual content set will change. (a bit of hand-waving here; could be more explicit)
If both are edge pieces, at best it is equivalent to swapping two of the pieces, but pairs of pairs must be swapped to make a solvable cube. At worst the count of color combinations will be thrown off.
The probability should be 8/53.
Correction made per Steven Lord's comment.
Edited on June 1, 2021, 9:16 pm
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Posted by Charlie
on 2021-06-01 12:25:59 |
solution -- I think
