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by amelius
2623 days ago
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How about these problems: Given a cube where two squares have been swapped (so it's not a valid cube anymore), how fast can you determine that it's the case? And how fast can you shuffle it to the state that is closest to the solved state? And for more than two swapped squares? |
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Two random squares being swapped isn't really something that happens in real life. But abstracting that, you'd notice as soon as you've placed the piece one of them is on, which is kind of hard to predict in general, it'll really depend on where it is wrt which side you started from.
What does happen in real life is two pieces ("cubies") getting swapped, say after assembling back up a popped cube. Using CFOP, you'd notice the inversion itself one sequence before the end, though you might notice the fact they're not oriented properly one sequence before that.
For more swapped pieces, (theory ahoy) there's two cases. Odd number of swaps: they're all equivalent to the case above. Even number of swaps: equivalent to no swap at all. Interestingly, you can cancel out an edge piece swap with a corner piece swap.
I hinted at it above, and it's related: no need for swaps, you can make the cube unsolvable by flipping an edge or rotating a corner in place. A CFOP solver would notice two sequences before the end. Edge flips cancel each other out by pairs; corner twists cancel each out by triplets of the same orientation. Contrary to swaps, there's no catching up a flip with a twist.
In all cases, almost-solving them is just a gasp of surprise away from solving an untainted one.