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Community Answers
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plasmid's post in blind bartender was marked as the answer
Very neat problem. The answer is certainly unintuitive.
Edit: renamed states for clarity
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plasmid's post in A plane old stuffed cube was marked as the answer
If no one else is going to go after this one... I can give an answer based on, well, working in the spirit of the best solution the program could find. But I can't prove that there aren't any larger squares that can fit in the unit cube.
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plasmid's post in Calculated was marked as the answer
Now we just need seven more of those to make the whole tongue-twister.
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plasmid's post in I got Charlie's number was marked as the answer
I don't know about everyone else, but I'm stuck.
If the "I" in this problem has information that we (the problem solvers) are not privy to by virtue of the OP, then I could come up with a solution.
If the "I" in this puzzle definitely does NOT have any more information than what's presented, then I am Stuck with a capital S.
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plasmid's post in How many faces? was marked as the answer
The shortest and most intuitive (although least formal) proof of the answer that I can come up with.
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plasmid's post in 4 points, 2 lengths was marked as the answer
Without any sort of formal proof, I think that these are all possible shapes. Although I haven't yet fully convinced myself that the last shape I drew actually exists. It seems like it should though.