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"such that the picture falls whenever any k out of the n nails get removed, and the picture remains hanging when fewer than k nails get removed."

I interpret that to mean exactly K nails; no more, no less.

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If it falls when exactly K nails are removed, I don't think it will go back up when you remove the K+1th nail. And if you remove k-1 nails first, and then remove 2 more simultaneously, the picture will fall anyway. So it's ">= K".

"At least k" might imply that removing more than k nails is necessary to make the picture fall, which is not the case. I think "exactly k" is perfectly clear.

"Exactly k" is not clear because the picture will still fall if you remove k+1 nails. I think it's harder to get clearer than the way it's stated in the paper: "the picture falls whenever any k out of the n nails get removed, and the picture remains hanging when fewer than k nails get removed."

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