The cool part is that 30 is valid over an arbitrarily large spread of sets. Yes, we need a statistician to show this (or I have to hunt down some 20-year-old notebooks), and of course you could generate more than 30 solutions, but there's a knee-of-a-curve thing about "30": basically it's hard to draw 30 "solutions" from a pot full of randomly arranged solutions and not have "one" come from each of the extreme edges of possible solutions, i.e. from better than 90th percentile and worse than 10th. That's what makes this the Law of Large Numbers--it is irrelevant how many possible solutions there really are.
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