So what do you hope *is* the aesthetic value of puzzles? I have my own thoughts on this, but I'd rather hear yours. (I say "aesthetic" value because that's what you said. They also have lots of other virtues.)
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Replying to @johncarlosbaez @Cshearer41
Well, if we are to benefit each other by discussing this matter, we must first agree on the meanings to be attached to the terms we use. I have tended to think of a puzzle as a question devised by one human being to divert another -- that is, as something essentially artificial.
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If we use the word "puzzle" here in this way, then we must recognize that many tantalizing mysteries are not puzzles. It's absurd, for example, to label the question, "Why does the fine structure constant have its observed value?" a puzzle. There's nothing artificial about it.
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I would likewise argue that we cannot label with the word "puzzle" such questions as, for example, "Which prime numbers arise as divisors of the values taken on by x^3 - x - 1, as x ranges over the positive integers?"
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As Poincare says, "There are questions that one poses, and questions that pose themselves." Certain problems have a species of inevitability that contrasts rather markedly with the contingency and artificiality that seem to me characteristic of puzzles, properly so called.
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If we can agree that a puzzle is essentially an artificial entertainment, like a play or a joke or a musical composition, then I presume we can also agree that the aesthetic value of a puzzle lies in its power to affect us emotionally. Art matters only because it moves us.
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I would argue that most of the aesthetic value of a mathematical puzzle lies in its power to surprise and delight us. Penrose makes this point eloquently; he notes that we're deeply affected when we discover at a glance an answer we feared we could get only by tedious struggling.
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But I would note that some surprises affect us more profoundly than others.
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Consider, for example, these two jokes: (1) As a man prepares to leave for work, his son says to him, "Father, the man next door kisses his wife every morning before he leaves for work. Why don't you do that?" The father answers: "Son, I scarcely know the woman."
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And: (2) A father, unsettled by his teenage daughter's long, darkness-shrouded visit with her suitor on the front porch, accosts her. "Daughter," he says, "your young man stays very late. What does your mother say about this?" "She says that young men haven't altered a bit."
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Perhaps the distinction here is clear. Both jokes surprise us with their clever conclusions, and in both cases it is surprise that moves us. But the second joke does something more than surprise us: it hints at a general principle -- and one of some depth, too.
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I think a great puzzle should have the same character as this second joke, of at least hinting at something beyond mere cleverness. I think it's possible for a puzzle to do this, and I always feel a vague pang of regret when a puzzle I've solved falls short of this standard.
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