*ahem* sorry for the delay, but below is the best written version of the solution I could find. I had to read it a couple of times, but it should lay out the answer to this puzzle. Woof!
How does one simplify this puzzle? By making it the 1 Green-Eyed Dragon Puzzle. If you tell a single green-eyed dragon that "at least one of you" has green eyes, that dragon would know instantly and unambiguously that she has green eyes. At midnight she would turn into a sparrow. (poor dragon!)
Now let's imagine 2 green-eyed dragons staring at one another, after being informed by you that at least one of them has green eyes. Each would look upon the other and, seeing a set of green eyes, think the following: "Do I have green eyes? I don't know. But if I do not, then this other dragon, upon seeing my non-green eyes, will know instantly and unambiguously that he is the one with green eyes, and at midnight will turn into a sparrow." Each dragon sits and waits to see what the other does. When, at midnight, neither dragon transforms into a sparrow, each one knows instantly and unambiguously that the other dragon did not leave because it, too, saw a dragon with green eyes. And so, on the second night, each transforms into a sparrow at midnight.
Let's expand the problem to 3 green-eyed dragons. Following your announcement, each dragon thinks to itself that if it does not have green eyes, then the other two dragons will determine their eye color by the reasoning laid out in the 2 green-eyed dragon scenario presented above. In this case, all three dragons wait for the other two dragons to transform into sparrows on the second midnight. When this does not happen, each of the three dragons concludes instantly and unambiguously that it has green eyes. On the third midnight, all three transform into sparrows.
Through the process of induction, we conclude that any number of green-eyed dragons, N, will all turn into sparrows on the Nth midnight following your seemingly inconsequential observation.
I hope this helps!
1 comment:
I didn't get it until the 3rd night. I like my solution better. It's much nicer. (what sound do otters make that would be like "woof"?)
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