Green-eyed dragons solution

Let’s start with a smaller number of dragons, N, instead of one hundred, to get a feel for the problem.

If N = 1, and you tell this dragon that at least one of the dragons has green eyes, then you are simply telling him that he has green eyes, so he must turn into a sparrow at midnight.

If N = 2, let the dragons be called A and B. After your announcement that at least one of them has green eyes, A will think to himself, “If I do not have green eyes, then B can see that I don’t, so B will conclude that she must have green eyes. She will therefore turn into a sparrow on the first midnight.” Therefore, if B does not turn into a sparrow on the first midnight, then on the following day A will conclude that he himself must have green eyes, and so he will turn into a sparrow on the second midnight. The same thought process will occur for B, so they will both turn into sparrows on the second midnight.

If N = 3, let the dragons be called A, B, and C. After your announcement, C will think to himself, “If I do not have green eyes, then A and B can see that I don’t, so as far as they are concerned, they can use the reasoning for the N = 2 situation, in which case they will both turn into sparrows on the second midnight.” Therefore, if A and B do not turn into sparrows on the second midnight, then on the third day C will conclude that he himself must have green eyes, and so he will turn into a sparrow on the third midnight. The same thought process will occur for A and B, so they will all turn into sparrows on the third midnight. The pattern now seems clear.

Claim: Consider N dragons, all of whom have green eyes. If you announce to all of them that at least one of them has green eyes, they will all turn into sparrows on the Nth midnight.

Proof: We will prove this by induction. We will assume the result is true for N dragons, and then we will show that it is true for N + 1 dragons. We saw above that it holds for N = 1; 2; 3.

Consider N + 1 dragons, and pick one of them, called A. After your announcement, she will think to herself, “If I do not have green eyes, then the other N dragons can see that I don’t, so as far as they are concerned, they can use the reasoning for the situation with N dragons, in which case they will all turn into sparrows on the Nth midnight.” Therefore, if they do not all turn into sparrows on the Nth midnight, then on the (N + 1)st day A will conclude that she herself must have green eyes, and so she will turn into a sparrow on the (N + 1)st midnight. The same thought process will occur for the other N dragons, so they will all turn into sparrows on the (N + 1)st midnight.

Hence, in our problem all one hundred dragons will turn into sparrows on the 100th midnight.

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