Vesneeten: Home to the Blue and Green Eyed Dragons

$\mathbf{Problem:}$ Vesneeten is the island home to a rare breed of dragons who have either blue or green eyes. These dragons live harmoniously, but are highly superstitious, and are afraid of having green eyes. If a dragon ever finds out it has green eyes, it will throw itself into a volcano at noon the next day. A friendly visitor come to Vesneeten and when they leave, tells the dragons that at least one of the dragons has green eyes. The dragons are all perfect logicians, naturally, so what happens?

$\mathit{Proof:}$ Eventually, we will generalize our solution to $n$ dragons, but for now, we can explore some smaller cases to get a feel for things. For the sake of simplicity, every dragon on this island is male because gendered pronouns are much easier to use.

$\mathbf{Case \, 1:}$ There is 1 dragon with green eyes, and the rest of the dragons, any arbitrary number, all have blue eyes. The 1 dragon with green eyes will see that every other dragon has blue eyes. With the knowledge that at least one of the dragons have green eyes, he can deduce the dragon with green eyes must be himself, and throw himself into the volcano the next day.

$\mathbf{Case \, 2:}$ There are 2 dragons with green eyes, call them A and B. Consider the perspectives of dragons A and B. Dragon A sees that another dragon has green eyes, so he can not deduce immediately that he has green eyes. The next day, after dragon A sees that dragon B still has not jumped, he can deduce that dragon B also saw green eyes the first day. Therefore, dragon $A$ deduces that there must another green-eyed, and seeing that there are no other green-eyes besides B, can confirm that he himself must have green eyes. The same logic applies to dragon B, and they both jump the next day.

$\mathbf{Case \, n \,(Induction):}$ In the small cases, we saw how deductive reasoning was used for the dragons with green eyes to confirm the color of their own. Also, note that the number of dragons with blue eyes is irrelevant, since any blue-eyed dragon always sees 1 more green-eyes dragon than any green-eyes dragon. Thus, we proceed using induction on the number of green-eyed dragons, $n$.

$\mathbf{Base \, Case:}$ We already showed the base cases for $n=1, 2$ dragons.

$\mathbf{Inductive \, Hypothesis:}$ I claim that for a population of dragons, $n$ of which have green eyes, all of the green-eyed dragons will jump into the volcano on the $n$-th day. Let this statement be denoted by $P(n)$

We wish to prove this scenario for $P(n+1)$. Choose an arbitrary dragon in the group of $n+1$ dragons with green eyes, call him Mike (No relation to Michael). For now, Mike can assume that he does not have green eyes (Poor Mike), leaving us with a group of $n$ green-eyed dragons. We know by our inductive hypothesis that a group of $n$ green-eyed dragons will all jump on the $n$-th day, and more specifically, each one of them sees $n-1$ other green eyes. However, since Mike also has green eyes, each one of the green-eyed dragons actually sees $n$ other green-eyed, so none of them jump on the $n$-th day. Mike, deducing that there must be more than $n$ green-eyed dragons, confirms that he must also have green eyes, and they all jump on the $n+1$-th day. This completes out induction since we chose Mike arbitrarily from the $n+1$ dragons.