Answer to Question 1: To help you understand the answer to the puzzle, I'm going to walk you through three different variations of the game.

First, imagine that there was only one blue-faced person on the island. What would happen? Well, clearly, after the stranger came to the island, he would know that he was the one with the blue face since he was the only one that could have a blue face. So he would kill himself the first night.

Ok, so, second: Imagine that there were only two blue-faced people on the island. What would happen? Well, the first day after the stranger came to the island, one of the guys (call him Rod) would look at the other guy (call him Todd) and think: "I know that Todd has a blue face, but I don't know that I have a blue face (otherwise I would've killed myself before today if I thought I did have a blue face), so Todd must be the one who has a blue face. Moreover, Todd must be looking at my NON-blue face and figuring out that he is the one with a blue face, so he will surely kill himself at midnight." But Todd, being just as rational as Rod, is thinking something similar about Rod (since Rod, contrary to his assumption, does indeed have a blue face). So both Rod and Todd are expecting that the other will kill himself by midnight on the first night. This is because each thinks that he himself does NOT have a blue face, and so each makes an assumption about what the other is seeing. However, when both realize the next day that neither has killed himself, what is the only thing they can conclude? That their assumption about their own faces is incorrect; that both Rod and Todd were each seeing a blue face, in which case they must both kill themselves by midnight the second night.

Ok, so now imagine that you are a silent observer, hanging out on the blue-faced island, watching a two-person blue-faced game. Given the above reasoning, you should predict that it will take two nights for the two blue-faced people to realize they each have a blue face. However, let's imagine that, much to your surprise, neither of the blue-faced people kill themselves on the second night. You should then realize that you yourself are part of the game, and that you must have a blue-face as well. For the only explanation for why two blue-faced people haven't killed themselves on the second night is that there is another blue-faced person around. They are all waiting for the other two people to kill themselves because they are all assuming that they themselves do not have a blue face. So, once you have realized that you have a blue face, and once the others have come to similar conclusions using similar reasoning, you should all kill yourself on the third night. And this is just what happens with the blue-faced people on the island.

To put the answer a slightly different way: Consider the first guy, Rod, the second guy, Todd, and the third guy (call him Maud). Now Rod thinks to himself: "I know that Rod and Maud have blue faces, and I know that Rod knows that Maud has a blue face, and I know that Maud knows that Rod has a blue face. But Rod probably thinks that Maud sees no blue faces, since I am assuming I don't have a blue face, and Rod will be assuming that he (Rod) doesn't have a blue face either. So when Rod thinks about what Maud is seeing, he assumes that Maud is seeing two NON-blue faces. And so Rod will expect Maud to kill himself on the first night. Maud will work through similar reasoning regarding Todd. And so both Rod and Maud will expect that the other will kill himself on the first night. When this doesn't happen, however, they will realize that they each have a blue face, and so they will kill themselves on the second night. If this doesn't happen, however, and if they are still both alive on the third day, then I will know that I, too, have a blue face. For Rod and Maud will have gone through the same sort of reasoning as I have. So, if everyone is still alive on the third day, then we should realize that we all have blue faces, and so we will all kill ourselves at midnight on the third night."

Answer to Question 2: The new information that the stranger provided was this: When one of the guys--take Todd again--thinks about what the others are seeing or what the others know, he thinks that one of them sees no blue faces. For example, when Todd assumes that he himself does not have a blue face, and he thinks about what it is that Rod thinks that Maud sees, given that Rod is also assuming that he (Rod) has a non-blue face, then he will conclude that Rod thinks that Maud sees two non-blue faces. Yet if this conclusion is correct, then the stranger's utterance will be news to Maud--namely, it will inform him that he is the one with the blue face. It will also be news to Rod, if Todd wants to speculate as to what Maud thinks that Rod sees.

Put another way: one of the things that each of Rod, Todd, and Maud did NOT know was what one of the others thought that the third person was seeing. The stranger's utterance let's them know that if one of them thinks that another is seeing no blue faces, then the one who is seeing no blue faces must have a blue face.

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