Antti

http://terrytao.wordpress.com/2011/05/19/epistemic-logic-temporal-epistemic-logic-and-the-blue-eyed-islander-puzzle-lower-bound/

Once more, I try to explain this as patiently and simply as I can. ydoaPs, you seem to keep missing points that I have stated multiple times already. But it's probably because I don't state my case clearly enough... imatfaal, thanks We cannot run the same solution for brown-eyed islanders the amount of information about the brown-eyed islanders is not equal to the amount of information about the blue-eyed islanders. Hence the solution should not be running simultaneously. Wikipedia: What's most interesting about this scenario is that, for k > 1, the outsider is only telling the island citizens what they already know: that there are blue-eyed people among them. However, before this fact is announced, the fact is not common knowledge. For k = 2, it is merely "first-order" knowledge. Each blue-eyed person knows that there is someone with blue eyes, but each blue eyed person does not know that the other blue-eyed person has this same knowledge. For k = 3, it is "second order" knowledge. After 2 days, each blue-eyed person knows that a second blue-eyed person knows that a third person has blue eyes, but no one knows that there is a third blue-eyed person with that knowledge, until the third day arrives. In general: For k > 2, it is "(k − 1)th order" knowledge. After k − 1 days, each blue-eyed person knows that a second blue-eyed person knows that a third blue-eyed person knows that.... (repeat for a total of k − 1 levels) a kth person has blue eyes, but no one knows that there is a "kth" blue-eyed person with that knowledge, until the kth day arrives. The notion of common knowledge therefore has a palpable effect. Knowing that everyone knows does "make a difference. When the outsider's public announcement (a fact already known to all) becomes common knowledge, the blue-eyed people on this island eventually deduce their status, and leave. There is no reason whatsoever why there would be two solutions running simultaneously. You see the difference between everyone knowing something and everyone knowing that everyone knows something. First and second order knowledge there. The thing is this puzzle requires 100th-order knowledge. So it is not enough that "they already knew that at least one blue-eyed islander existed [and that] they also already knew that at least one brown-eyed islander exists." So what happens is that the explorer's words increase the stock of 100th-order knowledge. Second-order knowledge is not enough. And as no-one increases the order of knowledge concerning brown-eyed islanders they can't reason as the blue-eyed islanders do.

It takes time to think it out. Now then, any new puzzles for me?

After the first suicidal event: Each islander left will reason that if they had have blue-eyes the first suicidal event would not have taken place, as it did take place they, without any doubt, must have brown eyes. I think I should maybe rename this thread... "Arguing about my favorite logic puzzle" LOL!

The explorer says that there exists one pair of blue eyes so at least one blue-eyed islander. Did you read my explanation, if you did, I would just be repeating myself, if you didn't, I suggest you do. Or I might just as well post it here. Here we go. Oh, before that, the reason why it doesn't break up is that the explorer introduces new knowledge, and it doesn't need to be directed. The thing is that everyone knows that everyone knows... If this explanation doesn't do I found another by xkcd's Randall: http://xkcd.com/solution.html And Wikipedia seems to use this in as an example in context of common knowledge: http://en.wikipedia....(logic)#Example No, the information the explorer gave to them is (in addition to what you said) that "everyone knows that there exists at least one blue-eyed islanders. See the difference.

Thank you. I don't mind at all; if anything it's helpful to think of ways to explain something to people, it only deepens one's understanding on the subject. And I like to defend my case so I think I will enjoy it here. I wouldn't mind discovering some new puzzles though. Oh, I might add this to enlighten the case with three blue-eyed islanders: After 2 days, each blue-eyed person knows that a second blue-eyed person knows that a third person has blue eyes, but no one knows that there is a third blue-eyed person with that knowledge, until the third day arrives. Okay, case with four islanders. Let's call them Andy, Bob, Cindy, Derek. After explorer speaks. Andy, Bob, Cindy, and Derek will all see three islanders* with blue-eyes and reason "if I don't have blue eyes, there will only be three blue-eyed islanders and they will all commit suicide on the third day". Well, the third day comes and no one commits suicide as none of them yet has proof for themselves being blue-eyed. After no-one commits suicide Andy must conclude that Bob, Cindy, and Derek all see three blue-eyed islanders. And since he can only see three (which would mean that each of them saw only two blue-eyed islanders) he must conclude that the third one they see is he himself. * Andy sees Bob, Cindy, and Derek. Bob sees Andy, Cindy, and Derek. Cindy sees Andy, Bob, and Derek. Derek sees Andy, Bob, and Cindy. See what I did there? Sure, you do. Here!

All blue eyed islander commit the suicide at the same day, thus all islanders that remain have to conclude that they have brown eyed (since there are only two possibilities either blue or brown eyes). The only think Johnny needs to know is that Hugh must see a blue-eyed islander as he hasn't committed suicide. As he can see only Hugh having blue-eyed he must reason that he has blue-eyes. (And after this surely Johnny knows that there are two blue-eyed islanders, himself and Hugh who he can see).

Well, the title is quite self-explanatory. So, what are your favorite logic puzzles and why? (I'm running short and need new ones!) My favorite so far is probably the blue-eyed islanders puzzle. It took me long enough to solve it, but I managed to do so without cheating. I like puzzles that seem counter-intuitive at first. So here's a version of the puzzle:

I'm vegetarian. I drink way too much -- coke. I don't really drink alcohol. I can taste if someone asks. I tend to eat whatever vegetarian food (there's two choices at least) the university's main cafeteria has to offer. I do consume the recommended amount of water as phosphoric acid. What else... No need to purify water as I live in Finland. Usually I don't add salt to anything. I think my diet is quite healthy except for the coke consumption. I definitely eat enough vegetables and I know that I get enough micronutrients.

Hi everybody! My first name is Antti, pronounced [ˈɑntːi] (though pronunciation doesn't really matter as this is an internet forum and I don't really mind anyway, anti (as in 'antimatter' is fine with me)). I'm studying mathematics and biochemistry at the University of Oulu. Other than that, I also enjoy theoretical physics and chemistry in general. I like to play around with Mathematica. I'm here to make friends. That's about it. EDIT. I forgot to mention, I also have a blog.