BTW would the same statement from a blue-eyed Guru make any difference to the outcome?
It shouldn’t–it would just mean the guru got to leave with the rest of the blue-eyes.
I would think it means nothing changes. If the guru was bound by the same rules as the rest of the islanders, I believe she doesn’t leave and also doesn’t affect the day the rest of them leave. They all know the person with blue eyes the guru sees is not the guru herself. So if there is only one blue eyed person, he leaves the first night. Etc.
Mithras, no, I’m not relying on the Islanders guessing. I’m relying on their powers of observation and ability to draw appropriate inferences. Say there were ten people with pink-eyes. Obviously everybody knows there are pink-eyes. Does everybody know that everybody knows? I believe the answer is yes. If true, they don’t need the guru to supply that information. There’s some minimum below which this isn’t true. I think it’s four, in which event each person of that color will see only three, not be able to infer that everybody knows, and thus need the guru’s observation. At five and above, the guru’s observation is superfluous. Except, as jtgain says, to start counting days. And, frankly, I think that could have been done more cleanly.
Stated a little differently, the important assumption upon which the puzzle turns is something other than what the guru says. It’s that the Islanders are ninja logicians. Each realizes there’s a solution (counting days), there’s only one solution (if this isn’t true, the whole thing collapses), all the other Islanders (being ninja logicians) will realize there’s a solution and only one and, therefore, all will apply the same logic. And the logic will work if everybody knows that everybody knows such-and-such color exists (which they will for all cases where N is five or more) and if everybody knows on which day to start counting. If there had been only one, two, etc. are just abstractions which don’t have to be true, and indeed are known not to be true if N is five or more.
Nah, you’re still not getting it. It’s easy to forget that we, as external observers, can know things that the islanders cannot. Try putting yourself in their shoes. Let’s say there’s five brown eyed islanders as you suggest - Alan, Bob, Charlie, Dave and you. You see four pairs of brown eyes, so you know there are either four or five browns. But what do you think Alan sees? If your eyes are not brown then he sees three pairs of brown eyes, and his chain of logic might be different to yours.
Extending this further - you believe, that Alan might believe, that Bob might believe, that Charlie might believe, that Dave might believe, that there might be no brown eyed people on the island! Or to put it another way, you *do not *know that everyone knows, that everyone knows, that everyone knows, that everyone knows, that there is at least one brown eyed person on the island. Changing to ten, fifty, or a hundred people just adds more iterations to those statements.
I’m loving this puzzle more and more. The solution seems so crazily counter-intuitive but once it clicks with you its just inescapably true.
No, this can be proved false almost the same way the actual solution is proved true.
If there is only one blue eyed islander, he will never leave. This is obvious since he will not even know a blue eyed islander exists.
Assume that if there were n blue eyed islanders they would never leave, and suppose there are n+1. Each of those n+1 can see that there are either n or n+1 blue eyed islanders. Each new day, only 2 things can provide information: “someone left last night” or “nobody left last night”. “Nobody left” is always consistent with only n blue eyed islanders. “Someone left” is impossible, because the situation is symmetric. So nobody can ever eliminate the possibility that there are only n blue eyed islanders, and nobody can ever leave.
By induction, nobody will ever leave. It is essential that the guru announces that a blue eyed islander exists.
First, while everyone would know that everyone knows there’d be someone with pink eyes if there were ten islanders with pink eyes, the point is that it’s not true that everyone would know that everyone knows that everyone knows (x10) that there is at least one islander with pink eyes.
Second, on what basis do you determine at what level the guru’s information is necessary? We might be able to intuitively grasp why it’s needed up to a certain level (four or five people, say) and think it’s not beyond that. But if we rely on that to determine when we think the islanders leave, it points more to us not being perfect logicians than anything to do with the puzzle.
Third, is the question I’ve asked several times. If all they’re doing is counting days and not gathering information, why would they choose to count from 0 to 200 instead of 200 to 0? What makes one direction more logical than the other?
There are many solutions where the islanders pick an arbitrary method and carry it out. They might seem reasonable, but they are not logical. Additionally, they don’t lead to the islanders gaining actual knowledge about their individual eye colors. They might consider your imaginary world where the islanders pick an event and start counting eyes and successive nights. But they would also consider the scenarios where they start the count the night before or the night after. Being perfect logicians, they would understand there are many possible worlds where the other islanders wouldn’t be counting days. Therefore they would only be guessing their eye color.
Finally, in the correct solution the islanders are not simply counting days. They are learning each night the minimum number of blue eyed islanders that each person knows that each person knows that each person knows (repeated forever) there are.
I still don’t get what you said?
You appear to have failed to comprehend that “Everyone knows that everyone knows that there are pink islanders.” is an entirely different statement than “Everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that there are pink islanders.”
The top one is true. The bottom one is not true until the guru gives his information; then it becomes true.
I have accepted the error of my ways. The announcement does serve the purpose of counting days, but ALSO the announcement of the specific color puts islanders on the lookout for that color and makes them put themselves in the “color” or “not that color” category.
That’s why when the guru says “blue” the blue eyed people eventually leave, and the brown people never leave.
If he says “brown” the brown people leave and the blue eyes stay forever.
I’m pretty sure I’m right, but I’m going to let this go.
What? No. He could’ve said “There may or may not be any blue eyes on this island” and everyone would’ve stayed. It’s got nothing to do with “being on the lookout” for anything.
Do you have a reason for thinking you’re right?
All those with blue eyes get to leave. When asked by the “gatekeeper” what color their eyes are, everyone (whether blue- or brown-eyed) says “blue”.
He probably thinks he has a reason, but why don’t you just save him some time and do a 100 layer version of your diagram?
And how, with no communication, are they going to coordinate this?
NO!
[stomps out of room]
I think just a 3 or 4 layer version is suffice to prove somebody where the think they’re wrong in the logic but even that takes a lot of effort to think about.
Unfortunately, PBear has stated explicitly that he thinks it works for small numbers like 3 and 4, but not for larger numbers like 100.
So for him, one might really actually have to go through the 100 person reasoning step by step–which would take, I guess, several weeks at least on the forum…
Well, the bottom one is also true from the start on an island with 100 islanders (I’ve lost track of the thread; have we changed the number of islanders to 10?). But the version with 100 "Everyone knows that"s is not true from the start on an island with 100 islanders.
I’m sure Chessic Sense implicitly meant that, since they certainly seem to understand the problem perfectly, but I just want to get everything explicitly pedantically correct in this thread, to stave off unnecessary misunderstandings…