Sorry, that’s what I meant of course… 
For N=1 I think the information provided is exactly what’s communicated directly by what the Guru said. (He sees at least one blue-eyed person). The single blue-eyed islander in this case will not need to resort to forming beliefs about what “everyone knows” in that scenario. (Being a perfect logician, he’ll form the belief. But the belief need not enter into his practical reasoning.)
Just as a follow-up, there’s a very good inductive proof of the solution on this page.
And on his solution page xkcd poses a few questions “that may force you to further explore the structure of the puzzle:”
He doesn’t provide any answers and even though I’ve just about wrapped my head around the solution now I couldn’t satisfactorily answer any of them!
Haha, yeah, I’ve struggled with so many of those riddles that involve understanding the truthfulness of each actor that I am always calling into question whether or not a statement is true, and whether or not the other actors know that the statement is true, etc.
Thank you to EVERYONE who has done such a fantastic job explaining this riddle! I just shared this riddle with a friend of mine, and helped him work his way to the solution with a few guided hints (think about an island with 1 person, 2 people, 3 people, etc), and he seemed to get it! So, I guess that means I understand it fairly well 
Man, I’m still messed up I think i’m like over thinking it because I can’t even get past the “no new information” from what the guru said way of thinking. But I’ll ask again and hope one day I’ll eventually get it (hopefully sometime this year? he he). If they were all perfect logicians and they all knew this already wouldn’t they already know that the guru sees at least one blue eyed person?. I still don’t understand how that changes anything? they already know he knows this?. I mean if this was straight mathematics with only numbers like a writing a program with el if statements maybe I wouldn’t take the person’s actions so seriously but I don’t like the fact that everyone already knew what he was thinking? am I just stuck on something so simple?.
No, this is not a simple point to be stuck on. In fact, it’s really quite difficult to grasp and even I can’t say that I could explain it very well for the 100 person case.
But if you can think back to the 2 person situation, I think you’ll agree that new information is added EVEN if both people have blue eyes.
If it’s just the 2 people with blue eyes, they are stuck there forever with the guru, because they can’t ever know what color their eyes are. Person A knows that the guru sees at least one person with blue eyes. And person B knows this as well. But person B doesn’t know that person A knows this, and that’s the crux of the matter.
But when the guru speaks and says that she can see at least one person with blue eyes, person A knows that if person B doesn’t leave on the first night, then he has blue eyes too and will leave on the second night with B.
Similarly you can work it out for 3 people, and so on and so on.
(For those of you who understand the solution better than I did, correct me if I made a mistake in my reasoning/explanation)
They knew the guru knows this. But because he hadn’t said it out loud, they didn’t know whether everyone knows whether everyone knows whether everyone knows … (do this a total of 100 times) … whether everyone knows whether everyone knows that the guru knows this.
Just like in the two-person case. Both know the guru knows there’s a blue eyed person. But neither knows whether the other knows the guru knows there’s a blue eyed person.
This type of puzzle tends to operate with the assumption that, absent some explicit qualification to the contrary, statements are true, But it would clearly be better if the problem included something along the lines of “… the Guru (known by all to always speak the truth)…”.
I also think it would be better if the paragraph that begins with “There are no mirrors or reflecting surfaces, nothing dumb. It is not a trick question…” were replaced with: “Valid solutions to this puzzle use only the information presented above.”
Okay, let me take my shot at explaining this, after a mostly sleepless night trying to understand it.
So, first off, damn you Randal (of XKCD), and damn you, Dave, who should know better than to point me to these things late in the evening. I need my sleep!
Now, get rid of some of the baggage around the story. There is nothing inevitable about this logic. It makes perfect sense for them all to look at each other after the Guru speaks, and think, “So do I.” Then they all turn on the Guru for wasting her opportunity to say something meaningful. This version ends badly for the Guru.
I’d prefer if the puzzle were stated as, on the 100th day, all the blue-eyed people left. Explain their logic. (This makes it sounds less like there’s an obvious answer and you’re an idiot for not immediately seeing it. I mean, come on, there are 200 people capable of individually intuiting the correct answer, and they couldn’t figure out any way to communicate? And I’m the stupid one?)
If you are on the island, the Guru’s statement changes your problem from, what color eyes do I have? into, do I have blue eyes, yes or no? And since a blue-eyed person sees 99 sets of blue eyes, their question is, are there 99 sets and I don’t have blue eyes, or are there 100 and I do have blue eyes? (The brown-eyed people see 100 sets of blue eyes, and wonder if there are 100 or 101 sets. But they’re already condemned to 100 days of suspense and ultimately crushing disappointment. Day 101 ends badly for the Guru as well, if there’s any justice.)
To put it another way, if I don’t have blue eyes, then all those other blue-eyed people only see 98 sets. If I do have blue eyes, all those other people see 99 sets, just like I do. Are they seeing 98, or 99? Because that would tell me if I have blue eyes or not.
Now, to reprise what’s been said earlier in the thread, if there were only one blue-eyed person, they could leave on the first day, because all they would see is non-blue eyes, so it has to be them. If there were two blue-eyed people, they could leave on the second day because they each see one set of blue eyes, and yet that other person didn’t leave. This builds each day.
The takeaway is that you can count the days, and if there were the same number of blue-eyed people as there were days, they could leave. The only reason you have to count days is because they can’t communicate with each other. I didn’t make up the rules, but there it is. All the blue-eyed people see 99 sets of blue eyes, so they already know on that first day that the only day that will concern them is day 99.
Day 99 comes, nobody leaves. All blue-eyed people now know every other blue-eyed person is seeing 99 sets of blue eyes, not 98, so there has to be 100 blue-eyed people, so they all leave the next day.
I hope this helps somebody.
I still keep hearing that originally “nobody knew that everyone knew that everyone knew that everyone knew…”. with them being individually holding that single piece of information which isn’t much (it’s saying there is at least one blue eyed person). I still don’t see this whole “now they know that they know that they know…”. I don’t see what domino effect could possibly continue all the way from any one person to any last person without any single kind of coordination (which requires SOME KIND of communication whether direct or indirect) between every one of them. OK the guru started with one single statement (which if there are at two blue eyed people would be the same starting statement for any other person). So after she said that, “everyone knows there is at least one blue eyed person”. After the first night if nobody left the island then it’s because they haven’t figured out their eye color yet?. That’s where I’m at still. BTW anyone know of some kind of drawing or diagram that kind of displays this shows maybe five people and who they are and stuff? maybe that would help me because I just don’t get it from reading, but I’m workin on it.
Did I mention I was up most of the night getting this straight in my head? Because what’s up with that Guru?
She is the only person who can never know her own eye color under these crazy rules, so she’s stuck on the island. So, instead of saying, ‘I see 100 blue eyed people and 100 brown eyed people’ so all 200 people could leave (because if you only see 99 of one color, you know what yours has to be), she makes it so 100 brown-eyed people are condemned to stay, to keep her company.
In fact, you could change the puzzle to something like, the Guru is in love with someone and wants to keep them near, but there is also a rival she desperately wants to leave the island. The beloved and the rival don’t share eye color. What color eyes does her beloved have?
It’s still a deeply dick move by the Guru.
Emoticorpse, we simulposted. Check my post just above yours, maybe it will help.
And maybe not, but worth a try.
Yeah I read your post merrily I’m still missing something. Question? do the people on the island know there are 100 blue eyed people in total? because that’s in the second paragraph and the end of the first paragraph says they know everything in the first paragraph? or is that information in the second paragraph just for us as people outside tryin to figure this out?
emoticorpse, no, they only know what they see. The brown-eyed people see 100 blue-eyed people and 99 brown-eyed people, but have no idea what color their own eyes are. For all they know, their own eyes are green – or blue – or brown – or, who knows, red.
The blue-eyed people are the same, seeing 100 brown-eyed people and 99 blue-eyed people, but clueless about their own eyes.
That’s why what the guru says helps to sort it out. It sets up a situation where you can tell if you are blue-eyed or not. If you aren’t, then you still don’t know what color eyes you have, only that it isn’t blue.
Except that the problem specifically states that they don’t know there are only two eye colors. And in fact there aren’t: the Guru has green, and none of the people know that the Guru is the only one.
The brown eyed people never leave, because in the problem as defined, they will never learn their eye color, only that it is not blue.
Um, no.
Everyone already knew that some of them had blue eyes. Everyone already knew that between 99 and 101 of them have blue eyes, as each of them can see either 99 or 100 others whose eyes are blue.
What the Guru’s statement does is give them all a mutually agreed upon place to start counting. If there were only one, he would leave on the first ferry after the Guru’s statement. If there were two, they would leave on the second ferry after the Guru spoke, and so on.
The Guru’s statement makes not getting on the ferry a communicative act. When you don’t get on the ferry on day N, you are telling everyone “I see at least N people who have blue eyes”. On the day that statement isn’t true, you get on the ferry.
But the Guru was not telling anyone anything they didn’t already know, as ALL of them knew some of them had blue eyes.
At least 49.
They don’t need to know their own eye color to know there are at least 99 people on the island with blue eyes.
None of which addresses the point that telling you “at least one of the people on this island has blue eyes” isn’t giving you any information you didn’t already have.
I’m not sure if you’ve read the whole thread, but we’ve been over this. The Guru does tell them something that they don’t know - with n blue-eyed islanders, they learn that
everybody knows
…that everybody knows
…that everybody knows that there is at least one islander with blue eyes
(repeat for n-1 levels of indentation)
which is something that they didn’t know before. In the trivial example of two blue-eyed islanders, they didn’t know that the other guy knows of a blue-eyed islander, but after the Guru’s pronouncement, they do.
Okay, we’re all perfect logicians on that island. I can see 99 blue eyed people. I don’t know if my eyes are blue, so I know that when those blue-eyed people look around, they see at least 98 other blue eyed people. And the brown eyed people see at least 99.
We all already knew that there were blue eyed people on the island, and we all already knew that we all already knew it. And we all knew that we all knew that we all knew it. Ad infinitum.
The Guru didn’t tell us anything new: we all knew that he could see people with blue eyes, and we all knew that we all knew. All he did was give us an event to use as a place to start counting.
So the guru could never have shown up and everyone would have figured it out the same in this same scenario?