Blue-eyes logic puzzle

Factual Questions
281

There are four people on the island. The guru, who has no eyes, and the three islanders Amy, Belinda and Carl, who have blue eyes. Amy, Belinda an Carl are perfect logicians. They never communicate with each other. They all can see the others’ eyes at all time. All three want off the island. The boat comes at midnight every night and only allows them off the island if they know their own eye color. Somehow, who knows how, there is nothing on the island at all–not even the water–which ever shows any of them their own reflections.

The guru never speaks. But one day, she says to no one in particular, but loudly enough for everyone to hear, “I see at least one blue-eyed islander.”

Amy, Belinda and Carl believe the guru. And now they each set to logickin.

Amy thinks to herself:

Suppose I have non-blue eyes.
In that case, Belinda sees one blue eyed person (Carl.)
And while I’m supposing that, I’ll also suppose Belinda thinks:
Suppose I (Belinda) have non-blue eyes.
In that case Carl sees no blue-eyed people.
And while I (Belinda) am supposing that, I’ll also suppose Carl thinks:
Suppose I (Carl, in Belinda’s imagination) have non-blue eyes.
Well, I (Carl) can’t have non-blue eyes, since no one else on the island has blue eyes and the guru said at least one person has blue eyes.
So, I (Carl) have blue eyes, and I get to go on the boat tonight.
Based on this, Belinda (in Amy’s imagination) concludes: If Carl doesn’t get on the boat tonight, then I (Belinda) have blue eyes.
So, if Carl doesn’t get on the boat tonight, then Carl and Belinda will get on the boat on the second night.
But remember, thta was all under Amy’s supposition that she herself doesn’t have blue eyes.
So, Amy realizes, if Carl and Belinda don’t get on the boat on the second night, she herself must have blue eyes as well.
So on the third night, she will be able to get on the boat.

The same logic applies for Belinda and Carl.

So, on the third night, all three of the islanders will leave the island on the boat.

282

Wow that’s confusing. Because if I think back to the case of only 2 islanders, both with blue eyes, if one gets struck dead right after the announcement, then the remaining guy is screwed. He will have no way of knowing what color his own eyes are. If the lightning bolt strikes on the second day, then he’s in the clear because now he knows his own eyes are blue since the other blue eyed guy stuck around.

Now when I consider the case with 3 blue eyed islanders, if one gets struck dead right after the announcement, the two that are left will be just like the case as if there were only 2 to begin with. So the remaining guys will leave on the 2nd day. If one gets struck dead on the 2nd day, then they just have to wait an extra day in order to be sure of their own eye color. So the two guys would leave on the 3rd day as if no one had been struck by lightning in the first place.

I’m not really confident if I’ve even got this understanding right so far, but I’m gonna go with

  1. Remaining 99 will leave on the 99th night

  2. Remaining 99 will leave on the 100th night.

But that doesn’t make sense to me so I’m pretty sure I’ve messed up somewhere.

283

With two blue eyed people (both logical) after the first night if the only one you see hasn’t gone it’s because he can’t confirm he’s the only blue eyed person. The only reason he can’t confirm he’s the only blue eyed person is because there must be another. since he knows the minimum count which is the 1 that he counted and the maximum which is that one he counted plus him he must be the other.

But with three blue eyed people one of them (take your pick) watches the others as if they were the only two in this process of elimination. those two have not figured it out as simple as it is with two blue eyed people and if they can’t figure it out like that it’s because they (each one) has a third blue eyed person confusing hte other two and it must be you so now you know your eye color is blue and confusing the others.

But with four blue eyed people any one of them is watching the other three…

… I hope I’m getting somewhere

284

“Come to the island!”, they said. “You’ll love it!”, they said. “It’ll be just like Lost”, they said, “but even more cerebral, and with fewer Nikki flashbacks”.
Yes, I did want to meet intelligent, like-minded, people. Yes I did want to ‘get away from it all’. But godammit, what’s the deal with this no talking rule?
Well, apparently that green-eyed hippy chick is going to make some important announcement tomorrow. All I can say is it better be worth the wait.
Worst. Cruise. EVER!

285

Yeah, the worst part was when they reassigned our eye colors and then wouldn’t let us look at any reflections or tell us what color our eyes were! Ugh. I hope I got blue. I’ve always wanted blue eyes. Can’t wait till tomorrow for the big announcement! (I hope she tells me my eye color)

286

I guess I did get a free cruise, and all I had to do was tell them I’m a ‘perfect logician’. I wonder what that means, anyway?

287

OK some key things I am understanding now logically for all this to work. Every one of them knows that every one is working this same process of elimination at the same time as the days pass (that’s the part of great logic that these islanders had that maybe I along with most other people don’t start with). supposedly 4 of them would leave early if they happened to figure out their eye color but that couldn’t happen with this system their using because it just works out that way that they must wait to see until their count is at least N many blue eyed people N being whatever they counted and then them the only one left all still there confused realizing their the one missing from the blues. This whole he knows that he knows that he knows has nothing to do with communicating what one knows to the next and so on it’s just the principle that this single systematic method is doing? I hope I’m getting somewhere…

288

I would be interested in the hijack. (There are senses of the phrase “I know that X” under which it is manifestly the case, to me, that “I know that X” entails “I know that I know that X”, and there are also senses of the phrase “I know that X” under which this is manifestly not the case. I am curious as to why you would say the entailment does not hold, but mostly because hearing your argument for this is crucial to my learning what you are construing the phrase “I know that X” to mean.)

289

Day 2.

Oh great. My ear infection has spread from the left ear to the right. How is that even possible? Jeez. If I only had some antibiotics. I know I’m not allowed to talk to anyone, but that hippy chick seems to be some kind of group leader so I walked up to her and started pointing at my ears and wincing. But she just turned away from me.
Damn – it’s just constant buzzing in my ears now, and I can’t hear anything. I clapped my hands together and I couldn’t even hear that. And boss-lady is giving her big speech tomorrow – I hope it’s nothing too important. Well, tough titties I guess. I need a doctor!

290

First example that comes to my mind might be a little cheeky, but a dog, I think, typically knows where its food is without knowing that it knows where its food is.

That might look like cheating. A dog arguably has no concept of knowledge, so of course it doesn’t know it knows where the food is because it doesn’t know anyone knows anything. Yet–I’d say that the dog doesn’t need to have a concept of knowledge in order to know someone knows something. Arguably it knows I (and other dogs) know where it is, for example.

Thinking of human examples, to kinds come to mind:

  1. Situations in which a person believes something for the right reasons, but is not sure his reasons are good. In this case, I think it’s possible for him to know a thing without knowing he knows it.

  2. Situations in which a person believes something, but believes he doesn’t believe it. If the thing he believes is believed for good reasons, then it’s knowledge–but since he doesn’t believe he believes it, he doesn’t know he knows it.

A scenario like the first would be: X acts on the visual reports of his eyes straightforwardly, though he has strong reasons to suspect they are decieving him. (His suspicions are incorrect.) In this case I’d say he knows the wall in front of him is white, though he doesn’t know he knows it.)

A scenario like the second would be: X believes he is the most important person in the room, but doesn’t believe he believes it. About his belief, he believes himself humble–he believes he believes he is of little importance. But the best explanation for most of his non-vocal actions is that he believes he is extremely important.

(Granted both of these kinds of scenarios put in me a strong desire to start talking about multiple agents within a single human organism, ascribing different knowledge to the different agents…)

What sense of “know” do you think entails that knowing X means knowing one knows X?

291

The three person case doesn’t work. I’m not sure why, but it can’t.

Imagine that the three blue-eyed people, A, B, and C, were on the island, following the rules in the problem. Guru is not there yet.

Now, take each one aside privately, and give them an exemption from the “no communication rule.”

Ask A the following questions:
Will the Guru, when he arrives, see a Blue-eyed person? (He answers yes)
Does B know that the Guru, when he arrives, will see a Blue-Eyed person? (B can see C, so the answer is Yes)
Does C know that the Guru, when he arrives, will see a Blue-Eyed person? (C can see B, so Yes)
Does B know that I know that the Guru can see a blue-eyed person? (B knows I can see C, so yes)
Does C know that I know that the Guru can see a blue-eyed person? (C knows that I can see B, so yes)

So, everyone has the information that the Guru will provide. And everyone knows that everyone has information.

Then the Guru arrives. And all three people wait for him to say, “I can see a blue eyed person,” cause all of them know he can say it.

Instead he takes a nap, and everyone is pissed off.

Five days later, he wakes up, and makes the announcement the he can see a blue-eyed person.

Then three days later, everyone leaves.

This scenario posits, without reasonable justification, that stating aloud something known to all parties and known by all parties, magically sets a ‘logic cascade’ into motion. It’s paradoxical on the face, and doesn’t make sense. I’m not sure if there is a flaw in the logic of the solution, or if it’s a paradox like the surprise test paradox that has no satisfactory solution, but claiming that everyone leaves on the hundredth day is not just counterintuitive, but logically impossible.

292

But the crucial question you didn’t ask is:

Does B know that C knows that the Guru, when he arrives, will see a Blue-Eyed person?

The answer? “I don’t know.” For A doesn’t know that A has blue eyes, so A does/n’t know that B knows A has blue eyes. Moreover, A knows B doesn’t know that B has blue eyes. So A doesn’t know whether B knows that C knows that the Guru will see a blue-eyed person. So A’s answer is “I don’t know.” Once the guru arrives, though, and speaks, A’s answer will be changed to “yes.”)

293

In the case of 3 blue eyed people
A does know that each B and C will know that the guru will see a blue eyed person because they can see that same blue eyed person the Guru will see when he eventually shows up. So his answer is always Yes.

294

A knows that B will know the guru will see a blue eyed person.
A knows that C will know the guru will see a blue eyed person.
But A does not know that B knows that C will know the guru will see a blue eyed person.

Then, after the guru speaks, A will know that B knows that C knows the guru sees a blue eyed person.

295

In the case of 3 people we may as well use numbers instead of assigned letters and working independently each person is each others third. So when the guru speaks they knew that his second guy was at least gonna be guru’s first. so now you know that the guru sees one out of your blue eyed count. so he see’s it? ok. That’s where I’m stuck and the logic everyone on the thread of the just doesn’t click to me. To me that’s where it stays at that point blue eyed people have no advantage whatsoever over any other group of eyed people with 3 people or more.

296

Can you elaborate on this? I don’t really understand what you’re saying here. (I bolded the parts that are especially unclear to me.)

297

Tell me how this illustration grabs you. It is intended to show why it is that A does not know whether B knows whether C knows there is at least one blue eyed person. It illustrates the way that:

For all A knows, (for all B knows, (for all C knows, there may not be any blue eyed people on the island)).

Tell me, in particular, whether you understand why both A and B have question marks next to them in imaginary-B’s thought balloon.

(In fact, now I kind of wish I’d used brown dots instead of question marks. If I had, then a thought balloon would mean “this is one way the world could be, for all I know.”)

298

I’ve made a version using brown dots instead of question marks, here.

A thought balloon in this illustration means: I am imagining one way the world might be for all I know.

When you see brown eyes inside a though balloon, this doesn’t mean the imaginer thinks those eyes are brown. It merely means he can imagine that the eyes could be brown, for all he knows. (Hence, for example, each one is able to imagine that his own eyes are brown, since he doesn’t know his own eye color.)

299

You may be able to sacrifice one eye and look at it.

Without communication it would be hard to count 199 people and categorize their eye color.

You could produce children, though it may be difficult without communication to find enough partners and for males verify paternity, and not 100% but if you are able to produce enough all one eye color offspring while having same and opposite eye colored mates could give a strong reason to accept that you have the offsping’s eye color.

This also assumes you don’t have a iPhone where you can take a pict of yourself.

300

Excellent - the best way I’ve yet seen to illustrate this.

And to extend this you can with little effort mentally draw as many surrounding balloons as you’d like.