Eye color riddle

Miscellaneous and Personal Stuff I Must Share
41

ok, so the guru doesn’t tell them anything new but is responsible for setting off the time bomb that the natives know everyone else will start working on the ‘2 blue eyed’ logic from the moment the statement is made up until blue eyes are no more.

and the reason the other eye colors don’t suffer is because there isn’t anyone to start them working out the logic at the same time, so they are safe. :o

still, i must meekly protest :stuck_out_tongue: that it’s an assumption that the ‘perfect logicians’ will all apply the same ‘2 blue eyed’ logic at the moment the statement is made instead of waving dismissively at the guru saying, ‘tell me something i don’t know’ before splashing into the sea for more beach action frolics :smiley:

42

But the guru is telling them something they do not already know. He is telling them that the other natives all know about the blue eyes. You may say that the other natives already know this as well. True, but the other natives don’t know that the other natives know this.

Actually the other other natives do already know this, but eventually you work to a situation where the other, other , other (repeat 99 times) natives are not really sure. Now they are. And the guru, by his statement, is saying so.

Gee…I thought I was going to be able to make it clearer.

43

I have to go alog with shijinn and Bob Cos on this. If they really are such perfectly logical beings then the whole society should have self-destructed long ago. The whole exercise strikes me as a reductio ad absurdum proof that such a society cannot logically exist. Therefore their very existence should logically prove to them that they are not perfectly logical! :smiley:

Here’s an interesting line of thought: Suppose such a society could exist and suppose they were given knowledge of the genetics of eye color.

44

With 100 sets of blue eyes on the island (and at least 99 on display for everyone on the island, including Gilligan, Maryann and the Professor), how can anyone not be aware that, (1) there is at least 1 person with blue eyes on the island, and (2) that everyone else on the island is already aware of this fact?

Again, what new information is the guru imparting? And if no one could be sure of his eye color before the guru’s great pronouncement, what the hell did he add to shed any light on the matter, the great busybody? If anyone on the island wasn’t sure of there already being at least one set of blue eyes in the vicinity, he was a chucklehead of the highest order.

45

It took me ages to get this, but:

Bob cos: your (1) and (2) are quite right. But there is (3) that everyone is aware that everyone is aware that someone has blue eyes. And so on to (99) or so. One of which people are not aware of.

This is isn’t the way to solve the puzzle, but it shows that some info is added.

I realise that it sounds very stupid, but it’s right. Think about the case with 3 people. And 4 people. Do you get these?

Shall I try and explain again, or just agree to differ.

46

Ok to murky the waters more in an effort to not do school work.

As has been pointed out in all cases past n=1 the blue-eyes get no new information but it clearly works in the n=2 and n=3 cases. Everyone gets that. The stumbling block really seems to be, “Ya so what it works for two consecutive low number cases? It can’t generalize, I mean they all KNOW that if the guru comes and looks at them all that he has to see someone with blue-eyes and so him saying that doesn’t give any new information.” Fine I see the objection, and I see the objection related to it about them going off on their own. I also realize that explaining it in terms of proofs in number theory isn’t working so well. I’m going to try and rehash the essential core of it yet again in a way that doesn’t involve insane sci-fi scenarios.

(1) I see n people with blue-eyes.
(2) There are either n people with blue-eyes or n+1 with blue-eyes.
(3) If I assume that I do not have blue-eyes then all individuals with blue-eyes see (n-1) individuals with blue-eyes.
(4) Given my assumption (3) and that all individuals are perfectly logical then any particular individual with blue-eyes can repeat my argument from step (1) with n=(n-1) for them.
(5) This argument iterates through the blue-eyed people until it posits a blue-eyed individual who sees no other individuals with blue-eyes. This individual should leave the island if and when the guru announces seeing a blue-eyed individual.
(6) If they do not leave the island it may be the blue-eyes=2 case and another individual is posited to be assuming that case.
(7) If this individual does not leave it propigates up the chain to my assumption either turning out to be the blue-eyes=n or the blue-eyes=n+1 case at which point I will know my eye-color.

You’ll notice that key to it is that everyone on the island is interchangable for the assumptions. If everyone isn’t interchangable then it won’t work out.

Was that at all clear? I’m really starting to believe I have no ability to communicate ideas effectively. I can only blaim this damn puzzle’s inherent annoyance factor for so long.

47

I am so confused. :dubious: I just don’t get it. I think the blue eyed people knew it the whole time.

48

This puzzle has been bothering me all day, and I finally found a way to explain it to myself. Maybe it’ll help.

There are exactly 100 blue-eyed people on the island. All of them look around and think, "I see 99 blue-eyed people. Either I have blue eyes or I don’t. If I don’t, then there are exactly 99 blue-eyed people. If there are exactly 99 blue-eyed people, then each of them is looking around thinking,

"'I see 98 blue-eyed people. Either I have blue eyes or I don’t. If I don’t, then there are exactly 98 blue-eyed people. If there are exactly 98 blue-eyed people, then each of them is looking around thinking,

“’“I see 97 blue-eyed people. Either I have blue eyes or I don’t. If I don’t, then there are exactly 97 blue-eyed people. If there are exactly 97 blue-eyed people, then each of them is…”’”

Et cetera. Until finally, there are two hypothetical people left, and they are both looking around thinking,

“I see one blue-eyed person. Either I have blue eyes or I don’t. If I don’t, then there is exactly one person with blue eyes.”

Here is were the guru’s information is important. Without it, the 100 blue-eyed people learn nothing from this line of reasoning, because the hypothetical final person with blue eyes doesn’t know there are any people with blue eyes on the island. With it, the final two hypothetical people learn that they have blue eyes because they each think,

“The blue-eyed person I see knows that there is at least one blue-eyed person on this island. If I do not have blue eyes, he will know that he has blue eyes, because he sees no blue-eyed people. However, if I have blue eyes, he will not kill himself, because he sees someone with blue eyes. Therefore, if he does not kill himself tomorrow, I will know that I have blue eyes, and must kill myself the next day.”

The third hypothetical person uses a similar line of reasoning:

“The blue-eyed people I see know that there is at least one blue-eyed person on this island. If I do not have blue eyes, they will realize that they have blue eyes [because of the above line of reasoning], and will kill themselves the day after tomorrow. If I do have blue eyes, they will not, because they see me. If they do not kill themselves the day after tomorrow, I will know I have blue eyes, and must kill myself the next day.”

The… let’s say sixth (this is as far as I will go, I swear!) hypothetical person uses a similar line of reasoning:

“The blue-eyed people I see know that there is at least one blue-eyed person on the island. If I do not have blue eyes, then there are exactly five that do. If there are exactly five that do, each of them thinks that, if they do not, there are four, and that if there are four, each of them thinks there may be three. [now see above reasoning]. If I do not have blue eyes, then the five that do will kill themselves five days from now. If I do have blue eyes they will not, because they see me. If they do not kill themselves five days from now, I will know I have blue eyes, and I must kill myself six days from now.”

As you can see, this line of reasoning from the hypothetical people will continue up to the hundredth, who are all real, and they will see that if the 99 they see do not kill themselves on the 99th day, they must have blue eyes, therefore they kill themselves on the 100th day.

I still think there must be a flaw in this line of reasoning somewhere because it seems wrong, but it sounds logical.

49

The Tim and Loopus, I believe you’re offering similar ways of looking at this. Am I correct? (By the way, I love stuff like this, so I appreciate the discussion.)

Aren’t your scenarios flawed in assuming iterations of observation (or iterations of virtual observation), ignoring the fact that without any further analysis or observation, everyone can see a veritable herd of blue-eyed people right at that moment. I infer this flaw from…

…and…

There are never “just two people left” who are wondering if everyone knows there are blue-eyed people running about the place. And no one could logically posit that there is an individual on the island who can’t see lots and lots of blue-eyed people. Let me put it another way: the flaw is that someone can posit a scenario that ignores the information right in front of everyone’s eyes. For example…

It is impossible for a perfect logician to posit that there is anyone on the island assuming someone else can think this, not when everyone knows that everyone else can see at least 98 sets of blue eyes. Knows it for sure. I don’t think your arguments have overcome my objections to the riddle, or I am missing it. Personally, I think the whole tribe should commit ritual suicide over their disgraceful lack of internal logic.

50

I’m sorry, but this logic falls apart with more than 2 blue-eyed people. Assume the case of 3 people with blue eyes. The thought process of person A with blue eyes is:

I can see both B and C. I know that there are at least 2 people with blue eyes.

I know that B can see C and C can see B. Therefore, both of them also know that there is at least one person with blue eyes.

Saying that there is at least one person with blue eyes does not give anyone information they did not already have, and everyone knows it.

51

Sigh. What falls apart, I’m afraid, is our collective inability to explain clearly the solution to this riddle. I can’t for the life of me figure out how to make it any clearer than has been done here by others.

My girlfriend currently refuses to even speak to me any more about this riddle. She agrees with the dissenters.

But she, and they, are wrong.

52

When the stranger/guru says that there is someone with blue eyes, everyone on the island (regardless of eye color) knows these facts:

  1. I can see lots of people with blue eyes.

  2. Everyone else here can see lots of people with blue eyes.

  3. DUH! Of course there is at least one person with blue eyes.

53

No, it still works.

Person A knows that if B and C were each seeing only ONE person (each other) with blue eyes, they would have known after the first night and killed themselves the second night.

When they do not, Person A has to realize that there must be more than just Person B and Person C, and therefore know that it has to be themselves.

Sorry to repeat what’s been said, but a colleague of mine (who hates all of you for exposing her to this) was able to understand it better for some reason when I gave people names, so see if this helps at all:

Let’s say there are 5 people on the island. 2 are brown eyed, and 3 are blue eyed. I am one of the 3 blue eyed people, and Steve and Frank are the others.

I can see that Steve and Frank are blue eyed, so I say to myself “Ha! Sucks to be them!”
I know that Steve knows Frank is blue eyed, and Frank knows Steve is blue eyed. I know that nobody will off themselves the first night, because I figure that Steve will expect Frank to go, and Frank will expect Steve to go, but they’ll realize the next day that there must be more than one blue and off themselves the second night.

But on the third day, when nobody has died, I realize that Steve and Frank must be seeing another person with blue eyes just like I was, and at that point I would look around at the other two folks, both brown eyed, and say “Flaming cheese!” and kill myself.

54

I’ve been thinking about this a lot. I’m not sure that I can totally explain it (or that I can explain my thoughts clearly) but I have what I think are some interesting thoughts on this.
I think that the whole thing may revolve around what we mean by “perfectly logical”. Obviously it means that they can deduce anything that can be deduced from the available information. Can we assume that it also means that ALL of them will ALWAYS deduce ANY logically possible situation and the logical implications of that situation? If so, than even if the stranger doesn’t show up, those who see at least three blue-eyed people can hypothesize the appearance of this stranger and the logical implications of his statement. They will also realize that his actual appearance or non-appearance is irrelevant because EVERYONE sees a least 2 blue-eyed people and EVERYONE will hypothesize the stranger and EVERYONE will realize that everyone else will make the same hypothesis and reach the same conclusions. So, if “perfectly logical” means that they always consider all logically possible situations then if there are n > 2 people of eye color X then all of those people will commit suicide on the nth noon.

If perfectly logical does not mean the above then they will not hypothesize the stranger (or some of them may but will not be assured that others will) and the whole line of reasoning will not occur. If this is the case then, if the stranger does appear, he is in fact introducing information. The information that he is introducing is the possibility of the n+1 line of reasoning. Of course that would mean that he could produce the same result by simply appearing and saying something like “there is at least one person on this island with mumblemumblemumble eyes”. Or he could simply describe this riddle to them.

55

OK, Elret, you are right about working with three, but what if Steve, Frank, Bill and Jim all had blue eyes?

Steve can see 3 people with blue eyes.
Frank can see 3 people with blue eyes.
Bill can see 3 people with blue eyes.
Jim can see 3 people with blue eyes.
All the brown-eyed folk can see 4 people with blue eyes.

No one is any wiser than they were before.

56

Nope, Fat Bald Guy, it keeps working the very same with each person you add. It just takes one day longer to realize it, that’s all.

In your example, each of them will assume that each of the other three are seeing only two other people with blue eyes. This means that Steve, for example, will assume that things will work out the way they did in my example above where there were in fact only 3 blue eyed people. When, on the fourth day, Frank, Bill and Jim are still kicking, Steve will realize that there must be one more.

57

OK, so they assume that each of the others are seeing two other people with blue eyes. Therefore they assume that everyone knows that the guru’s statement is correct. Why do they have to be concerned about working out how many blue eyes there are?

58

This is a cute riddle.

In the interest of murkying the atmosphere, how about some more possible scenarios:

  1. Some or all islands have exactly one blue eye and one brown eye. These ones thus correctly have neither 2 blue eyes nor 2 brown eyes.
  2. Eye color of some of the islanders changes with time. maybe eyery minute even.
  3. Some islanders have two blue eyes and two brown eyes. they are 4 eyed freaks.
  4. Neither eye number nor eye color is constant for some of the islanders.
  5. Some natives know the color of one of their eyes but not the other one.
    etc…
59

FBG I am with ya, buddy. But this is still a pretty cool puzzle.

OK… Elret, what you are saying is true, and I can follow the logic of that, but that doesn’t change the fact that the stranger gave them no new information!

The stranger could have shown up and said “Ready set GO!” with the same result. Therefore, the current state of affairs on the island is impossible. Forget the stranger for a second. Let’s say you’re a native on the island. If you can see N number of blue-eyed people, and N days go by with no one killing themselves, its time to off yourself! (this is really really bizarre)

60

I saw this post fall off the front page and I thought “Finally! no more of this damn puzzle!” But alas, it lives.

Anyway, what if one of the islanders said “I see someone with blue eyes.” Would it be different? Is the Guru just the catalyst for something that should have happened all along? Everyone was always aware that someone had blue eyes.