Eye color riddle

Miscellaneous and Personal Stuff I Must Share
1

Forgive me if this riddle has been posted before. I did a search and couldn’t get any matching results.

A co-worker, knowing my fondness for puzzles, presented me with this:

On a certain island it is completely forbidden to know your own eye color. No one may talk about eye color and there are no reflective surfaces whatsover. Even the water is too murky to see one’s reflection clearly. Ergo, no one knows his/her own eye color.

The rule of honor is that should you discover your own eye color, you must commit ritual suicide at noon the following day at the altar in the center of the island. Bizarre as it seems, everyone agrees to this.

Everyone on this island is otherwise completely logical and expertly logical. Everyone on the island has either blue or brown eyes. Though it is not common knowledge, we know there are exactly 100 people on the island with blue eyes. The brown eye number is not given, but there are several natives with brown eyes. No one is blind. The island is small enough that everyone knows everyone else. The island is impossibly remote. Strangers do not come to this island. At least they didn’t…until…

One day a stranger arrives on the island. In front of the entire population of the island he states: “There is at least one person on this island with blue eyes.”

The stranger then leaves.

100 days later, at noon, all 100 of the blue-eyed people commit ritual suicide at the altar in the center of the island. Why?

I think I know the answer, but I am having a hard time explaining it. Can someone help?

2

It is important to know that everyone knows everyone else on the island is also completely logical. They pride themselves on logic, despite their bizarre eye-color laws.

3

So, the inhabitants themselves do NOT know there are exactly 100 people with blue eyes?

4

The answer to that riddle can be found here.

5

the blue eyed folks can see their eyes reflected in the eyes of brown eyed folk, but not vica-versa.

That’s the Inksters guess anyway.

6

Q.E.D.: I just read that solution, and it sort of concurs with mine. Still, it sure seems that the stranger told the islanders nothing they didn’t already know themselves. Sad to say, I’m still having trouble explaining the solution clearly.

7

Inky- if that were the case the suicides would have started sooner.

For this puzzle them not knowing is also key. If they knew the number of blue eyed individuals then they would just by encountering eveyone else know to kill themselves, if someone told them there are 100 people with blue eyes and they all know 99 people with blue eyes then they clearly all have to kill themselves.

The real difficulty with this puzzle is a clear statement of the answer. After this point the actual answer will become pretty clear so if you are still working on it don’t read more unless you really feel you have the answer but can’t explain it well.

Ok so I think lots of people have the intuition that it has to do with the blue eyed people noticing what the other blue eyed people are doing, or rather failing to do. From there the explination gets hard, and like the link Q.E.D. gives it is best to think about it in terms of 2-blue eyed individuals to get the statement of the answer. Also the riddle is phrased better in the link and that helps state a clear answer.

8

I must admit I don’t quite understand it either. Why wouldn’t the brown-eyed people realize the same thing (that their own eyes were brown) and leave the island or kill themselves or whatever the particular version of the riddle calls for? Excuse me while I stuff my brain back into my head…

9

Does this mean that all the brown-eyed people kill themselves the next day?

10

I would think so. If all the blue-eyed people know what their own eye color is, wouldn’t the brown-eyed people be able to apply the same logic to themselves? Ack, my head hurts now!

11

Well clearly there is a brown-eyed conspiracy. :wink:

Excuse my school addled mind if this isn’t clear but the brown-eyed folk don’t do it because they can see the whole drama unfold. That is when the information is reveiled (and I’m only using the n=2 case because god damn if I’m going to force myself to think of it in higher numbers) the brown-eyed see 2 blue eyed individuals. Being perfect logicians they understand the riddle in the hypothetical about any n and any set of colors. Therefor when the blue-eyes logic themselves to death on day N the brown eyes realize that they aren’t blue-eyes.

Now if the color distribuition was common knowledge that everyone has brown or blue eyes, then yes they off themselves next. On the other hand with unknown colors and numbers they stay around because for all they know they have purple eyes. in the guru case this is clear to see because any brown-eyed individual can think they are green-eyed like the guru, they just know they aren’t blue-eyed because of the number of days it takes for the blue-eyed folk to flip out.

12

Other interesting questions that kept me awake last night when I should have been sleeping:

  1. Everyone on the island would agree that there are somewhere between 99 and 101 blue-eyed people (if they were allowed to agree to such a thing, which they are not). Suppose the stranger had said, “There are between 98 and 102 blue-eyed people.” What then?

  2. Suppose the stranger had said, “There are both blue and brown-eyed people on the island.”

  3. Finally, suppose the stranger had said, “If there is a blue-eyed native on the island, then there is a brown-eyed native on the island.”

Ugh.

13

The link is bothering me. The question says

Then the answer states that If they know that there are 100 people on the island with blue eyes, and 99 of them don’t leave, then they know that the last one must have blue eyes. However, wouldn’t this require that they know how many people have blue eyes?

14

monica:

Here’s my understanding…still somewhat shaky, but getting firmer:

An Islander does not know how many members have blue or brown eyes. However, blue-eyed Islanders DO know that there are either 99 or 100 blue-eyed people. Because they are able to determine that there are not 99 blue-eyed natives, they therefore know that there are 100 blue-eyed souls–by adding themselves.

In the “Guru” version, the remaining Islanders still do not know their own eye color. They knew that there were either 100 or 101 blue-eyed Islanders. The mass exodus confirmed the total to be 100 for the remaining Islanders, but because they themselves still might have green eyes (or some other color), they will not have to leave the island until the Guru interferes again.

In the “Suicide” version, because the only Islander options are blue and brown, the remaining Islanders realize that with the mass suicide there must be only 100 Islanders with blue eyes. That leads them to their own eye color…and their own demise on the next day.

Does that make sense?

15

You’ve already answered your own question. Since each person can see 99 blue-eyed people, they already know there are at more than 98 blue-eyed people. Similarly, they know there are at most 100 (99 + maybe themselves). Therefore, there are less than 102 blue-eyed people. Therefore, the stranger has told them nothing new.

That depends on how many brown-eyed people are on the island. If there are two or more, then each person can see [more than] one blue-eyed person, and at least one brown-eyed person. Therefore, they learn nothing new. If however, there is exactly one brown-eyed person on the island, he sees only blue-eyed people, and deduces his eyes must be brown. He kills himself the next day, and the rest kill themselves the day after (since they’ve figured out that the brown-eyed guy must have been the only one with brown eyes, so they must all have blue eyes).

The same thing happens here as in case 2. If there multiple brown-eyed people, then nothing new is revealed. If however, there is only one with brown eyes, he figures out that since there is at least one blue-eyed person, there must also be one with brown eyes, namely, himself. And death ensues for the entire population within the next two days.

16

Ok so here is the clearest presentation of the logic of this puzzle that I can give, and I will also hopefully be able to answer your three questions Biotop.

Everyone, being a perfect logician, knows that in the N=2 case the 2 blue-eyed individuals they see are engaging in this thought process “I see someone with blue-eyes so that may be who the guru is refering to.” (next day) “The blue-eyed person is still here so he must see a blue-eyed person. No one else I see has blue-eyes and we see all the same people except ourselves. Therefor I have blue-eyes.” The two of them reach this conclusion simulatiniously and go on to meet their fate. Now these perfect logicians also know that it works out such that in N days all N blue-eyed people go off to meet there fate because they understand how it extends to N+1. That is in the case of 3 the third blue-eyed person watches those two and sees on the next day they are still here. All of them are thinking (as each can be called the third blue-eyed preson) “There must be a third blue-eyed individual as the N=2 case did not occur. I see no-one with blue-eyes except those two. I must be the third blue-eyed person.” and so it goes to any N.

Now the guru version is far clearer as the guru introduces the information directly as opposed to the visitor declaring something that is technically already known. Everyone can see blue-eyed individuals so they know that bit of information but the point of the decleration comes across and it is the same as the guru’s. All blues expect the N-1 case, while all browns expect the N case, and when on the N-1 day the other blues are around the blues realize that they’ve gotta go. Thus all the browns confirm they aren’t blue eyed.

If browns know that all individuals come in two categories they will off themselves after the blues do, realizing that they must be brown-eyed. If they can be any color then they do nothing as they have no way to logically confirm their eye color.

Giving them the 98 to 102 range of blue-eyes gives no new information. They can confirm that there are not 98 and not 102, leaving them with the 99 to 101 range already known. That is telling them the range cannot have an effect unless they automatically off themselves without input. Lying to them and telling them there are a 101 blue-eyed individuals produces the amusing effect of all brown-eyes killing themselves while all blue-eyes going into comas as they can’t figure out what’s up with the extra blue-eyed pairs.

Telling them that there are both blue-eyed and brown-eyed individuals doesn’t give them any new information unless you mean it to say that there are only two eye colors which only matters if someone sets of the logic based suicide spree by doing the guru statement as well. Also anytime you give information about an eye-color that has only one member you pretty much ensure their death.

Finally the conditional is the same situation but I suspect you mean what if the guru decides to be a real ass and say “I see a brown-eyed person and a blue-eyed person.” This looks tricky but it actually isn’t.

If there are fewer of one color than another people know to watch the smaller group. The smaller group runs through the problem and all kill themselves on day n. The big group continues through with their problem and all kill themselves on day N.

Now for n=N I thought you had a pain in the ass but lets look at the case of 2 blues and 2 browns.
On the second day both blues are still there, they saw the other blue and knew that he could be the one the guru ment when he said he saw a blue-eyed person. Clearly if the other blue is around he is seeing another blue, and since a particular blue sees only one other blue they simulatiniously realize they both have blue-eyes. The browns experience a similiar reasoning. They both off themselves on the same day so it is really just the same thing as the != case. Basically when it turns out that all the individuals of an eye-color that you see are still around on a day equal to one more than the number of the eye-color than you can see you realize that you must have that eye-color. It doesn’t matter how many eye colors are working throuhg it at the same time or what the numbers are. (The you refers to a hyptohetical islander) You know that the only reason individuals with a particular eye color stay is that they don’t know their eye-color. You also know that it takes the number of days equal to the number of people with a particular eye color to know for sure. When they show up on the day number of all those of the color you can see plus 1 it must be because it isn’t sorted out yet and the only way that is possible is if you are of the eye-color. Different eye-colors running through it don’t conflict because each individual has a prediction about the numbers based on their own eye-color. That is the problem either resolves on your predicted day and you are safe or it goes one past and you aren’t.

17

man i don’t understand the logic at all. how can anyone apply the ‘2 blue eyed then add more’ logic when everyone knows there are at least 90+ of them around? knowing from the onset that there are so many blue eyes around renders that particular logic invalid.

18

Impressive, The Tim. Impressive.

But with regard to the range of 97 to 101 blue eyes…

It seems to me as if this speeds up the process. For if there were 97 people with blue eyes on the island, then the blue-eyed people could look around and say, “I only see 96 blues. I must be the 97th.” So they off themselves the next day. Because no one kills themselves (or leaves the island) on the following day, there can’t be only 97 blues. Therefore, if there are 98 blue eyes, they’d figure it out the next day. etc.

Doesn’t giving the range of 97 to 101 mean that the 100 blue-eyed people are history in 4 days instead of 100? Am I missing something?

19

Thanks Biotop. As for the range they know that there are only two possible blue-eyed numbers, the amount they see and the amount they see plus one. The critical day can only change by telling them how many blue-eyed people you see, that increments it differently. Also if you give a specific number they either go into a tizzy from the lie, being overly logical, or off themselves immediately because they do the math.

The key is that the critical day is the number of individuals with the eye color. So when the day rolls around equal to the number you can see and they are still there you know that you are one of them. The only thing that will change this is how fast it increments based on how man people of the eye color are said to be seen.

20

It just occured to me that you meant your ranges to be a specific sight thing, in which case yes it does change the speed of things. The thing about this puzzle that’s so damned annoying is that pretty much any vaguely relevant statement sets them off because they have so much information already. This also makes a lot of statements look redundant.