How'd they figure that? (logic puzzle)

[belliott4488]

Okay, here's my favorite brain teaser. Sorry if an equivalent version has appeared here before.

On a remote island somewhere there was an ancient country whose population consisted of 200 people with green eyes and 800 people with blue eyes. The citizens of this country were all happy and lived harmoniously, dutifully obeying the only two laws that governed them:

1: At noon every day, all 1000 citizens must gather in the town square for ten minutes of silent contemplation. If, at the end this period, any citizen knows the color of his own eyes, he must immediately commit ritual suicide.

2. Under no circumstances is any citizen ever to speak of or otherwise make any reference to eye color, his own or anyone else's.

Naturally, given the dire nature of the first law, there were no mirrors or other reflective surfaces to be found anywhere on the island, nor any other means for a citizen to discover the color of his eyes accidentally.

Life went on peacefully for years, since no ever learned or wanted to learn the color of his eyes. All that ended, however, after the fateful day when a stranger washed up on shore, the only survivor of a doomed pleasure cruise. As he staggered up on to the beach, he looked about him and declared, "whoa, cool - there are dudes with green eyes here, man!" Naturally, the citizens were shocked by this breach of protocol, and threw stones at the stranger, driving him back into the sea, where he was promptly devoured by a hungry shark.

Now, since he did not specify which people had green eyes, no one on the island had learned his own eye color, so no one committed suicide the next day. Or the next, or the next after that. This went on for a long time, but on the 800th day after the stranger's arrival, all the green-eyed people committed suicide after the town meeting at noon.

The question, of course, is how did they know their eyes were green? Secondarily, why did it take 800 days after the only revelation of new information (which wasn't really new, since everyone on the island already knew that there were people with green eyes on the island)?

This isn't too difficult if you've seen this type of problem before, but it is kind of fun to work through the necessary thought processes of these poor doomed simps.

 

[Gokul43201]

Wow, knowledge kills, eh? This is another a version of a teaser that has been here a long time ago: https://www.physicsforums.com/showthread.php?p=221817

 

[quetzalcoatl9]

nice problem, i enjoyed that one

even more interesting:

lets say there are only people with green eyes on the island (no blue-eyed people). after the stranger arrives and delivers his edict, the island population will be fine, free of suicides...until a single blue-eyed person arrives.

(btw, you have a typo in your description of the problem, they all commit suicide 200 days after the stranger's arrival - the number of blue-eyed people is irrelevant , there need only be at least one blue eyed person).

also, to be complete, i would say that rules 1 and 2 must be evaluated in that order. technically you could argue that by committing suicide you would be indirectly telling someone their eye color! having rule 1 preempt rule 2 prevents that.

 

[belliott4488]

nice problem, i enjoyed that one

even more interesting:

lets say there are only people with green eyes on the island (no blue-eyed people). after the stranger arrives and delivers his edict, the island population will be fine, free of suicides...until a single blue-eyed person arrives.

Really? I would have thought that after a number of days equal to the population, everyone would have to commit suicide, each citizen having figured out that he's not the single blue-eyed person on the island.

(btw, you have a typo in your description of the problem, they all commit suicide 200 days after the stranger's arrival - the number of blue-eyed people is irrelevant , there need only be at least one blue eyed person).

Aughh! You're right - I messed that up! I hope I didn't confuse anyone and prevent his or her solving the puzzle ...

also, to be complete, i would say that rules 1 and 2 must be evaluated in that order. technically you could argue that by committing suicide you would be indirectly telling someone their eye color! having rule 1 preempt rule 2 prevents that.

I'm not sure I'm understanding what you mean by evaluating the rules, let alone the order in which they're done. The second rule applies at all times, and the first applies once ever day. Why would you have to think of one applying first?
Also, you are right about suicide informing others about their own eyes. In fact, if the citizens are aware that there are only two eye colors available, all the blue-eyed citizens would have to commit suicide the day after the green-eyed guys. If there were the possibility of brown-eyed people, then I guess they'd still be okay.

 

[quetzalcoatl9]

Really? I would have thought that after a number of days equal to the population, everyone would have to commit suicide, each citizen having figured out that he's not the single blue-eyed person on the island.

yes you're right, scratch that, i don't know why i thought that..

I'm not sure I'm understanding what you mean by evaluating the rules, let alone the order in which they're done. The second rule applies at all times, and the first applies once ever day. Why would you have to think of one applying first?
Also, you are right about suicide informing others about their own eyes. In fact, if the citizens are aware that there are only two eye colors available, all the blue-eyed citizens would have to commit suicide the day after the green-eyed guys. If there were the possibility of brown-eyed people, then I guess they'd still be okay.

the act of NOT committing suicide will reveal someone's eye color, which is prohibited by rule 2. if rule 2 took precedence over rule 1, then i would not follow rule 1. as long as rule 1 takes priority over rule 2 there is no conflict. this is similar to how robocop went nuts :)

also, we could say that they commit suicide "if and only if" they discover their eye color, thus disallowing suicide for any other reason.

 

[belliott4488]

the act of NOT committing suicide will reveal someone's eye color, which is prohibited by rule 2.

Sorry ... still not getting it. Before the revelation by the stranger, life went on with no suicides and no problems - how would the lack of suicides have indicated anyone else's eye color, and whose eyes would those have been?

Or, are you saying that would happen only after the stranger's revelation? I guess on day N-1, the fact that all the green eyed guys didn't do themselves in would, in fact, reveal to the other greenies what their eye color was. Of course, that's how they know to commit suicide the next day ... so maybe I do get it - there's a fundamental contradiction between rule #2 and any action or inaction that allows anyone to deduce the color of his eyes, so suicide must always result from someone having violated Rule #2. Of course, we could also take a more literal reading of that rule, which would be to say that no one may overtly or explicitly discuss anyone's eye color, but merely failing to prevent someone else from a logical deduction doesn't count.

also, we could say that they commit suicide "if and only if" they discover their eye color, thus disallowing suicide for any other reason.

Yeah, that would help, too!

 

[quetzalcoatl9]

Or, are you saying that would happen only after the stranger's revelation? I guess on day N-1, the fact that all the green eyed guys didn't do themselves in would, in fact, reveal to the other greenies what their eye color was. Of course, that's how they know to commit suicide the next day ... so maybe I do get it - there's a fundamental contradiction between rule #2 and any action or inaction that allows anyone to deduce the color of his eyes, so suicide must always result from someone having violated Rule #2. Of course, we could also take a more literal reading of that rule, which would be to say that no one may overtly or explicitly discuss anyone's eye color, but merely failing to prevent someone else from a logical deduction doesn't count.

yes precisely

 

[omg precal]

you guys think way too advanced for me...

*quits forum*

 

[belliott4488]

you guys think way too advanced for me...

*quits forum*

ha ha - that was something else I forgot in the original post: the citizens of this country all think way too much!

 

[quetzalcoatl9]

ha ha - that was something else I forgot in the original post: the citizens of this country all think way too much!

yes, but it is to the citizens detriment! they would be better off if they were not so rational!

or, alternatively, they would be better off not living within a theological system

 

[ArielGenesis]

hey but before all of that, could you just explain the problem in a simple way first, they you can all argue about the detail. i kinda get it... that because after 200 days, the green found out that they must be green.

but if these people can count, and these people think too much. they could simply count the number of green people in the society and know their own color assuming that they know that there have to be 200 blue and 800 green and 1000 total...

 

[quetzalcoatl9]

but if these people can count, and these people think too much. they could simply count the number of green people in the society and know their own color assuming that they know that there have to be 200 blue and 800 green and 1000 total...

but they don't know exactly how many people have green eyes, they only know what they see around them

 

[Abyss]

could someone spell it out for me?

 

[belliott4488]

The best way to understand it is to use mathematical induction, i.e. figure it out for a small number and then show that the reasoning extends to larger numbers. So in this case, start with two blue-eyed guys and one green-eyed guy. Once you're convinced that the green-eyed guy must die on day 1, see if you can see why two green-eyed guys would have to die on the 2nd day. Carry on from there.

 

[Wonderballs]

A. Why are they trying to kill themselves? You all assume they want to know what colour their eyes are and want to commit.

B. I don't get either part of the question.

 

[belliott4488]

Wonderballs:

This is a logic puzzle, not a study in sociology. The obviously absurd premises of the laws of this community and the citizens' willingness to abide by them are simply given. The task is to determine the logical reasoning that would allow the citizens to reach the conclusions they do, when they do. It's not about why people behave in any particular way - that's just a humorous construct.

 

[Wonderballs]

I still don't understand why after 800 days they commit, or after the foreigner gets washed away they suddenly know what colour theier eyes are even though it stated that he didint specify whose eyes were green.

Which is exactly why its a brain teaser

 

[belliott4488]

You're right - that's the puzzle! ;-)

Did you try my hint in an earlier post? That's a good way to start.

 

[Wonderballs]

I have been looking over this in my head for quite some time, and I just can't figure it out. I'm not convinced that even if ONE of them commited suicide that the rest would follow suite... and I definately don't understand why they ALL do it on the 200th day.

 

[quetzalcoatl9]

spoiler:



consider the case of only 2 ppl, one having green eyes. pretend that you are one of them - if you see that the person across from you does not have green eyes (and you know that at least one person has green eyes) then there is only one possibility - you have green eyes. therefore, you do yourself in the next day. so, 1 person had green eyes and then commit suicide 1 day later. (note the number of ppl with blue eyes is irrelevant).

consider the case of only 3 ppl, with 2 ppl having green eyes. let's say that you have green eyes: you will look around and see 1 person with green eyes and 1 person with blue eyes. now there are only two possible choices here (a) you have green eyes or (b) you have blue eyes. if you have blue eyes, then there is only one person on the island with green eyes in which case we have the situation described in the paragraph above. but if you wait one day and that other green-eyed dude did not commit suicide, then you know that he must be seeing another green-eyed person out there - YOU! therefore you will kill yourself after 2 days, as will he since he is in exactly the same boat. so, 2 people with green eyes and then commit suicide 2 days later.

continue for N people and N days...


[/color]

 

[2muchsleep]

If the people either know that there are 800 blue or 200 green couldn't they have just walked around counting before the stranger comes, ie if they count only 199 green they are green or count 799 blue they are blue

 

[Huckleberry]

There are alternatives.

1. Could always pluck out your eyes. Then you wouldn't have green eyes any more and wouldn't be in violation of the rules when the time came for the ritual.

2. Could just assume that the cast away was a dirty liar who likes to cause trouble for communities by breaking the rules. (ofcourse I would conveniently assume he followed the same rules I did.

3. Change the laws.

Would the following options work?

4. Could sacrifice yourself at the first ritual the following day, sparing everyone else from discovering their eye color.

5. There could be an new villager with green eyes, such as a newborn, who hadn't heard the castaways words.

Honestly, I don't understand why would all kill themselves after 200 or 800 days. If they had never seen anyone with green eyes and heard that comment then I can understand they would know they had green eyes. But since they can see 199 or 200 other people with green eyes, then ofcourse they already know that some people have green eyes. In this case the strangers words aren't revealing anything new. Why would him telling them what they already know start mass suicides?

 

[Huckleberry]

If I see 0 people with green eyes and I discover that someone has green eyes, then I can be sure that I am the one with green eyes.

If I see 1 person with green eyes and he does not kill himself at the first ritual then I know that there must be more than one person with green eyes. If I see nobody else with green eyes then I know I have green eyes.

If I see 2 people with green eyes then at the first ritual I would assume they see either 1 or 2 people with green eyes, with myself being that possible 2nd person.

How would I know if I had green or blue eyes after the second ritual when I see 2 people with green eyes? How can I conclude that if they don't kill themselves, I must have green eyes? How does the number of rituals factor into the logic?

 

[quetzalcoatl9]

see my post #20 above, and all will be told...

 

[Huckleberry]

I explained as much in my post, but I still don't understand how it applies when I can see 2 or more people with green eyes.

And I still don't understand why some castaway telling them what they already know would begin this fatal chain reaction. I would think day 0 would begin from the moment they passed the two laws. Seems a ridiculous thing to do, sentencing themselves to death like that.

 

[belliott4488]

Huckleberry,
No one is suggesting that this is a true story nor that it would make much sense in the real world. It's a logic puzzle.

Here's the answer to your question re: seeing two green-eyed people:
You've figured out what your response would be if you saw only one green-eyed person. If you saw two green-eyed people, then you'd hope that they each see one green-eyed person and will do exactly as you would in that case, i.e. they'd kill themselves on day 2. If they don't, then the only explanation is that they each see two green-eyed people, not just one. You must be the second one. You learn this on day two, so on day three you do yourself in.

The same logic applies on day 3 if you see 3 green-eyed people: if you're blue-eyed, then the green-eyed guys kill themselves on day 2 as you would have in the previous paragraph; if they don't, then they must see three green-eyed people, so you must be green-eyed.

 

[Huckleberry]

Huckleberry,
No one is suggesting that this is a true story nor that it would make much sense in the real world. It's a logic puzzle.

Dude, I totally thought this was for real! You scared me.

I know it's a logic puzzle. I'm saying I think the puzzle is broken. The addition of the mysterious stranger who washes ashore one day to reveal to the villagers that some of them have green eyes creates a paradox. Since the villagers have been living with these laws for a long time and they already know that some people have green eyes, by the time the guy washes ashore there would be 0 or 1 person on the island that has green eyes.

So, obviously the guy is a liar, and people shouldn't commit suicide based on what he says. As far as I can see the arrival of this stranger has no meaning to the logic puzzle. The purpose of adding him seems to be the same as a guy walking into the room when there are only 0 or 1 people with green eyes and letting everyone know that someone has green eyes. I don't how this works when there are already 2 or more people with green eyes.

How would this puzzle work with no mysterious stranger? Or can you explain to me why the stranger is necessary in this puzzle?

edit - since the guy doesn't specify how many people have green eyes I don't see how what he says makes any difference. And since the villagers don't speak of their eye color they would be perfectly safe. Now if he said 200 of you have green eyes then 200 people would commit suicide the next day. Otherwise the paradox that they are alive at all would continue.

 

[Huckleberry]

And what's stopping these people from looking around and saying to themselves, "Hey, I see 199 other people with green eyes. Certainly they must see either 199 or 200 people with green eyes also. So if they don't all kill themselves tomorrow then I must have green eyes also and we can all kill ourselves the day after that.

Why do they need to wait 200 days at all? I think the puzzle is broken because they know if everyone else has green eyes. They only don't know what color their own eyes are.

 

[belliott4488]

Well, Mr. Huck, you've certainly got the problem down! Your questions are exactly the statement of the problem: how can it be (logically) that these guys kill themselves when they do, given all the considerations you've named? That's precisely what makes this an interesting problem - to me, at least.

I think the only problem you're having is that you haven't yet fully absorbed the logic that leads to the stated result. I'll give it another shot here (more spoiler):

Suppose there's one green-eyed guy. So long as no one says, "hey, there's a green-eyed guy here," he never knows his eyes are green, nor do any others know that their eyes aren't green. The significance of the arrival of the stranger is that he does, in this case, bring a new piece of information to one person, namely the green-eyed guy, who now knows that 1) there's a green-eyed guy and 2) he's it.

Now suppose there are two green-eyed guys. Yes, they both know that there's at least one green-eyed guy, just as everyone else knows that there are at least two. No one knows the exact number, though, not knowing his own eye color. Now the stranger arrives and makes his announcement. Each green-eyed guy looks at the other one (surreptitiously), hoping that that guy is the only green-eyed guy so that he'll kill himself on day one, for the reasons in the preceding paragraph. They both eagerly await this on day one, but of course, neither one kills himself, so they are both very disappointed. That's because each one knows that the only reason the other green-eyed guy didn't kill himself must be because that guy himself sees another green-eyed guy, who must be himself. That's how they both discover that they are green-eyed. Everyone else, by the way, was eagerly hoping to see both of them kill themselves on day 2, which is how they would learn that they were not green-eyed.

If there are three green-eyed guys, they each wait until day two to see if the two green guys they see will kill themselves as in the previous paragraph. When they don't, then each of them realizes that he must be a third green-eyed guy.

The same reasoning applies for N green-eyed guys on the Nth day.

It is truly weird, since you can well ask, as you did, why should they all wait till the 200th day, since they all know that there are at least 199 green-eyed people? The answer is, how would they agree on this? They couldn't say how many green-eyed people they see without immediately letting everyone else know whether or not he had green eyes. So, maybe they could just pick a safe number, say 2 less then the number they see. Well, the blue-eyed guys might think it's safe to say, "let's skip the first 198 days, since everyone agrees that we all see at least 198 green-eyed people, right?" But that immediately gives away the fact that they see 200, because each green-eyed guy sees only 199, so each of them would have said, "we can agree to skip 197 days, since I know everyone see at least 197." But if he were smart enough, he wouldn't even say that, because he'd realize that if there really were only 199 green-eyed people (i.e., if he were not green-eyed), then each of the 199 green-eyed people would be wondering if there were really only 198 green-eyed people, so that those guys would recommend skipping196 days -- except that they couldn't, because they'd be wondering if those 198 green guys were seeing 197 green guys, in which case they would suggest skipping 195 days, and so on and so forth ...

It's called recursion, and it's not easy to get your brain to wrap around it. I once forced myself to write out the thought process of each green-eyed guy on Day 199, just as I did above for the simpler cases above - it's not easy, but it is logical!

 

[quetzalcoatl9]

are you people really that slow, or just can't read?? there IS new information being introduced by the stranger, which is the following:

"there is at least one person here with green eyes"

which was NOT known before the stranger's arrival

 

[Huckleberry]

are you people really that slow, or just can't read?? there IS new information being introduced by the stranger, which is the following:

"there is at least one person here with green eyes"

which was NOT known before the stranger's arrival

How was this not known? Can the people not look around and see for themselves that other villagers have green eyes?

 

[Huckleberry]

Hmm, so far all of the explanations have relied on an expectation that one person would look around and see either 0 or 1 other green-eyed person. Then the descriptions just trail off and say extrapolate for N days. I'm not ready to assume that is a valid method yet.

Here's what I got so far.
If I see 0 people with green eyes and I discover that someone has green eyes, then I can be sure that I am the one with green eyes.

If I see 1 person with green eyes and he does not kill himself at the first ritual then I know that he sees someone with green eyes. If I see nobody else with green eyes then I know I have green eyes.

If I see 2 people with green eyes then at the first ritual I would assume that the other people with green eyes believe they are looking at the only other green eyed person. When that person doesn't commit suicide they will hopefully realize that they also have green eyes. Then the next day, if there are no other people with green eyes and they do not commit suicide then I must have green eyes.

So far all of these explanations have an assumption in them that relies on drawing a conclusion from one person. If there are 4 people with green eyes then the minimum number of green-eyed people that anyone can see is 2. In no case will anyone assume that any other person sees less than 2 other people with green eyes. How can they draw a conclusion as to their own eye color without being able to make an assumption based on someone seeing only one other person with green eyes?

In order to explain this to me, please tell me how 4 people with green eyes could discover they all had green eyes. Please do not use N to extrapolate.

The answer is, how would they agree on this

I'm pretty sure this is wrong. They can agree on it without communicating because they are all logical. The minimum number of people with green eyes is 199. Thus N = 199.

edit - Sorry. It would be 198. The minimum number of people with green eyes that anyone would see would be 199, but the minimum number that it is possible to assume anyone could see would be 198.

 

[belliott4488]

are you people really that slow, or just can't read?? there IS new information being introduced by the stranger, which is the following:

"there is at least one person here with green eyes"

which was NOT known before the stranger's arrival

I guess I am slow ... everyone on the island already saw either 200 or 199 green-eyed people and should therefore have been aware that there was at least one person there with green eyes. Why do you say that that was not known before the stranger's arrival?

 

[belliott4488]

** Total spoiler below ... **
Sorry, but this thread is now entirely about the solution to this problem, so I think trying to disguise it any longer is somewhat pointless. I hope that's okay...

Hmm, so far all of the explanations have relied on an expectation that one person would look around and see either 0 or 1 other green-eyed person. Then the descriptions just trail off and say extrapolate for N days. I'm not ready to assume that is a valid method yet.

We didn't invent it; it's called induction and it's been around in mathematics and logic for quite a long time. If it weren't valid, I'm pretty sure someone would have pointed that out long ago ...

Here's what I got so far.
If I see 0 people with green eyes and I discover that someone has green eyes, then I can be sure that I am the one with green eyes.

If I see 1 person with green eyes and he does not kill himself at the first ritual then I know that he sees someone with green eyes. If I see nobody else with green eyes then I know I have green eyes.

If I see 2 people with green eyes then at the first ritual I would assume that the other people with green eyes believe they are looking at the only other green eyed person. When that person doesn't commit suicide they will hopefully realize that they also have green eyes. Then the next day, if there are no other people with green eyes and they do not commit suicide then I must have green eyes.

So far all of these explanations have an assumption in them that relies on drawing a conclusion from one person. If there are 4 people with green eyes then the minimum number of green-eyed people that anyone can see is 2.

? I would have said the min. number is 3. If there are 4 greenies, who sees only 2?

In no case will anyone assume that any other person sees less than 2 other people with green eyes. How can they draw a conclusion as to their own eye color without being able to make an assumption based on someone seeing only one other person with green eyes?

Okay, you're convinced that if there are three greenies, then they should kill themselves on day 3, right (that's what you said above, i.e. each person goes through the thought process you've described in the 3rd paragraph of your explanation)? So, if you see 3 greenies, but they don't kill themselves on day 3, what's the only possible explanation? It must by that they see more than the two greenies they would see if there were only 3 greenies in all, IOW, there must be 4, and you must be number 4. Therefore, if there are 4 greenies, they must kill themselves on day 4. Therefore, if you see 4 greenies, but they don't kill themselves on day 4, you must be greenie #5. Therefore if there are five greenies, they must kill themselves on day 5. Therefore, if you see 5 greenies, but they don't kill themselves on day 5, you must be greenie #6. See where this is going?

In order to explain this to me, please tell me how 4 people with green eyes could discover they all had green eyes. Please do not use N to extrapolate.

I'm pretty sure this is wrong. They can agree on it without communicating because they are all logical. The minimum number of people with green eyes is 199. Thus N = 199.

edit - Sorry. It would be 198. The minimum number of people with green eyes that anyone would see would be 199, but the minimum number that it is possible to assume anyone could see would be 198.

I think you just stumbled across the problem. The blue-eyed guys would want to say 199, while the green-eyed guys would want to say 198. Either way, they give away the number that they see, which let's everyone else know his own eye color.
A bit more explicitly: if the green-eyed guys tried to explain why they prefer 198 as the min. number, the blue-eyed guys all say, "whoa, 198? I see 200, so he can't see 201, which means I'm NOT green-eyed." The green-eyed guys say, "hm ... 198? that's what I would have said, since I see 199, so I guess he sees the same as I do: 199 greenies, so that means I'm green-eyed, since that guys obviously doesn't see his own green eyes (but I do)."
Any clearer?

 

[Huckleberry]

** Total spoiler below ... **
We didn't invent it; it's called induction and it's been around in mathematics and logic for quite a long time. If it weren't valid, I'm pretty sure someone would have pointed that out long ago ...

That statement is an example of logical induction and isn't necessarily true. This is very different from mathematical induction, which I'm, unfortunately, not familiar with. It's been ages since my last math class, but I did realize that some mathematical principle was being used here. I wish others would confirm or deny the validity of mathematical induction in this logic problem. It might not really help me understand the answer, but I would feel more comfortable accepting it. In the meantime, I'll consider how this functions in the problem.

The people with green eyes and the people with blue eyes would each see a different number of people with green eyes. When N = the number of greenies that they see and nobody kills themselves then they can be sure that they are a greenie also. All the greenies would realize this at the same time. I get that. I think at this point I'm trying to reason how the induction can be true if there is never a point when someone can look around and see only one other greenie. If they can't verify the premise then how can they use induction? Does it matter?

? I would have said the min. number is 3. If there are 4 greenies, who sees only 2?

Sorry again. I worded my statement incorrectly. I should have said that when there are 4 greenies then the minimum number of greenies that can be assumed to be seen by anyone is 2.

I think you just stumbled across the problem. The blue-eyed guys would want to say 199, while the green-eyed guys would want to say 198. Either way, they give away the number that they see, which let's everyone else know his own eye color.
A bit more explicitly: if the green-eyed guys tried to explain why they prefer 198 as the min. number, the blue-eyed guys all say, "whoa, 198? I see 200, so he can't see 201, which means I'm NOT green-eyed." The green-eyed guys say, "hm ... 198? that's what I would have said, since I see 199, so I guess he sees the same as I do: 199 greenies, so that means I'm green-eyed, since that guys obviously doesn't see his own green eyes (but I do)."
Any clearer?

Not really. Since everyone could look around and see at least 100 greenies they could arbitrarily say N = 100. I think there is a minimum number of greenies that can be assumed to exist that can be arrived at mathematically. N could be set to that number.

If there are 200 greenies and 800 blueys

If I have blue eyes I would see 200 greenies. I am looking for a minimum number of greenies that everyone can see so I'll assume I have blue eyes. (Without knowing it, I am fortunate that I am correct.) Then I would consider the perspective of a greenie.

From the perspective of any greenie he would see 199 greenies. He is looking for a minimum number of greenies that everyone can see so he'll assume he has blue eyes. (Without knowing it, he is unfortunate that he is incorrect.) So he would consider the perspective of one of the 199 greenies that he sees.

From his perspective he would consider himself a bluey and see 198 greenies. The first greenie knows this greenie is incorrect about the color of his own eyes, but this greenie doesn't know that. He would need to consider someone elses perspective. Any other greenie will also see this guy has green eyes and will calculate the same total number of greenies. Thus the minimum number of greenies that anyone can be assumed to see is 198. N = 198.

So on the 2nd morning after the laws were passed they would all kill themselves? If induction doesn't work backwards in this problem, might it not work forwards also?

edit - nevermind. I see the problem with this now. The answer would always be a range of 2 numbers. I didn't use the same logic for the bluey as I did for the greenie. They took a different number of steps. They could still choose some arbitrary number below that range, but they would have to communicate to do so. I think I understand how induction works in this problem now. Thanks for the help and the patience, belliott.

I still don't understand what significance the stranger holds in this puzzle. I would think they would start counting the days as soon as the laws were passed, since they already know that some villagers have green eyes.

 

[belliott4488]

That statement is an example of logical induction and isn't necessarily true. This is very different from mathematical induction, which I'm, unfortunately, not familiar with. It's been ages since my last math class, but I did realize that some mathematical principle was being used here. I wish others would confirm or deny the validity of mathematical induction in this logic problem. It might not really help me understand the answer, but I would feel more comfortable accepting it. In the meantime, I'll consider how this functions in the problem.

I hope we're not getting too far afield, here, but ... I don't know how "logical induction" and "mathematical induction" are two "very different" things. I would have thought the second is just a specific example of the first.

In any case, induction is a way of proving the truth of a statement for all values of some parameter in the statement. To do it you have to prove two things: 1. If the statement is true for any given value n of the parameter, then it is also true for n+1, and 2. It is true for a specific value of the parameter, typically one or zero.

The people with green eyes and the people with blue eyes would each see a different number of people with green eyes. When N = the number of greenies that they see and nobody kills themselves then they can be sure that they are a greenie also. All the greenies would realize this at the same time. I get that. I think at this point I'm trying to reason how the induction can be true if there is never a point when someone can look around and see only one other greenie. If they can't verify the premise then how can they use induction? Does it matter?

I'm afraid I'm not following you. What do you mean, "there is never a point when someone can look around and see only one other greenie"? If there are two greenies, then each one looks around and sees only one greenie. If there are three, then they each one wonders if the other two see only one, but they learn otherwise one day two.
Beyond that, you don't need anyone to look around and see only one greenie. All you need to know is that if there is only one greenie, then he dies on day one - that gives you point 2. above. Point 1. above comes from realizing that if N green guys kill themselves on day N, then N+1 green guys must kill themselves on day N+1 (because they observed that the N greenies that they see didn't kill themselves on day N, and thus conclude that there must be N+1 greenies, i.e. they are green-eyed themselves).

Sorry again. I worded my statement incorrectly. I should have said that when there are 4 greenies then the minimum number of greenies that can be assumed to be seen by anyone is 2.

Not really. Since everyone could look around and see at least 100 greenies they could arbitrarily say N = 100. I think there is a minimum number of greenies that can be assumed to exist that can be arrived at mathematically. N could be set to that number.

If there are 200 greenies and 800 blueys

If I have blue eyes I would see 200 greenies. I am looking for a minimum number of greenies that everyone can see so I'll assume I have blue eyes. (Without knowing it, I am fortunate that I am correct.) Then I would consider the perspective of a greenie.

From the perspective of any greenie he would see 199 greenies. He is looking for a minimum number of greenies that everyone can see so he'll assume he has blue eyes. (Without knowing it, he is unfortunate that he is incorrect.) So he would consider the perspective of one of the 199 greenies that he sees.

From his perspective he would consider himself a bluey and see 198 greenies. The first greenie knows this greenie is incorrect about the color of his own eyes, but this greenie doesn't know that. He would need to consider someone elses perspective. Any other greenie will also see this guy has green eyes and will calculate the same total number of greenies. Thus the minimum number of greenies that anyone can be assumed to see is 198. N = 198.

This is incorrect, although I admit to having a difficult time following your argument, especially in terms of which guy your pronouns are referring to ("he" and "his").
In any case, whatever reasoning you apply to the blue eyed guy will apply equally well to the green-eyed guy so that whatever number the blue-eyed guy suggests will be one more than the number the green-eyed guy suggests, simply because the blue-eyed guy counts one more green-eyed guy. That will allow them to figure out which of them sees more greenies, which is all they need to know to figure out their own eye color.
You even state that one of them (I didn't follow which) knows that the other is incorrect. That is true, but it's the key to the fact that they will be able to determine exactly how many greenies that other guys see.
I have a feeling this won't convince you, so let's try something: Suppose you're in this position, only let's say you see 153 green-eyed guys. What number would you suggest as the minimum that we all see? If you tell me what number you'd suggest, then I'll tell you how many greenies you see and the color of my own eyes.
In case you think it matters, you can see that I'm blue-eyed (although I don't know that, of course). Of course, I know your eye color, but I'm not telling you. (I've already decided what color eyes I see that you have, as well as how many greenies I see, but I'm not telling you that until after you respond and I tell you my reasoning - I promise I won't change my mind!)

So on the 2nd morning after the laws were passed they would all kill themselves? If induction doesn't work backwards in this problem, might it not work forwards also?

No, that's one of the weird things. On the day the laws are passed, no one states that there are green-eyed people, so the process doesn't begin.
Again, if there were only one green-eyed guy, he wouldn't have any reason to kill himself on day 1, since no one told him there is a green-eyed guy.
That means if there were two, they wouldn't expect the other guy to kill himself on day 1, so they wouldn't have any reason to kill themselves on day 2.
That means if there were three, they wouldn't expect the other two to kill themselves on day 2, so they wouldn't have any reason to kill themselves on day 3.
That means if there were four, they wouldn't expect the other three to kill themselves on day 3, so they wouldn't have any reason to kill themselves on day 4.
Etc.

 

[belliott4488]

I still don't understand what significance the stranger holds in this puzzle. I would think they would start counting the days as soon as the laws were passed, since they already know that some villagers have green eyes.

Sorry - I posted my long response before seeing the edits you added.
The thing about the stranger is that it's the announcement that there are green-eyed people that is the key, not the simple knowledge. The announcement makes the knowledge general, whereas people's observations give them different pieces of information, i.e. any green-eyed people are aware of one fewer green-eyed person than any non-green-eyed people. The lack of shared understanding seems to be key, here.
Again, the simple case of one green-eyed guy makes this clear. In that case, he's the only one who is explicitly unaware that there is a green-eyed guy. For the two or more, you have to do the inductive thing.

 

[Huckleberry]

A common example of logical induction would be a statement like "Every crow I've ever seen has been black, therefore, all crows are black." They are often reasonable statements, but they are not inherently factual. They seem different to me from mathematical principals.

No, that's one of the weird things. On the day the laws are passed, no one states that there are green-eyed people

If there were one person he would never know his eye color.

If there were two greenies then when they look at each other they would wonder why the person doesn't kill themselves at the next ritual. They would realize that he doesn't know his eyes are green. He may be the only greenie.

If there are 3 greenies then they would all see 2 other greenies. They might assume that those greenies are the only 2 on the island, who are looking at each other thinking the last greenie doesn't know his eye color.

Ok, this may be an example of induction also. Hmm, but once the stranger makes his statement then 1 no longer is true. Then 1 greenie would know he is the only greenie on the island. And if there were 2 then the second would know he had green eyes after the first doesn't kill himself, etc. It seems that after the stranger makes his statement it can't be assumed that people don't know their eye color any more. So the only option when they don't kill themselves is that one must have green eyes. I'm still not sure that's correct, but it is very strange. I have a hard time believing that people would look around and see many green-eyed people and not come to the conclusion themselves that there are green-eyed people on the island.

What do you mean, "there is never a point when someone can look around and see only one other greenie

I think this is the last sticking point for me on this riddle and the reason I question the validity of induction here.

If there are 4 greenies then they all can assume they have blue eyes and look at the perspective of another greenie. They see the 2nd greenie and know that he is wrong to assume he has blue eyes. The most they can assume is that the 2nd greenie sees 2 other greenies. So all the greenies would realize that every greenie knows that there are more than 1 greenie on the island.

Also, if there are 4 greenies and none of them ever see only 1 other greenie then there is no original greenie to draw the first conclusion that they are the only greenie on the island. If all the greenies know that there never was the original greenie who could deduce they had green eyes, then how can they use induction to be certain if they have green eyes.

I don't know. Very possible I'm wrong here. I'm not sure what you could say that would be new to somehow explain it to me. You have been very patient so far, but I'm sure you must be tired of me by now and I think I need a break from this problem for a while. Thanks for sharing the puzzle. I enjoyed it very much.

 

[belliott4488]

I think this is the last sticking point for me on this riddle and the reason I question the validity of induction here.

If there are 4 greenies then they all can assume they have blue eyes and look at the perspective of another greenie. They see the 2nd greenie and know that he is wrong to assume he has blue eyes. The most they can assume is that the 2nd greenie sees 2 other greenies. So all the greenies would realize that every greenie knows that there are more than 1 greenie on the island.

Also, if there are 4 greenies and none of them ever see only 1 other greenie then there is no original greenie to draw the first conclusion that they are the only greenie on the island. If all the greenies know that there never was the original greenie who could deduce they had green eyes, then how can they use induction to be certain if they have green eyes.

I don't know. Very possible I'm wrong here. I'm not sure what you could say that would be new to somehow explain it to me. You have been very patient so far, but I'm sure you must be tired of me by now and I think I need a break from this problem for a while. Thanks for sharing the puzzle. I enjoyed it very much.

Well, I agree, we've probably spun this out about as long as we should. ;-) Before I let it go, however, let me address your remaining question, in case you do come back to this thread at some point.
If I understand you, you're concerned that if there are more than 2 greenies, then there is no one to make that initial declaration, "whaddaya mean there are green eyed people - I don't see any! Uh-oh, it must be ME!" That is true, but it's also unnecessary for 2 or more, because all anyone needs to know is that the N green-eyed people he sees didn't kill themselves on Day N. Let me walk through this for the case of 4 greenies that you asked about.
First, the quick answer is that each of the four greenies sees three greenies, and when they don't kill themselves on Day 3, that is all that they need to see, because if there were only three then they'd have to kill themselves on Day 3, for the reasons we've already gone over. They don't have to reason all the way back to the case of one greenie. But ... they could. Here's how:
This is what the blue-eyed people think:
"I hope that I'm not green-eyed, and since I see only four greenies,
..that means I hope they each see three greenies and think to themselves,
...'I hope I'm not green-eyed, and since I see only three greenies,
... that means I hope they each see only two greenies and think to themselves,
..."I hope I'm not green-eyed, and since I see only two greenies,
... that means I hope they each see only one greenie and think to themselves,
..."I hope I'm not green-eyed, and since I see only one greenie,
... that means I hope he sees no greenies and thinks to himself,
..."Oh, crap, I'm screwed."
..."
..."
..."
"
Follow that? ;-) It's not actually that any of them are thinking those things beyond the first two, but rather that they are thinking about what the others are thinking, and specifically, what the others are thinking that the others are thinking (and so forth).

 

[Huckleberry]

They don't have to reason all the way back to the case of one greenie. But ... they could. Here's how:
This is what the blue-eyed people think:
"I hope that I'm not green-eyed, and since I see only four greenies,
...'I hope I'm not green-eyed, and since I see only three greenies,
... that means I hope they each see only two greenies and think to themselves,
..."I hope I'm not green-eyed, and since I see only two greenies,
... that means I hope they each see only one greenie and think to themselves,
..."I hope I'm not green-eyed, and since I see only one greenie,
... that means I hope he sees no greenies and thinks to himself,
..."Oh, crap, I'm screwed."

I think you are wrong here. If there are 4 greenies then all the villagers will know that the other villagers know that there are greenies on the island. That is all that is needed to start counting days to N=3 when all the greenies realize they have green eyes. So the strangers words aren't necessary, and the counting of days would begin at the passing of the law.

All the people with blue eyes will see 4 greenies and all the people with green eyes will see 3 greenies. Since the greenies will always see one less person with green eyes than a bluey does, I'll show how every greenie knows that the other greenies know there are other greenies on the island.

There are 4 greenies.
Greenie4 hopes he has blue eyes and sees 3 other greenies.
Greenie4 hopes that Greenie3 only sees 2 other greenies.
Greenie4 realizes that Greenie2 will see that Greenie3 also has green eyes,
so Greenie2 will include Greenie3 in his count and will hopefully arrive at a total of 2 greenies.

The reason of Greenie4 will be the same reasoning of all the greenies. They can all assume that the other greenies see other greenies. I think it's kind of silly to assume that when there are hundreds of greenies on this island that they can't agree that greenies exist. Induction isn't appropriate for this part of the problem.

 

[belliott4488]

I think you are wrong here. If there are 4 greenies then all the villagers will know that the other villagers know that there are greenies on the island. That is all that is needed to start counting days to N=3 when all the greenies realize they have green eyes. So the strangers words aren't necessary, and the counting of days would begin at the passing of the law.

Nope. That will work only if someone says out loud, "I see green people" - that's critical - it's not enough that everybody knows it. Think again about the case of only one greenie - I think you agree that he won't kill himself. Without that, the reasoning changes for all other cases, starting with the case of two greenies, neither of whom will expect the other to kill himself on Day 1, and who thus won't learn anything about their own eye color when he doesn't. It doesn't matter that they both knew that there was (at least) one greenie when the law was passed - they don't expect him to do anything special on Day 1. The point is that there has to be some statement made that they all agree on. Until it's voiced, however, it has no effect. Weird, but necessary.

All the people with blue eyes will see 4 greenies and all the people with green eyes will see 3 greenies. Since the greenies will always see one less person with green eyes than a bluey does, I'll show how every greenie knows that the other greenies know there are other greenies on the island.

There are 4 greenies.
Greenie4 hopes he has blue eyes and sees 3 other greenies.
Greenie4 hopes that Greenie3 only sees 2 other greenies.
Greenie4 realizes that Greenie2 will see that Greenie3 also has green eyes,
so Greenie2 will include Greenie3 in his count and will hopefully arrive at a total of 2 greenies.

The reason of Greenie4 will be the same reasoning of all the greenies. They can all assume that the other greenies see other greenies. I think it's kind of silly to assume that when there are hundreds of greenies on this island that they can't agree that greenies exist. Induction isn't appropriate for this part of the problem.

Not quite!
Yes, with four greenies, they all know that there are at least three greenies, because that's what they see. If they could say this aloud, then they could skip the first three days of waiting, but then again, doing so would violate the law, so they'd have to die the next day, anyway - and with good reason, since they could not say this without revealing how many greenies they, themselves, actually see.
Here's how:
You've said that the four greenies all know that everyone sees at least two greenies, so they should just say this aloud and get it over with. Let's say Greenie 4 takes the plunge and makes this prounouncement. Here's what everyone else reasons in response:

Blue guys: "hm ... I see four greenies, and if I'm green-eyed then Greenie 4 should also see 4, but then he'd suggest that we all see at least 3, not 2 - so he must see only three, which means I'm blue-eyed!"

Green guys: "uh-oh - That guy with green eyes just said we all see at least 2 greenies, but if I'm blue-eyed, then he should see only two greenies himself, and he should have said everyone sees at least one greenie, but since he didn't, that means he must see three greenies, which mean I'm green-eyed!"

They can't suggest a minimum number without giving away the number that they see, since everyone else sees either the same number. or one more or less than they do (depending on whether the guy making the suggestion is green-eyed or blue-eyed).

 

[Hurkyl]

So the strangers words aren't necessary

Yes they are. In order to conclude your own eyes are green, you need to know:
(1) You see N green-eyed people.
(2) N days ago, if there was only 1 green-eyed person person, he would have killed himself.

 

[Sleek]

Lets say there is a green eyed person and 2 blue eyed person. If someone says (like the stranger did) that he sees green eyed people, then the green eyed person, knowing there are only three people in total with other two blue eyed, would know he is the only green eyed.

If there are two green eyed people. Then at the first day, the green eyed duo will look at each other, assuming themselves to be blue. But they both won't die the next day. Thus it would imply that the other one thinks someone else is green eyed, but the rest two are blue eyed, so he should also be green eyed. Thus they both die on the second day.

Irrespective of the no. of blue eyed people (>0), n green eyed people would die on nth day.

 

[Huckleberry]

Yes they are. In order to conclude your own eyes are green, you need to know:
(1) You see N green-eyed people.
(2) N days ago, if there was only 1 green-eyed person person, he would have killed himself.

I would think they have all this information before the stranger tells them there are green-eyed people on the island. They only need look around and see that there are other green-eyed people on the island. That satisfies the requirement for (1). If there are 4 green-eyed people then in no case can anyone assume that another person sees less than 2. Since everyone knows that everyone knows that there are more than one green-eyed people it satisfies the requirement for (2).

It doesn't matter if the green-eyed people and blue-eyed people can't agree on N. As soon as there are 4 green-eyed people it can no longer be assumed that someone is the only green-eyed person.

 

[Hurkyl]

I would think they have all this information before the stranger tells them there are green-eyed people on the island.
...
If there are 4 green-eyed people then in no case can anyone assume that another person sees less than 2. Since everyone knows that everyone knows that there are more than one green-eyed people it satisfies the requirement for (2).

But that's not what (2) says. (2) says that if only one person had green eyes, he would have killed himself N days ago.

 

[Huckleberry]

But that's not what (2) says. (2) says that if only one person had green eyes, he would have killed himself N days ago.

As soon as it is known by everyone that 2 greenies see each other then someone should start killing themselves.

 

[Hurkyl]

As soon as it is known by everyone that 2 greenies see each other then someone should start killing themselves.

Why?

 

[Huckleberry]

If it is known that nobody sees less than two greenies then everyone knows the information that the stranger told them, that there are greenies on the island.

 

[Hurkyl]

If it is known that nobody sees less than two greenies then everyone knows the information that the stranger told them, that there are greenies on the island.

But how does that prove people start killing themselves?

 

[Huckleberry]

As long as it is certain that everyone on the island knows that there are greenies then they should start induction. The induction goes on until N days where N is equal to the number of people with green eyes that they see. If those people don't kill themselves the next day then that individual knows he must have green eyes.

If it is not certain that everyone on the island knows that there are greenies then the process of induction can't begin. But since as long as there are 4 greenies everyone is certain that the others know, they can begin induction.

I just don't understand why they need a stranger to tell them anything before they can begin induction.