# How'd they figure that? (logic puzzle)

1. Aug 11, 2007

### 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.

2. Aug 11, 2007

### Gokul43201

Staff Emeritus
3. Aug 15, 2007

### 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.

Last edited: Aug 15, 2007
4. Aug 15, 2007

### belliott4488

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.

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

5. Aug 16, 2007

### quetzalcoatl9

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

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.

6. Aug 16, 2007

### belliott4488

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.
Yeah, :tongue2: that would help, too!

7. Aug 17, 2007

### quetzalcoatl9

yes precisely

8. Aug 22, 2007

### omg precal

you guys think way too advanced for me...

*quits forum*

9. Aug 26, 2007

### belliott4488

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

10. Aug 26, 2007

### quetzalcoatl9

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

11. Aug 28, 2007

### 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...

12. Aug 28, 2007

### quetzalcoatl9

but they dont know exactly how many people have green eyes, they only know what they see around them

13. Sep 5, 2007

### Abyss

could someone spell it out for me?

14. Sep 6, 2007

### 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.

15. Sep 20, 2007

### Wonderballs

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

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

16. Sep 20, 2007

### 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.

17. Sep 20, 2007

### 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

18. Sep 20, 2007

### belliott4488

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

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

19. Sep 20, 2007

### 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 definetly don't understand why they ALL do it on the 200th day.

20. Sep 21, 2007

### 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....