[Guest]

Note that this puzzle is a variation of the puzzle Upon solving the classic puzzle I thought of an interesting variation to it that I will share here. Here is my variation:

A group of 200 people who have either blue or brown eyes live on a volcanic island. They are all perfect logicians -- if a conclusion can be logically deduced, they will do it instantly. No one knows the color of their eyes. Every night at midnight, a ferry stops at the island. Any islanders who have figured out the color of their own eyes then leave the island (note: they're forced to leave if they've determined their eye color), and the rest stay. Everyone can see everyone else at all times and keeps a count of the number of people they see with each eye color (excluding themselves), but they cannot otherwise communicate. Everyone on the island knows all the rules and information in this paragraph.

One day, when the volcano rumbles at noon as a sign to everyone that there will be a massive, deadly, volcanic eruption in exactly two days, one person is randomly selected to be allowed to make a single statement. The only restriction on the statement is that the person saying it must be sure that everybody else on the island already knows that the statement is true.

What single statement does the person say in order to save as many people as possible before the volcano erupts in two days? How many people of each eye color are saved on each night? (Note: Everybody wants to save as many people as possible and everybody knows that everybody else wants to save as many people as possible.)

Also, for the sake of making your answer explanation simpler, let there be 100 people with blue eyes and 100 people with brown eyes on the island and let the person who is randomly selected to give the statement be one of the people with blue eyes. Note that everyone knows which person gave the statement (i.e. everyone but the statement-giver knows the statement-giver's eye color).

[Guest]

if he is allowed to use numbers

he could say "i see at least 99 blue eyed people and at least 99 brown eyed people."

this would be true since a blue eyed sees 99 blue and 100 brown,

and a brown eyed sees 99 brown and 100 blue.

that night each blue eyed will wait for the 99 to leave the island. this is because if there is in fact 99 in total then each will believe that there is 98 since he does not know his own eye colour. on hearing there is at least 99 he will realize that he is the 99th blue eyed.

therefore noone will leave and the next day, on still seeing 99 blue eyed people, they will realize that they are the 100th person and that night they will all leave.

the same logic applies to the brown eyed persons.

sorry if my answer is confusing. i'm not good at explaining things

Your answer violates the restriction that everybody on the island must already know that the statement is true.

[Guest]

if he sees let's say 99 blue eyes, he could just state:

"49 + 50 = 99"

Thus messaging to the others that there are 99 blue eyes in front of him, that is not restricted by the rules cause they all know 49+50=99...

unless they may not plan a strategy

They cannot plan a strategy, no. They do not communicate in any way ever except for seeing how many people on the island there are and what their eye colors are (except their own eye color). However, perhaps your statement would be enough to lead these intelligent logicians to decide that that is the information that the statement-giver is trying to convey. So your answer would possibly work, but I have a better statement that won't leave so much guesswork. I like your thinking, however.

[Guest]

He can pick up a colour.. Say Brown... and suppose he sees n pairs of eyes... He needs to say he sees atleast n-1 (99) pairs of brown eyes... No one either blue or brown can cast any doubt on the statement.

Whichever colour he choses... All the people of that colour will leave on the night following tonight... So he will choose the colour he could see maximum...

Just working on a way to include others...

Good, but yes, more can be included. You're not far off.

[Guest]

The speaker's statement will be "There are minimum 99 brown and minimum 98 blue eyed people here". This statement can be validated by everyone.

This will help everyone but for the speaker leave the same night as he will not know whether he is brown or blue... But please some one take him as well for saving others

Explanations: The speaker could not tell 99 blue and 100 brown as he does not know his eye colour. Even if he know his eye colour as Blue, he cannot tell 100 brown as if he is not sure if he could be brown. If he is not brown, then the maximum brown eyed men a brown eyed fellow can see is only 99.

A CORRECT ANSWER Found ^:

I would say that your answer is correct. It is not the answer that I had intended for the problem, but is very similar to the answer that I intended. The difference is that my statement is shorter (you almost have two statements in one... I didn't intend for that). So I'll say you're right. But, let's see if you (or anyone else) can find the logic and reasoning that makes my shorter statement still allow for 199 people to get off the island on the very first night (and 0 people get off on the second night).

Here is my answer:

"There are at least 98 brown-eyed people here."

When trying to understand the logic that allows for the other 199 people to get off the island, note that:

- Everyone must have either blue or brown eyes.

- The statement-giver has blue eyes (he, himself does not know that, but everybody else does).

- The statement-giver sees 99 blue-eyed people and 100 brown-eyed people (everyone else doesn't know that, except through logic and reasoning).

I'm pretty sure that that statement alone is enough for everybody else to figure out their eye color. I'll allow all of you to attempt to figure out my reasoning and then tomorrow I will post my explanation (or perhaps the day after tomorrow if I think somebody is close and I would rather give them a hint instead) for you all if you haven't figured it out already.

Good job TheKing! And good job in advance to whoever can figure out the logic that will allow for my short statement to get 199 people of the island on the very first night!

Edited April 23, 2010 by Use the Force

[Guest]

^ It's same as we've explained before...

If he says "There are minimum n brown eyed people here" everyone should figure out that he is seeing n+1 brown eyes, now everyone else who sees n+1 brown is bluem and those who see n brown are brown...

But you said "There are at least 98 brown eyes" are you sure that is true or is it just a spelling mistake? cause stating that there are at least 99 brown is good enough cause if he is seeing 100 brown and supposed his own color is not brown every other brown person would be seeing 99 browns...

Edited April 23, 2010 by Anza Power

[Guest]

"I will die" you should say. Everyone should know that everyone dies... A baby, teen, a grown-up, or a child.

[Guest]

Is it a correct assumption that the people do NOT know that there are 100 of each color? If they did know that, they would know the color of their own eyes, because if they saw 99 blue and 100 brown, they would know they must have blue.

However, every person would know that they (themselves) see 99 of one and 100 of the other.

At that point, everyone would then know that there are either 100/100, or 101/99, since they don't know their own eye color. They also know everyone else knows this fact. Thus, they know that everyone ELSE must see at least 98 of each color, because if it's 101/99, those in the 99 will only see 98/101.

Let's take a blue eyed person for example. That person will see 99 blue, 100 brown. He ALSO knows that EVERYONE will see at least 99 brown, and at LEAST 98 blue.

Since it can be assumed that he will want to save as many people as possible, any statement he makes will be one that reveals information about what he DOES see, but no more than can be guessed (but no less, either).

Thus, if he said "everyone sees at least 99 brown eyes and at least 98 blue eyes", I think people would be able to deduce that he actually DOES see 100 brown and 99 blue, and can thus decipher their own eye color accordingly.

I think the big question is, is this deduction "safe", knowing the speaker will be trying to save as many people as possible... That I'm not sure of, but I think it may just work.

[Guest]

^ It's same as we've explained before...

If he says "There are minimum n brown eyed people here" everyone should figure out that he is seeing n+1 brown eyes, now everyone else who sees n+1 brown is bluem and those who see n brown are brown...

But you said "There are at least 98 brown eyes" are you sure that is true or is it just a spelling mistake? cause stating that there are at least 99 brown is good enough cause if he is seeing 100 brown and supposed his own color is not brown every other brown person would be seeing 99 browns...

That is correct. And yes, I did mean 99 brown (well actually I was thinking 98 blue, but it's the same thing).

[Guest]

Is it a correct assumption that the people do NOT know that there are 100 of each color? If they did know that, they would know the color of their own eyes, because if they saw 99 blue and 100 brown, they would know they must have blue.

However, every person would know that they (themselves) see 99 of one and 100 of the other.

At that point, everyone would then know that there are either 100/100, or 101/99, since they don't know their own eye color. They also know everyone else knows this fact. Thus, they know that everyone ELSE must see at least 98 of each color, because if it's 101/99, those in the 99 will only see 98/101.

Let's take a blue eyed person for example. That person will see 99 blue, 100 brown. He ALSO knows that EVERYONE will see at least 99 brown, and at LEAST 98 blue.

Since it can be assumed that he will want to save as many people as possible, any statement he makes will be one that reveals information about what he DOES see, but no more than can be guessed (but no less, either).

Thus, if he said "everyone sees at least 99 brown eyes and at least 98 blue eyes", I think people would be able to deduce that he actually DOES see 100 brown and 99 blue, and can thus decipher their own eye color accordingly.

I think the big question is, is this deduction "safe", knowing the speaker will be trying to save as many people as possible... That I'm not sure of, but I think it may just work.

Your statement violates the restriction that the statement-giver must be certain that everybody already knows the statement is true. Your statement was, "Everyone sees at least 99 brown eyes and at least 98 blue eyes."

The statement-giver (blue eyes, we're saying) sees 99 blue eyes and 100 brown eyes. If he considers the possibility that he himself has brown eyes, then that would mean that blue-eyed people would see 98 blue-eyed people. Thus, these blue-eyed people would only be certain that everybody sees at least 97 blue eyes. Thus, the statement-giver is not sure that everybody else knows that "everybody sees at least 99 brown eyes and at least 98 blue eyes." Do you see the problem?

In other words, the statement-giver does not know that he has blue eyes. So, he must consider the possibility that he has brown eyes. If he does have brown eyes, then the 99 blue-eyed people that the statement-giver sees would only see 98 blue-eyed people (due to the fact that they can't see themselves). So, would these 99 blue-eyed people who see only 98-blue eyed people know for a fact that everybody else sees at least 98 blue-eyed people? No, because these 99 blue-eyed people do not know that they have blue eyes. For all they know, they could have brown eyes and thus the blue-eyed people would only see 97 blue-eyed people. I hope that helps.

So, your statement is worded in such a way that it violates the restriction. While it is true that the blue-eyed statement-giver who sees 99 blue eyes and 100 brown eyes would know for a fact that everybody else sees at least 98 blue-eyed people, he does NOT know for a fact that everybody else knows that everybody sees at least 99 brown eyes and 98 blue eyes.

[Guest]

The speaker's statement will be "There are minimum 99 brown and minimum 98 blue eyed people here". This statement can be validated by everyone.

This will help everyone but for the speaker leave the same night as he will not know whether he is brown or blue... But please some one take him as well for saving others

Explanations: The speaker could not tell 99 blue and 100 brown as he does not know his eye colour. Even if he know his eye colour as Blue, he cannot tell 100 brown as if he is not sure if he could be brown. If he is not brown, then the maximum brown eyed men a brown eyed fellow can see is only 99.

This is a correct answer. All that is required for the statement is, "There are a minimum of 99 brown-eyed people." Due to the fact that everyone on the island knows that everyone has either brown or blue eyes and also due to the fact that everyone knows that the statement-giver is trying to save as many people as possible, there is no need to say what the minimum number of both colored eyes there are. All 199 other people (not the statement-giver) can immediately determine their own eye color and leave on the very first night.

[Guest]

Actually I disagree with your reasoning. The statement does not violate your rules, however the people on the island do not know how many of each color there are. For example, what if a blue-eyed person sees exactly 100 brown-eyed people? He could think that he's either the 99th blue eyed person or the 101st brown eyed person. Both account for the "minimum" of 98 people.

I think my original answer is correct -- if the speaker says "All blue eyed people see 99 other blue eyed people", they know this is true -- the brown eyed people see 100 blue eyed people and know that each can't see his own eye color. The blue eyed people all see 99 blue eyes so this statement is still true.

The end result is that the brown eyed people know that if the blue eyed people only see 99 people (and they are blue eyed), that person MUST be brown eyed.

So all brown eyed people leave. The blue eyed people know that all brown eyed people have left and then the next day they can deduce they are blue eyed and leave.

A CORRECT ANSWER Found ^:

I would say that your answer is correct. It is not the answer that I had intended for the problem, but is very similar to the answer that I intended. The difference is that my statement is shorter (you almost have two statements in one... I didn't intend for that). So I'll say you're right. But, let's see if you (or anyone else) can find the logic and reasoning that makes my shorter statement still allow for 199 people to get off the island on the very first night (and 0 people get off on the second night).

Here is my answer:

"There are at least 98 brown-eyed people here."

When trying to understand the logic that allows for the other 199 people to get off the island, note that:

- Everyone must have either blue or brown eyes.

- The statement-giver has blue eyes (he, himself does not know that, but everybody else does).

- The statement-giver sees 99 blue-eyed people and 100 brown-eyed people (everyone else doesn't know that, except through logic and reasoning).

I'm pretty sure that that statement alone is enough for everybody else to figure out their eye color. I'll allow all of you to attempt to figure out my reasoning and then tomorrow I will post my explanation (or perhaps the day after tomorrow if I think somebody is close and I would rather give them a hint instead) for you all if you haven't figured it out already.

Good job TheKing! And good job in advance to whoever can figure out the logic that will allow for my short statement to get 199 people of the island on the very first night!

[Guest]

Actually I disagree with your reasoning. The statement does not violate your rules, however the people on the island do not know how many of each color there are. For example, what if a blue-eyed person sees exactly 100 brown-eyed people? He could think that he's either the 99th blue eyed person or the 101st brown eyed person. Both account for the "minimum" of 98 people.

I think my original answer is correct -- if the speaker says "All blue eyed people see 99 other blue eyed people", they know this is true -- the brown eyed people see 100 blue eyed people and know that each can't see his own eye color. The blue eyed people all see 99 blue eyes so this statement is still true.

The end result is that the brown eyed people know that if the blue eyed people only see 99 people (and they are blue eyed), that person MUST be brown eyed.

So all brown eyed people leave. The blue eyed people know that all brown eyed people have left and then the next day they can deduce they are blue eyed and leave.

I still disagree with what you are saying so I'll try to explain it better.The restriction is this: "The only restriction on the statement is that the person saying it must be sure that everybody else on the island already knows that the statement is true."

Your statement is, "I see 99 blue eyes, and so does every other blue eyed person."

First I'll say that I just noticed that you assumed that the statement-giver is aware that he has blue eyes. This is not true. The statement-giver does not know his own eye color. Thus, the statement-giver does not even know that his own statement is true because he does not know if the blue-eyed people see 99 blue eyes like he, himself, does or whether they see only 98 blue eyes (because he (the statement-giver) actually has brown eyes).

In my question I said, "let the person who is randomly selected to give the statement be one of the people with blue eyes." I then attempted to clear up that statement that I knew some people would misinterpret. I thus said, "Note that everyone knows which person gave the statement (i.e. everyone but the statement-giver knows the statement-giver's eye color)."

So the statement-giver does not know his own eye color and thus does not know that blue-eyed people see 99 pairs of blue eyes due to the fact that he does not know that his own eyes are blue. For all he knows, they could be brown.

So you assumed that the statement-giver knew that he had blue eyes even though he didn't know he had blue eyes. The reason why I said to answer the question as though he has blue eyes was to make it so that everyone answered the question using the same numbers, etc. I thought it would making explaining answers easier for people. Sorry I misled you.

[Guest]

I understand the given solution, however...

If the speaker were to tell everyone to choose a color one night then the other the next. The statement is known to be true to all parties on the island since each knows that they have either blue or brown eyes.

If the color choices are static (i.e. the speaker says blue then brown) then all 100 of blue will leave that night and all 100 brown the next and still escape half a day before the eruption.

If the colors are chosen at random, then you have a range of 0 (all wrong) to 200 (all right) leaving on the first night and the remainder leaving the second night.

This solution, however, exploits the fact that no restrictions were put upon an inhabitant's willingness to engage in trial and error. A way to prevent this would be to have a penalty for wrongly choosing your eye-color that is more severe than being burned alive by the volcanic eruption.

[Guest]

the statement giver is able to count 100 pairs of brown eyes then he knows that his are blue. his statement would be if you count 100 pairs of a brown colour then yours are blue.

if you count 100 pairs of blue then yours are brown. as they then all know there colour they all leave on day one

[Guest]

the statement giver is able to count 100 pairs of brown eyes then he knows that his are blue. his statement would be if you count 100 pairs of a brown colour then yours are blue.

if you count 100 pairs of blue then yours are brown. as they then all know there colour they all leave on day one

No because the statement giver doesn't know that there are exactly 100 people on the island with brown eyes.