[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).

[MissKitten]

This is the first thing I thought of.

He could say "I have blue eyes."

Of course, this is only true if everybody is still on the island.

[Guest]

"I see an even (or odd if that is the case) number of blue eyed people"

Edited April 21, 2010 by Simon Legree

[Guest]

Well the person could say, "I see an odd number of (so-and-so) pairs of eyes."

Now if he said "I see an odd number of blue eyes," then you would count the number of blue eyes you see. If the number was even, that had to mean that your eyes were blue. If you counted the number was odd, then you know that your eyes must be brown, and vice versa.

The statement-giver knows the color of his eyes because he is the color of the odd number of people.

*Note* This only works if the statement-giver knows that there are a hundred each of eye colors.

[Guest]

See my prior answer. If statement is he sees an even number of blue eyed people-any person seeing an odd number of blue eyes not including the statement maker--knows he must have blue eyes--and vice versa.--unfortunately the only person who is sacrificed is the "winner'" of the random selection who has no way of determining his eye color

Edited April 21, 2010 by Simon Legree

[Guest]

This is the first thing I thought of.

He could say "I have blue eyes."

Of course, this is only true if everybody is still on the island.

He does not know he has blue eyes so he cannot say that. Besides, everybody else on the island besides him already knows that he has blue eyes (they see him when he makes his statement) so there would be no point in saying that even if he did know his own eye color (which he doesn't... nobody knows their own eye color to begin with).

Edited April 22, 2010 by Use the Force

[Guest]

"I see an even (or odd if that is the case) number of blue eyed people"

You missed the key description in the question: "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."

So your statement wouldn't be allowed because everyone else doesn't know if he sees and even or odd number of blue-eyed people due to the fact that they don't know what their own eye color is.

[Guest]

Well the person could say, "I see an odd number of (so-and-so) pairs of eyes."

Now if he said "I see an odd number of blue eyes," then you would count the number of blue eyes you see. If the number was even, that had to mean that your eyes were blue. If you counted the number was odd, then you know that your eyes must be brown, and vice versa.

The statement-giver knows the color of his eyes because he is the color of the odd number of people.

*Note* This only works if the statement-giver knows that there are a hundred each of eye colors.

1. You missed the key restriction as well: "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."

2. The statement-giver does not know that there are 100 blue-eyed people and 100 brown-eyed people. Those numbers are just for the sake of you explaining your answer. The people on the island don't know that. All they know is everyone's eye color but there own and that their own eye color is either blue or brown.

[Guest]

See my prior answer. If statement is he sees an even number of blue eyed people-any person seeing an odd number of blue eyes not including the statement maker--knows he must have blue eyes--and vice versa.--unfortunately the only person who is sacrificed is the "winner'" of the random selection who has no way of determining his eye color

Again, that would work except for the fact that everybody doesn't already know whether the statement-giver sees an odd or even number of pairs of blue eyes. Thus, the statement doesn't meet the restriction: "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." If there wasn't this restriction (so far the answers have chosen to ignore it...), then the logic riddle would be far too easy. Think harder, people!

[Guest]

Nobody has gotten the answer yet! This is a complicated one, but I know someone will be able to get it.

Just make sure you all remember the restriction on the statement. The statement-giver must be certain that everybody already knows that the statement is true. For example, the statement could be "We are on an island" or it could "I don't know if I have blue or brown eyes." Be creative! It's a riddle!

[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

Edited April 22, 2010 by ana_stassia

[Guest]

If you think about it, there really is nothing that he could say that will get the most prople to leave the island because they already all know that what he will say is true so since they are perfect logicians they should've figured it out already. basically there is nothing more he can tell them.

he can tell them all to tell each other what their eye color is.

[bushindo]

How about this statement, which everybody on the island should agree is true

"There are either 99, 100, or 101 people with brown eyes on this island"

If you think about it, there really is nothing that he could say that will get the most prople to leave the island because they already all know that what he will say is true so since they are perfect logicians they should've figured it out already. basically there is nothing more he can tell them.

You are right in that the person chosen to save the logicians can not tell them any new information explicitly. However, the fact that such a statement, call it Q, is spoken in front of all logicians lead to a new piece of information: that is, all logicians now know that every other logician knows about Q. This new implicit piece of information allows for a solution to the puzzle.

Edited April 22, 2010 by bushindo

[Guest]

Here is my solution:

With 100 brown eyed and 100 blue eyes, assume that the speaker has blue eyes. The speaker would then say "I see 99 blue eyes, and so does every other blue eyed person".

The blue-eyed people know this is true because they also see 99 blue eyes (including the speaker but excluding themself).

The brown-eyed people know this is true because they see 100 blue eyed people and know the speaker can't see his own eyes.

Therefore, all brown eyed people must know that they are brown eyed, and leave.

The next day, the blue eyed people all see that there are only 100 people left and they see 99 blue eyed people, so they can conclude they are also blue eyed (they know that all brown eyed people left).

[Guest]

assuming they've stayed on that island for long enough to see everyone else (and in the process duly noted down the eye colours).

the guy should just read out his stats and ask everyone to crosscheck, they know his eye colour so they should be able to find out theirs.

OR

he can say mutiny and lead a revolution that overthrows the boat's captain and takes them all to safety.

perfectly logical.

i know im wrong.

[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

Edited April 22, 2010 by Anza Power

[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...

[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.

[Guest]

hi everyone,Making wild guesses from 'be creative' Please don't blame me as I am sure i am WAY younger than all of you

Say " I have no idea what is my eye colour but I know yours. " This maybe will work as everybody knows that this is true. If this is correct , everyone will escape by saying the same statement.

[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.

I think you've got it, let's say you're one of the logicians and you know everyone on the island thinks the same way, one of them is chosen and gets up and sees x blue eyes and y brown eyes, so he'd say "There are at least (x-1) blue eyes and at least (y-1) brown eyes here", if he said 98 blue and 99 brown, then you'd know that he is seeing 99 blue and 100 brown, if you also see 100 brown that means you are blue, if you see 99 brown that means you are brown, everyone else will follow the same logic and everyone (except the one who talked) will know their colors...

[Guest]

can't he say something like

"at least x number of you have blue eyes" (he excludes himself from the statement) where x is the number of people he sees with blue eyes minus one. That statement will be true to everyone.

Everyone knows his statement had to be known information to everyone and they know he must have made a logical choice to save the island. They will be able to look around and if they look at the 198 people other than the man who made the statement they can either see x blue eyes, or x+1 blue eyes.

After the first day, if no one leaves the island, it is safe to say that the number of people with blue eyes on the island was not x (which it was not). Therefore logically, there must be x+1 people with blue eyes in the population (remember the statement maker is out so you are only dealing with 199 islanders now).

Now that it is known that the total number of people with blue eyes is x+1, everyone can look around. If they see x blue eyes then they know they have blue eyes. If they see x+1 blue eyes then they know they have red eyes.

This saves 199 people on the second day, just in time!

[Guest]

can't he say something like

"at least x number of you have blue eyes" (he excludes himself from the statement) where x is the number of people he sees with blue eyes minus one. That statement will be true to everyone.

Everyone knows his statement had to be known information to everyone and they know he must have made a logical choice to save the island. They will be able to look around and if they look at the 198 people other than the man who made the statement they can either see x blue eyes, or x+1 blue eyes.

After the first day, if no one leaves the island, it is safe to say that the number of people with blue eyes on the island was not x (which it was not). Therefore logically, there must be x+1 people with blue eyes in the population (remember the statement maker is out so you are only dealing with 199 islanders now).

Now that it is known that the total number of people with blue eyes is x+1, everyone can look around. If they see x blue eyes then they know they have blue eyes. If they see x+1 blue eyes then they know they have red eyes.

This saves 199 people on the second day, just in time!

That is correct but why do you need 2 days? as I said before he says "There are at least x people with blue eyes" and they all understand that x is one less than the number of blue eyes that he sees, so knowing what he sees and what you see, you can determine your color instantly and so will everyone else on the island except the speaker himself who won't be able to figure it out since he doesn't get any info from anyone else...

[Guest]

I think you've got it, let's say you're one of the logicians and you know everyone on the island thinks the same way, one of them is chosen and gets up and sees x blue eyes and y brown eyes, so he'd say "There are at least (x-1) blue eyes and at least (y-1) brown eyes here", if he said 98 blue and 99 brown, then you'd know that he is seeing 99 blue and 100 brown, if you also see 100 brown that means you are blue, if you see 99 brown that means you are brown, everyone else will follow the same logic and everyone (except the one who talked) will know their colors...

'i see a minimum of 98 blue and a minimum of 99 brown'

all brown eyed know that he really saw 100, but said 99 to ensure that each brown knew this to be true. (each brown sees 99 brown)

so now that each knows there is 100, he will realize that he is the 100th person. that night all brown will leave. (if the speaker was brown eyed they would not leave since they would also see 100 brown eyed persons and cannot conclude that night that they have brown eyes.)

the next day, only blue eyed people are left. as perfect logicians, they understand that the brown eyed people cannot leave behind another brown eyed. (all brown eyed will know their eye colour at the same time)

then the rest of them can conclude that they are blue. that night (the second night) all blue eyed will leave the island.

so noone is left.

(if there are 99 blue and 101 brown, then the blue will leave first and the brown will leave second)

please correct me where my reasoning is wrong

[Guest]

That is correct but why do you need 2 days? as I said before he says "There are at least x people with blue eyes" and they all understand that x is one less than the number of blue eyes that he sees, so knowing what he sees and what you see, you can determine your color instantly and so will everyone else on the island except the speaker himself who won't be able to figure it out since he doesn't get any info from anyone else...

Anza, you are correct. You are using more implicit information than I was with my solution. Although there may be some complex debate about what is logical behavior.

- Sky

Edited April 22, 2010 by skyblueshu

[Guest]

Anza, you are correct. You are using more implicit information than I was with my solution. Although there may be some complex debate about what is logical behavior.

- Sky

Thats pretty much what I said :-)

[Guest]

ok, disregard my last post. i see what Anza is saying now