How'd they figure that? (logic puzzle)

[Hurkyl]

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

Because induction requires a base case.

 

[Huckleberry]

There is a difference between knowing that there are green-eyed people on the island and knowing that everyone else knows it also. When the stranger tells everyone that there are green-eyed people on the island then everyone is certain that everyone else knows that information. But as long as there are 4 green-eyed people then everyone can already be certain of that information. He is telling them nothing they don't already know. In fact, since the puzzle states that they had been living this way for years then the stranger would be lying, because there would be nobody on the island to make his statement to.

I don't see how the villagers conclude anything different from the strangers statement than they would from the information they could deduce when there are 4 greenies on the island.

 

[Hurkyl]

I don't see how the villagers conclude anything different from the strangers statement than they would from the information they could deduce when there are 4 greenies on the island.

Because of the base case of the induction.

Or, working from the top down (i.e. descent), with the 4 greenies, the stranger's statement affects the outcome of a hypothetical within a hypothetical within a hypothetical.

 

[Huckleberry]

But it is the same information.

If everyone knows that there are greenies then someone must be a greenie. If a greenie doesn't kill himself then there must be more than 1. etc, until everyone knows they are a greenie or not. The base case is formed when all the greenies know that all the other greenies are aware that there are more than one greenie. This is what they gather from the information that the stranger tells them, which is the same information that they deduce from 4 or more greenies existing on the island.

If there is any possibility that not everyone sees another greenie then there is no case for induction. This will happen if there are 3 or less greenies. In this case the 3rd greenie is looking at the 2nd. The 3rd realizes the 2nd doesn't know his own eye color. The 3rd is hoping that the 2nd greenie is looking at the 1st greenie and thinking that the 1st greenie is the only greenie on the island and just doesn't realize it. There may be at least one person that is unaware that there are greenies on the island.

When there are 4 greenies this is no longer possible. The 4th greenie looks at the 3rd. The 4th greenie realizes the 3rd doesn't know his own eye color. The 4th greenie is hoping that the 3rd greenie is looking at the 1st and 2nd greenie. But then the 1st greenie would be looking at the 2nd and 3rd greenie, and the 2nd greenie would be looking at the 1st and 3rd greenie. Every greenie will see 3 other greenie and realize that the other greenies see at least 2 other greenies.They must now all realize that there are greenies on the island, which is the exact same thing as the stranger told them. If I'm correct, and these villagers have infallible logic, then they wouldn't need to communicate this information to each other.

How does the stranger's statement create a base case for induction?

 

[belliott4488]

<edit> Whoops, I followed an email link that took me to the previous page, and I responded before I realized that there was another whole page of posts ... sorry ... anyway, here's how I responded to that <end edit>

Huck, you're just not getting this, are you? ...

<snip> If there are 4 green-eyed people then in no case can anyone assume that another person sees less than 2.

True, but also irrelevant, since there is no need for anyone to assume that anyone else sees fewer than 2.

Since everyone knows that everyone knows that there are more than one green-eyed people it satisfies the requirement for (2).

No, it doesn't. If no one has explicitly stated "There are green-eyed people," then why would anyone assume that 1 green-eyed guy would have killed himself 4 days ago (Requirement 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.

Again, true, but irrelevant. What is necessary is that the four green-eyed people all know that the three green-eyed people they see would have killed themselves on Day 3 if they were the only green-eyed people.

 

[belliott4488]

HUCK! You're missing something crucial here: the "base case" Hurkyl is referring to is the case of one greenie.

Do you agree that if there is only one greenie, he will not kill himself until the stranger arrives to make his pronouncement? If so, then by induction you know that no one else will kill himself unless the stranger makes his pronouncement, because the base case - i.e. hypothetical case of one greenie - fails.

The assertion that N people die N days after the announcement depends entirely on the truth of the statement that 1 hypothetical greenie would kill himself. If he does, then they all do; if he doesn't, none of them do.

 

[belliott4488]

Arrgh! You're so close it's driving me mad!

But it is the same information.

If everyone knows that there are greenies then someone must be a greenie. If a greenie doesn't kill himself then there must be more than 1. etc, until everyone knows they are a greenie or not.

No, that's not how it works. No one expects one greenie to kill himself on day 1, unless that greenie is the only greenie he sees, which is possible only for the cases of 1 or 2 greenies.

The base case is formed when all the greenies know that all the other greenies are aware that there are more than one greenie. This is what they gather from the information that the stranger tells them, which is the same information that they deduce from 4 or more greenies existing on the island.

No, you're misunderstanding what is meant by a "base case". It's the case where there is exactly one greenie. In the case you're describing, the 4 greenies kill themselves on Day 4 because they didn't see the other 3 three greenies kill themselves on Day 3. Period.

If there is any possibility that not everyone sees another greenie then there is no case for induction. This will happen if there are 3 or less greenies. In this case the 3rd greenie is looking at the 2nd. The 3rd realizes the 2nd doesn't know his own eye color. The 3rd is hoping that the 2nd greenie is looking at the 1st greenie and thinking that the 1st greenie is the only greenie on the island and just doesn't realize it. There may be at least one person that is unaware that there are greenies on the island.

ALMOST! That last sentence is false, since all 4 greenies see 3 other greenies, and everyone else sees 4 - NO ONE is unaware that there are greenies. BUT - you're right, the 3rd guy is hoping that the 2nd guy is hoping that the 1st guy sees 0 greenies. Get the difference? - it has to do with the "nested" hypotheticals. Each guy has hopes for what the other guy is thinking that the other guy is thinking that the other guy is thinking ...

When there are 4 greenies this is no longer possible. The 4th greenie looks at the 3rd. The 4th greenie realizes the 3rd doesn't know his own eye color. The 4th greenie is hoping that the 3rd greenie is looking at the 1st and 2nd greenie. But then the 1st greenie would be looking at the 2nd and 3rd greenie, and the 2nd greenie would be looking at the 1st and 3rd greenie.

The 4th guy is hoping that the 3rd guy is going through the exact thought process you described above (i.e. as if there are only 3 greenies), right down to the 2nd guy hoping that 1st sees no greenies.

Every greenie will see 3 other greenie and realize that the other greenies see at least 2 other greenies.

Yes, but they're hoping the other greenies don't know that.

They must now all realize that there are greenies on the island, which is the exact same thing as the stranger told them. If I'm correct, and these villagers have infallible logic, then they wouldn't need to communicate this information to each other.

Again, their knowledge of this is not sufficient. It is crucial that there be some kind of spoken statement, to "start the clock", i.e. to set the condition that would be necessary for one greenie to kill himself, if he were the only greenie.

How does the stranger's statement create a base case for induction?

Think about the necessity of the stranger's statement for the base case, i.e. the case of only one greenie. It ALL stems from that.

 

[Huckleberry]

Arrgh! You're so close it's driving me mad!

Haha, me too! I'm not even sure why I care so much.

ALMOST! That last sentence is false, since all 4 greenies see 3 other greenies

Well, the example this quote was referring to only had 3 greenies in it. That is why the final statement is true.

Again, their knowledge of this is not sufficient. It is crucial that there be some kind of spoken statement, to "start the clock", i.e. to set the condition that would be necessary for one greenie to kill himself, if he were the only greenie.

There is already a written statement to start the clock, from the moment they created the laws.

Think about the necessity of the stranger's statement for the base case, i.e. the case of only one greenie. It ALL stems from that.

I'll think about it. Right now I'm trying to figure out how descending induction works to create the belief that it is possible that there may be only one greenie who doesn't know his eye color, especially when there are 200 greenies on the island.

But I did think that if there were 3 greenies and 1 blue, it would be the same case as if there were 4 greenies. So I'm starting to think the whole 4 greenie theory is wrong. I think all I need to understand this puzzle is to know that the descending induction works and the 4 greenie theory doesn't. I'm out of time right now though, so I'll have to work it out later. I've got an Australian Pink Floyd show to go to.

 

[belliott4488]

Haha, me too! I'm not even sure why I care so much.

Well, the example this quote was referring to only had 3 greenies in it. That is why the final statement is true.

My mistake - I should have referred to one fewer greenie in each case, but your statement is wrong, regardless. Your "final statement" was (for the case of 3 greenies),

There may be at least one person that is unaware that there are greenies on the island.

Not so - in this case everyone sees either 2 greenies (if he is himself a greenie) or 3 greenies (if he's not), but no one is unaware that there are greenies. What there are, are people who think that there might be people who think that there is a person who is unware that there are greenies. That's a very different thing. It does, however, work for any number N of greenies; you just have to stick as many "people who think there are people who think that" phrases in there as it takes to get back to the case of one greenie.

There is already a written statement to start the clock, from the moment they created the laws.

No, not just any statement will start the clock; it has to be one that would cause a single greenie to kill himself on day one, if he were the only greenie. THAT is the base case.

I'll think about it. Right now I'm trying to figure out how descending induction works to create the belief that it is possible that there may be only one greenie who doesn't know his eye color, especially when there are 200 greenies on the island.

No - that's not the result. First, it's not that there is "only one greenie who doesn't know his eye color" - none of the greenies should know their eye color (at first, anyway - they'll figure it out on Day 198, when the greenies they see don't kill themselves). Second, it's not that there really is one greenie who sees no other greenies (the base case guy); it's that the 199 greenies each hope that the 198 they see each hope that the 197 that they see each hope that the 196 that they see ... each hope that the 2 greenies they see each hope that one greenie they see doesn't see any other greenies.

But I did think that if there were 3 greenies and 1 blue, it would be the same case as if there were 4 greenies. So I'm starting to think the whole 4 greenie theory is wrong. I think all I need to understand this puzzle is to know that the descending induction works and the 4 greenie theory doesn't. I'm out of time right now though, so I'll have to work it out later. I've got an Australian Pink Floyd show to go to.

Whoa ... Australian Pink Floyd? It that like, a bunch of Aussies in a Pink Floyd tribute band? Or maybe just a Pink Floyd concert in Australia ... either way, I hope it was fun! I'd say "Wish You Were Here," but really, I wish I were there!

 

[ThienAn]

or i know why they committed suicide is they had eyes. They can see each other' eye color. You should have posted that they are color-blinded, or mute

 

[belliott4488]

or i know why they committed suicide is they had eyes. They can see each other' eye color. You should have posted that they are color-blinded, or mute

Or, better yet, you should have read the original statement of the puzzle more closely!

 

[country boy]

Okay, now how many days until the blue-eyes self-inflict?

 

[Oerg]

Okay, now how many days until the blue-eyes self-inflict?

They do not know the colour of their eyes as blue, all they know is that there are green eyed people on the island, and that their eye is of different colour. In other words, they do not know the word to decribe their eye colour.

 

[Oerg]

I still think its a rather funny puzzle, with a few flaws, with one of the major flaws in that

If say there were 2 green and 1 blue, on the first day of comtemplation at the town centre, if there wasnt any hesitation in all 3 to do a ritual suicide, then they would have known at that moment that the 2 greens were green in eye colour. Why don't they commit suicide then and there?

And if all of the greeny citizens were to think this way, all would commit suicide on the first day.

I think this puzzle can only be solved on the basis of many assumptions.

Perhaps it would be better to say it this way,

1. All citizens would gather at the towncentre every day to contemplate the colour of their eye for 5 minutes. After 5 minutes of contemplation, all citizens who know the colour of their eyes are to commit ritual suicide immediately and the rest of the citizens are to return to their daily chores right away and return to the town centre again the next day.

 

[country boy]

Okay, now how many days until the blue-eyes self-inflict?

They do not know the colour of their eyes as blue, all they know is that there are green eyed people on the island, and that their eye is of different colour. In other words, they do not know the word to decribe their eye colour.

I don't think they are language-challenged. They will know about colors and have names for them.

To see the symmetry between green and blue, start with 200 of each (or two of each).

Some of the difficulties talked about here could be solved by stating that the islanders don't know how many colors there are until Robinson Crusoe tells them "Hey, there are only green and blue eyes on this island!"

 

[Oerg]

I don't think they are language-challenged. They will know about colors and have names for them.

Thats the problem with this question, it leaves a lot of information to assumption

 

[belliott4488]

I still think its a rather funny puzzle, with a few flaws, with one of the major flaws in that

If say there were 2 green and 1 blue, on the first day of comtemplation at the town centre, if there wasnt any hesitation in all 3 to do a ritual suicide, then they would have known at that moment that the 2 greens were green in eye colour. Why don't they commit suicide then and there?

And if all of the greeny citizens were to think this way, all would commit suicide on the first day.

I think this puzzle can only be solved on the basis of many assumptions.

Perhaps it would be better to say it this way,

1. All citizens would gather at the towncentre every day to contemplate the colour of their eye for 5 minutes. After 5 minutes of contemplation, all citizens who know the colour of their eyes are to commit ritual suicide immediately and the rest of the citizens are to return to their daily chores right away and return to the town centre again the next day.

I'm not sure I'm following your reasoning. Are you suggesting that if the green guys commit suicide on a given day, then since the blue guys might then learn their own eye color they could also commit suicide, i.e. on the same day? I see how your suggested statement of the rule for meetings solves that, but I don't see how my original statement was not equivalent. The reason I said that they must silently contemplate for ten minutes and only after this period commit suicide was specifically to prevent this problem.

I also don't see why anyone would kill himself on Day 1 in your example; the stranger guy announced only that there were green-eyed people, of which there are two, right? So the green guys would each be hoping that the other green guy was the sole green guy. (Unless you're using the fact that the stranger used the plural, which I think is an unnecessary muddling of the problem - in this case just have him say something stupid like, "there are one or more green-eyed people on this island, although I am not specifying anything about the actual number." )

As for country boy's question about how many days till the blue-eyed guys kill themselves - as the problem was stated in the original post, they never should, since each one can hope that he's the one brown-eyed guy on the island. If the stranger had said, "there are some green, some blue, but no other color of eyes ..." then they'd do themselves in on the next day after the green-eyed guys.

 

[country boy]

As for country boy's question about how many days till the blue-eyed guys kill themselves - as the problem was stated in the original post, they never should, since each one can hope that he's the one brown-eyed guy on the island. If the stranger had said, "there are some green, some blue, but no other color of eyes ..." then they'd do themselves in on the next day after the green-eyed guys.

Actually, law 1 requires that "At noon every day, all 1000 citizens must gather... ." After the 200 die, it is not longer possible to comply. So, is everyone else off the hook?

 

[belliott4488]

Actually, law 1 requires that "At noon every day, all 1000 citizens must gather... ." After the 200 die, it is not longer possible to comply. So, is everyone else off the hook?

Oh, for goodness' sake ... of course the law wouldn't be tied to to a specific population. This is a logic puzzle, not a legal puzzle! You have to exert a little effort to get into the spirit of the problem.

Change that to "all citizens must gather ..."

 

[country boy]

Oh, for goodness' sake ... of course the law wouldn't be tied to to a specific population. This is a logic puzzle, not a legal puzzle! You have to exert a little effort to get into the spirit of the problem.

Change that to "all citizens must gather ..."

Sorry, I thought I was in the spirit. Wording is important in logic puzzles, but we can also have a little fun.

I've enjoyed your puzzle and the discussion. Have a nice holiday!

 

[belliott4488]

Sorry, I thought I was in the spirit. Wording is important in logic puzzles, but we can also have a little fun.

I've enjoyed your puzzle and the discussion. Have a nice holiday!

Thanks - enjoy your holiday as well. That's the best spirit to be in!

I think this puzzle is plenty challenging enough without injecting additional complications. People tend to ask questions like, "well, what if not all the citizens have the same IQ - they can't all reach the same conclusions!" True enough, if the point were to be discussing a real-world problem, but I'd almost rather just reduce this to a problem in pure logic or information theory - not that I'm really sure how to do that!