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Comment by albinofrenchy | Hacker News Reader

I think the puzzle as phrased is slightly incomplete. The stranger must communicate something _new_ each day to make the induction work.

For instance, if the number of blues is 25, and the (merciful) stranger says that the number of blues is not prime every day, no one ever has enough information.

And in fact, even that doesn't seem to be enough. The non-trivial part must be that at least one person knows more than they did before the statement, otherwise a merciful stranger could say 'The number of blues is less than <big number>' each day, where the big number is more than the population.

Unfortunately that stringent of a definition for non-trivial sort of ruins the problem since in the formulation is the idea that at least one person in the village gets (at least) one number closer to knowing the exact number of blues everyday, and as soon as they know the exact number of blues they are eliminated.

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The clock is certainly necessary. But for information, I think its as easy as "somebody has blue eyes!" said once. If only one person has blue eyes, they see no one else with blue eyes so kill themselves. If two people have blue eyes, they see the other guy and think "maybe its only him". Next day, that guy is still alive; they now know the only remaining blue-eye is themselves; midnight comes and they both suicide. Then nobody else has to - the two guys died on day 2 so only 2 blue-eyes can exist. And so on.

If two people have blue eyes, "Somebody has blue eyes" is not new information, because everyone has already seen that someone has blue eyes.

> If two people have blue eyes, "Somebody has blue eyes" is not new information, because everyone has already seen that someone has blue eyes.

The new information in that scenario is that the other guy with blue eyes knows that someone has blue eyes.

If they have seen someone with blue eyes they would already know that.

They know that you know that. That's the new info.

It's not new information to either person individually. But while both people already knew it, they didn't know that the other one knew it. That's the new information.

> The non-trivial part must be that at least one person knows more than they did before the statement.

I think that's what they meant by non-trivial. If the stranger tells them something they already know, that doesn't count.

You can assume that an individual person can count the blue dots they see, but they can't count their own dot. If they count 9 dots, then the true number must be either 9 or 10, so they already know the number can't be prime.

I agree. If the stranger says something along the lines of, "The number of blue dots is greater than 50." on one day, and on some subsequent day says, "The number of blue dots is not an odd number less than 45." or "The number if blue dots is not a number less than 40 which is divisible by 3.", then no new information will have been conveyed by these latter example statements. I imagine that the stranger could keep stringing them along like this for quite some time, and if multiple statements referring to the same subset were allowed (ex: "The number of blues is not a number which is divisible by 37 which is less than 45." and "The number of blues is not 37.") then the stranger could continue this indefinitely.

The puzzle doesn't involve the stranger saying multiple things. He just shows up, says one thing that everyone already knows to be true (e.g. "At least one of you has blue eyes"), and then leaves forever. And that's enough to cause everyone to (eventually) commit suicide.

The fact that this happens even though it doesn't sound like new information is what makes it an interesting puzzle!

You are overlooking that all the residents gather together each day. Once a visitor has said something nontrivial, after that the new information each day comes when the residents gather, and discover who did or didn't commit suicide the night before.

Also:

> For instance, if the number of blues is 25, and the (merciful) stranger says that the number of blues is not prime every day, no one ever has enough information.

This is not correct. Saying "the number of blues isn't prime" just once would be sufficient to start the process, as would "there is at least one blue", etc. It doesn't matter that it's not new information to any one villager.

> This is not correct. Saying "the number of blues isn't prime" just once would be sufficient to start the process, as would "there is at least one blue", etc. It doesn't matter that it's not new information to any one villager.

Sorry, can you explain this part? If there were 10 blues and 10 reds, and the stranger said "there is at least one blue", why would that result in a suicide in any of the dot-town people?

That's exactly the puzzle!

If you'd like a hint, imagine that there are only two villagers - you're one of them, and the other one has a blue dot. A visitor tells you both "there is at least one blue". You already knew that, but did he? And what happens the following day, when you see him still alive (or don't)?

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