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Comment by millstone | Hacker News Reader

In the Dot-town suicides, say the stranger says "Alice has a blue dot." Alice then kills herself. Why would the other residents die?

I think this satisfies the requirements: there's certainly some number of blue dots for which the statement would be false, namely zero.

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That's not what the author intended by "anything about the number of blue dots". The author means "anything about the number of blue dots, but not who has them", because the proof of the solution relies on the fact that each person ("Alice") doesn't know whether their own dot is in the set the stranger speaks about but everyone else does know whether "Alice" is in the set.

"Alice has a blue dot" adds extra information (It's equivalent to "There is at least one blue dot", and Alice has a blue dot"), and that extra information is critical in that, even though it adds some information, it removes useful information by reducing the information entropy of the situation, which stymies efforts to make inferences.

The problem wording is (arguably) ambiguous on that point.

I think that's right. So the author's description of "anything non-trivial" is weakened to "anything non-trivial maintaining information symmetry", which feels constricting, certainly no longer "frighteningly general."

This also reveals a strategy for townsfolk who wish to survive: simply inform a blue-dot of their color, breaking the logic chain. That winning move ought to factor into the strategy of these perfect logicians!

I suppose then they will all simultaneously do it and all die. Or perhaps being perfect logicians they will hold a lottery and inform the winner of their state causing them to die alone for the good of the group.

Yeah, I was thinking about if the stranger said something like, “not all the dots are blue.” This is non-trivial by the definition given, and no one would have to kill themselves at all.

"not all the dots are blue" - in the case where there are n people with n-1 blue dots and 1 red dot, the person with the red dot kills themself, and then everyone else kills themselves because they know that the red dot person killed themself because they did not see any red dots.

Right, but if there were two red dots the town is fine.

In a one red dot town, you could say “there is at least one blue dot”

By induction and perspective-taking, 1 person dies each time everyone sits and thinks for a while.

Would they be fine?

So let's say there are two people with red dots, let's name them Ruth and Rudy. Ruth now knows that "not all dots are blue" (which is equivalent to "there is at least one red dot"). She sees Rudy with the red dot: Fine, here's the person with the red dot.

But wait a minute, why is Rudy not killing himself? If Rudy is the only person with a red dot, he should have seen only blue dots... however he knows there has to be at least one red dot, so if he only saw blue dots, the red dot must have been him, so he would have killed himself.

But he didn't, so that means he has seen at least one red dot, which isn't him, and isn't anybody else Ruth has seen because she only saw blue dots otherwise.

Oh no, it must be Ruth with the second red dot!

Ok now it is making sense. The same would be true if you went to three dots.... the third person would think, “wait, why aren’t the two blue dotted people killing themselves? There must be a third blue dot... wait, the third blue dot must be me”

That makes sense. What about the other commenter, who said something like “Alice has a blue dot.”

Wouldn’t that not lead to everyone’s death?

I think that "Alice has a blue dot" is technically not a statement about the number of blue dots.

In your three dots logic, this would have a red dot person kill themselves if there are two blue dots: The two blue dotted people aren't killing themselves because they are still thinking "wait, why isn't he killing himself" about the other.

I feel like some timing procedures need to be defined.

I think the 'they meet every night' is supposed to be the 'timing procedure'

Given 2 red dots, they would already know that not all dots are blue even before the stranger comes. At their first town meeting, everyone would know "not all dots are blue" and they would know that everyone else knew that information too.

Would this mean the colony would self-implode?

If you have a red dot, you only know that there is at least 1 red dot, and you don't know if that one person knows "not all dots are blue". Hence, it's new information when that person doesn't commit suicide.

But how would they know how long to wait before they kill themselves? What if Rudy hasn't killed himself yet because he's slightly slower at solving logic problems?

Since they meet every night the implicit idea is that whoever is to kill themselves will do it by the end of the night or next morning -- so after that, in the next-next night's meeting they can take whether someone committed suicide or not into account.

Whether someone is "slower at solving logic problems" goes into IQ etc, and is not relevant to the logic problem (the same way "what if someone refuses to kill themselves over something so superficial" is -- one should assume they can all do the math it takes equally well, else we would have been told of that.

If that were said to a population of 2 people, one with a blue dot and one with a red dot, the person with the red dot would kill themself immediately

Right, but isn’t the challenge supposed to be to prove that in every case, everyone dies? Not that ‘in at least one case, everyone dies’

If it was the latter, a one person town would always die.

I think that statement is still subject to the proof given. "Not all the dots are blue" just initialize K to {n}, and the induction proceeds as written.

The flaw in the proof is that it assumes that townsfolk only learn their dot color based on counting arguments. If the stranger's "anything non-trivial" statement includes information beyond just a raw count, then a townsfolk may learn their dot-color prematurely (e.g. by being told in the statement) and the proof fails.

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