4 ms·

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

• cortesoft 8y ago
  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.
  • uiri 8y ago
    "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.
    • cortesoft 8y ago
      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”
      • gowld 8y ago
        By induction and perspective-taking, 1 person dies each time everyone sits and thinks for a while.
      • anyfoo 8y ago
        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!
        • cortesoft 8y ago
          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?
          • uiri 8y ago
            I think that "Alice has a blue dot" is technically not a statement about the number of blue dots.
          • Deestan 8y ago
            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.
            • cortesoft 8y ago
              I think the 'they meet every night' is supposed to be the 'timing procedure'
        • Deestan 8y ago
          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?
          • fydorm 8y ago
            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.
  • zodiac 8y ago
    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
    • cortesoft 8y ago
      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.
  • millstone 8y ago
    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.
• gowld 8y ago
  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.
  • millstone 8y ago
    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!
    • neaden 8y ago
      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.