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As a logic puzzle it is very tenuous. The proposed solution (which is lower down the blog post) is that because any blue eyed Islander is able to count the numb

by ErrantX 16y ago

As a logic puzzle it is very tenuous. The proposed solution (which is lower down the blog post) is that because any blue eyed Islander is able to count the number of blue eyed people (99), if on day 99 those people do not kill themselves it means that there is one further blue eyed Islander, and it is them. Brown eyed Islanders see 100 blue eyes, and so know that if mass suicide occurs on day 100 then they are not blue eyed. It's only a short reprieve though; because either the browns know that only two colours exist (and so on day 100 immediately know they are brown) or they face the same conundrum as blue, and commit suicide on day 900. The problem with the puzzle is not really the logical reasoning (which is sound, as described in the puzzle). The problem is that the Islanders, who are described as very logical, should know this information already. That they are still alive when the foreigner arrives is illogical. EDIT: to downvoters; consider the problem & solution very very carefully (or read my subsequent comment). Initially it seems a little illogical that the browns would die too, but it is ultimately the same basic problem that the blues face :)

andrew1 16y ago
> It's only a short reprieve though; because either the browns know that only two colours exist (and so on day 100 immediately know they are brown) or they face the same conundrum as blue, and commit suicide on day 900. In the case where they aren't told that only two colours exist, why would they kill themselves on day 900? Suppose instead that one of the brown eyed people had green eyes, I don't see how the brown eyed people or green eyed person could deduce the colour of their own eyes.
- ErrantX 16y ago
  Imagine the problem like this. The island is populated by just 100 Blue eyed people. An individual can therefore see 99 blue eyes. Their logical reasoning is that if they are not blue eyed then on day 99 all of the blue eyed people will commit suicide (because they can see 98 blue and 1 of another colour). Because the others don't commit suicide on day 99 the only logical inference is that everyone else can see 99 blue eyes too. And so they too must be blue. The reason the solution is so complex to comprehend is because the logical inference has little to do with what eye colours exist in the tribe. It is, ultimately, simple mathematics. If a tribe has n blue eyes, y brown eyes and z green eyes any of the individuals in the tribe will eventually be able to logically infer which colour they have, regardless of having any starter information. Another way to think of it is this; they know that there are either n people with X eyes. Or n+1. Where n is the number of people they can see with X eyes and the unknown factor is their own eye colour. For everyone without X eyes n is greater than those with X eyes.
  - andrew1 16y ago
    You originally said that the people with brown eyes would commit suicide after 900 days. Can you explain to me how any of them can know for sure that they don't have green eyes?
    - ErrantX 16y ago
      I have done; I was using the "blue eyes on their own" as an example to demonstrate it. I'll try again (this can take a long while to get your head round :D). Once the Blue eyed people are dead each islander now knows that there are definitely at least 899 brown eyed people, with the unknown factor their own eye colour. Their hypothesis must be as follows (imagine this is the logical reasoning of ONE islander laying out the scenario): - If I do not have Brown eyes that means each of the brown eyed people can see 898 browns. If that is the case, then on day 899 all of those people will kill themselves. - If I did have Brown eyes then each of the brown eyed people can also see 899 brown eyes. If that is the case then no one will kill themselves on day 899, therefore as I can only see 899 browns, I must have brown Here is where most people struggle to understand: The number of brown eyes each individual can see is fixed at 899. The only unknown factor is their own eye colour. The solution is that everyone knows each other unknown factor. You know that either n or n+1 brown eyes exist (where n is the number of brown eyes they can observe). And so does everyone else (and, crucially, you know they know). If everyone's n matches yours, then on day n+1 you can infer what the unknown factor is. Another way of thinking about it is this; an individual can logically reason that only two types of people exist on the island. Those who see 899 browns (him, at the least), those who see 898 browns (potentially everyone else; on the hypothesis that his eye colour differs). If on day 899 no one commits suicide then each individual knows that only the former exists. And therefore they have brown eyes. (this is horribly hard to put across :S which is why it is such a delicious problem! The main thing to remember is that to the knowledge of one individual there is a known number of brown eyes, and a figure for the minimum and maximum number of brown eyes any other individual can see. If the number of days exceeds the number of browns you can see, you must be brown)
      - andrew1 16y ago

by ErrantX 16y ago

andrew1 16y ago

> It's only a short reprieve though; because either the browns know that only two colours exist (and so on day 100 immediately know they are brown) or they face the same conundrum as blue, and commit suicide on day 900. In the case where they aren't told that only two colours exist, why would they kill themselves on day 900? Suppose instead that one of the brown eyed people had green eyes, I don't see how the brown eyed people or green eyed person could deduce the colour of their own eyes.

ErrantX 16y ago
Imagine the problem like this. The island is populated by just 100 Blue eyed people. An individual can therefore see 99 blue eyes. Their logical reasoning is that if they are not blue eyed then on day 99 all of the blue eyed people will commit suicide (because they can see 98 blue and 1 of another colour). Because the others don't commit suicide on day 99 the only logical inference is that everyone else can see 99 blue eyes too. And so they too must be blue. The reason the solution is so complex to comprehend is because the logical inference has little to do with what eye colours exist in the tribe. It is, ultimately, simple mathematics. If a tribe has n blue eyes, y brown eyes and z green eyes any of the individuals in the tribe will eventually be able to logically infer which colour they have, regardless of having any starter information. Another way to think of it is this; they know that there are either n people with X eyes. Or n+1. Where n is the number of people they can see with X eyes and the unknown factor is their own eye colour. For everyone without X eyes n is greater than those with X eyes.
- andrew1 16y ago
  You originally said that the people with brown eyes would commit suicide after 900 days. Can you explain to me how any of them can know for sure that they don't have green eyes?
  - ErrantX 16y ago
    I have done; I was using the "blue eyes on their own" as an example to demonstrate it. I'll try again (this can take a long while to get your head round :D). Once the Blue eyed people are dead each islander now knows that there are definitely at least 899 brown eyed people, with the unknown factor their own eye colour. Their hypothesis must be as follows (imagine this is the logical reasoning of ONE islander laying out the scenario): - If I do not have Brown eyes that means each of the brown eyed people can see 898 browns. If that is the case, then on day 899 all of those people will kill themselves. - If I did have Brown eyes then each of the brown eyed people can also see 899 brown eyes. If that is the case then no one will kill themselves on day 899, therefore as I can only see 899 browns, I must have brown Here is where most people struggle to understand: The number of brown eyes each individual can see is fixed at 899. The only unknown factor is their own eye colour. The solution is that everyone knows each other unknown factor. You know that either n or n+1 brown eyes exist (where n is the number of brown eyes they can observe). And so does everyone else (and, crucially, you know they know). If everyone's n matches yours, then on day n+1 you can infer what the unknown factor is. Another way of thinking about it is this; an individual can logically reason that only two types of people exist on the island. Those who see 899 browns (him, at the least), those who see 898 browns (potentially everyone else; on the hypothesis that his eye colour differs). If on day 899 no one commits suicide then each individual knows that only the former exists. And therefore they have brown eyes. (this is horribly hard to put across :S which is why it is such a delicious problem! The main thing to remember is that to the knowledge of one individual there is a known number of brown eyes, and a figure for the minimum and maximum number of brown eyes any other individual can see. If the number of days exceeds the number of browns you can see, you must be brown)
    - andrew1 16y ago