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)
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)