[Jehar]

This is the missing element from most explanations I see of this problem. We all look at the n cases from the POV of a blue-eyed person. An outside observer with brown eyes has the same level of information available to him, so it seems to me just as likely that after the first day, each person, regardless of eye color, could reason that nobody left the previous day, so the visitor must have been referring to me. So either eventually everyone dies, or they all realize the paradox and forgo the ritual.

[Mindless2112]

Each blue-eyed person observes 99 other blue-eyed people in the tribe, thus reasoning that he/she has blue eyes on the 100th day. However, each brown-eyed person observes 100 blue-eyed people in the tribe, thus reasoning that he/she has blue eyes on the 101st day (however this does not happen because on the 100th day all the blue-eyed people commit ritual suicide.)

[IanDrake]

Doesn't that presuppose they know there are 100 blue eyed people in their tribe? When that information is presented to the reader, it's presented as outside knowledge.

If they knew the color counts, they would know their eye color and all would have to commit suicide. The fact that the tribe still existed means, they didn't know the totals.

For example, if I know there are 100 people with blue eyes and I can count as many without including myself, then I must have brown eyes and must kill myself.

So again, there is no possible way the tribe had any idea what the exact counts where.

As a brown eyed person, there are either 100 blue eyed people meaning I have brown eyes, or there are a 101 blue eyed people and I have blue eyes. If a census was ever taken and the exact number known everyone would have to commit suicide.

Since the visitor didn't mention an exact number then there is still no way to know if you have blue or brown eyes.

However, the tribe now knows that the visitor knows he himself has blue eyes. Will they make him follow their ritual?

Update: OK, after reading the link in the first comment, I get it.

[Mindless2112]

It is not necessary for the blue-eyed people to know the total number of blue-eyed people in the tribe, they can deduce it at day 100:

  * A blue-eyed person observes 99 blue-eyed people.
  * On day 99, the blue-eyed people do not commit ritual
    suicide.
  * Thus each blue-eyed person learns that all the blue-eyed
    people also observe 99 blue-eyed people.
  * Thus the blue-eyed person knows that the other blue-eyed
    people must observe that he/she has blue eyes.

[benjohnson]

>> The fact that the tribe still existed means, they didn't know the totals.

That's the crux of it right there.

If I may also add: the outside observers remark didn't tell the tribe any information they didn't already know - that there is a blue-eyed tribe member.

[maxerickson]

In the case that there is at least 1 member of the tribe with blue eyes, all the people with brown eyes can see his eye color. So they satisfy the question of who the traveler is talking about with the blue eyed member.

The blue eyed member doesn't see anyone else with blue eyes.

[Jehar]

That's a good point, but then the critical factor is that nobody knows how many people have blue eyes. More specifically, each person has to hold the possibility that n = observed persons with blue eyes + 1, which is himself.

[maxerickson]

That's why it takes n days.

Each blue eyed person can see n-1 others. If nothing happens on day n-1, they learn that they have blue eyes.