100 blue-eyed people riddle N=4

Question

The solution of this riddle is usually given with explicit reasoning for two cases:

• Case where there is only 1 blue-eyed person,
• Case where there are 2 blue-eyed people.

That logic is clear, and I can see how it extends to the case where there are 3 blue-eyed people.

When there are only 2 blue-eyed people, after the first day, they can each assume that the other person would have left the first day if they were the only person with blue eyes.

When there are 3 people with blue eyes, after 2 days, the third person can assume they have blue eyes, because otherwise the other 2 blue-eyed people would have left using the same logic for the 2-person case.

But, once you get to 4 people with blue eyes, the logic doesn't seem to hold up anymore. They all see that there are at least 3 people with blue eyes. Each day the guru says there is someone with blue eyes and we all know it. That doesn't seem to change anything, each day they can still see there are at least 3 other people with blue eyes. And it doesn't seem there is any reason that any number of days would change their opinion of their own eye color.

Can someone give an explicit reasoning for the case where there are 4 blue-eyed people?

Answer 1

The same logic really does work inductively for every case based on the previous case.

Let's say there are 4 people on the island. You're one of them, so you can see the other three all have blue eyes.

• IF you don't have blue eyes, then you know that, from the point of view of any of the others, they can see just two people with blue eyes. So they should leave on the 3rd day, for the same reason as in the case of only 3 people with blue eyes. Essentially, any non-blue-eyed person is discounted from the reasoning: whether there are 3 people with blue eyes and no-one else, or 3 people with blue eyes and 1 person with brown eyes, the logic is the same and those 3 will leave the island on the 3rd day.

• Therefore, by contradiction, you do have blue eyes. But you can't realise that until the 4th day, because it takes you that long to know that the blue-eyed people you can see didn't leave on the 3rd day.