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?