This issue does not add private information; it adds something subtler and more powerful: common knowledge. That is why its solution seems slow and, at the same time, relentless.
The island of blue eyes
Riddle statement
On an island live perfectly logical people. Exactly 100 of them have blue eyes. The others have brown eyes, but no one knows the color of their own eyes.
Everyone can see everyone else's eye color. There are no mirrors or reflective surfaces. There is also a strict law: anyone who becomes certain of their own eye color must leave the island at dawn the next day.
One day a visitor arrives and says out loud: "I see at least one person with blue eyes."
What happens next?
Show solution
Solution
Answer: The 100 blue-eyed people leave the island at dawn on the 100th day. another and I would think: "If I weren't blue, the other one would be the only one and she would be gone tomorrow." Since no one leaves on day 1, they both conclude on day 2 that they also have blue eyes, and they leave together.
The same reasoning is chained by induction for any number.
With 100 people with blue eyes, each one sees 99 and waits for 99 sunrises without anyone leaving. Since no one leaves during those 99 days, the 100 simultaneously conclude that they also have blue eyes. They all leave the island at dawn on the 100th day.