The island of blue eyes

1. The public announcement does not add an individual "new" fact, but it does raise it to common knowledge, and that triggers induction.
2. For n=100: since this does not occur on day 99, each one concludes that she is also blue.
3. The 100 people with blue eyes leave the island at dawn on the 100th day.

Answer: The 100 people with blue eyes leave the island at dawn on the 100th day.
Inductive scheme (common knowledge):

• If there was 1 person with blue eyes, upon hearing "at least one", they would leave on day 1.
• If there were 2, each one would wait for day 1; Since no one leaves, both deduce their color and leave on day 2.

Generalizing: with $n$ blue-eyed people, they all leave on $n$ day.
For $n=100$:

• Each person sees 99 blue-eyed people and hopes that, if they were not blue, those 99 would leave on the 99th day.
• Since this does not happen on day 99, each one concludes that she is also blue.
• They all leave at dawn on the 100th.

The public announcement does not add an individual "new" fact, but it does raise it to common knowledge, and that triggers induction.