Let's apply the Base Case and Build approach. Assume that there are n people on the island and c of them have blue eyes. We are explicitly told that c > 8.
Case c = 1: Exactly one person has blue eyes.
Assuming all the people are intelligent, the blue-eyed person should look around and realize that no one else has blue eyes. Since he knows that at least one person has blue eyes, he must conclude that it is he who has blue eyes.Therefore, he would take the flight that evening.
Case c = 2: Exactly two people have blue eyes.
The two blue-eyed people see each other, but are unsure whether c is , or 2. They know, from the previous case, that if c = " the blue-eyed person would leave on the first night. Therefore, if the other blue-eyed person is still there, he must deduce that c = 2, which means that he himself has blue eyes. Both men would then leave on the second night.
Case c > 2: The General Case.
As we increase c, we can see that this logic continues to apply. If c = 3, then those three people will imme- diately know that there are either 2 or 3 people with blue eyes. If there were two people, then those two people would have left on the second night. So, when the others are still around after that night, each person would conclude that c = 3 and that they, therefore, have blue eyes too. They would leave that night. This same pattern extends up through any value of c. Therefore, if c men have blue eyes, it will take c nights for the blue-eyed men to leave. All will leave on the same night.