I'm currently working through MIT's 6.042 practice problem sets.
One of the problems, called "Surveyevor" has me completely flummoxed. I tried and failed, so had a look at the solution and still couldn't make any sense of it. The problem and solution can be found here:
The solution is "on day p all the purple eyed people leave the island" - this doesn't make sense to me.
I don't understand why n changing would have any impact at all on the number of purple eyes the people can see.
For example, say I'm on the island and I can see 3 people with purple eyes. Each of these people can see at least 2 others with purple eyes. A person would only have reason to believe they have a purple eye if they see no other purple eyes. So as long as each purple-eyed person can see one other purple-eyed person, nothing changes, so no one ever leaves the island.
I can't understand why n (the number of days) would matter here at all, or even why induction would be used here. Am I missing something?