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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:

https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-042j-mathematics-for-computer-science-fall-2010/recitations/MIT6_042JF10_rec02_sol.pdf

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?

dwight
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    "Nothing changes" is wrong. Information changes. Namely, after each day you have the new information that yet another day nobody has left, and you can deduce something from that. The best is to first consider the cases where there are 1 or 2 people with purple dots. What happens then? – Michal Adamaszek Apr 16 '20 at 21:47
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    You’re missing the hypothesis that all of the contestants are master logicians: you’re to understand that all of them have proved Theorem $1$. Thus, when no one leaves on day $1$, they all understand that $p$ cannot be $1$ and therefore must be at least $2$. Thus, someone who sees only one purple eye will realize that there are only $2$, and that he or she must have the other one. – Brian M. Scott Apr 16 '20 at 21:49
  • Ah, I get it now. So if I see 2 purples and neither of them leave after the second day, I must also be a purple, or else the others would have left on the previous day. So we all leave on day 3. And so on. Makes sense, thanks for your help! – dwight Apr 16 '20 at 22:22

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