I thought I understood your answer and liked it, but just read a different explanation at http://thesilentknight.info/2016/flawed-logic-blue-spot-problem/ that suggests that the 100 people would all leave on Day 2. Now I'm confused again. Is that a slightly different problem?

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    $\begingroup$ The problem is the same, but the logic on the linked page is incorrect. The answers to this question explain why. $\endgroup$ – f'' Jul 8 '16 at 5:04
  • $\begingroup$ Sure, they know that everyone sees either 10 or 9 other blue-eyed people (if there are 10 blue-eyed people on the island). But they don't know that the others know that. Some people could think that everyone sees either 9 or 8 blue-eyed people. And they might think that other people would think that everyone sees 8 or 7 blue-eyed people. $\endgroup$ – Deusovi Jul 8 '16 at 5:19
  • $\begingroup$ OK, if they see 9, they know there are only two possibilities - either 9 or 10. They will reason that if there are 9, they know the minimum sighting will be 8. If there are 10, they know the min sighting will be 9. Either way, they know nobody will see only 7. They also know that there may be people who think the minimum sighting will be 9. Since they've covered all of the possibilities, and they know the others are capable of only a slightly different set of possibilities (which they've already listed), they know that everybody knows nobody will see as few as 7. What did I just miss? $\endgroup$ – Bruce Jul 8 '16 at 6:11
  • $\begingroup$ I may have it now (amazing what a little sleep will do for you). If you see 10 with spots, it is not a matter of thinking that anybody really thinks someone could have 3. It is a matter of each of the two groups (the group that saw 9 and the group that saw 10) being in a parallel universe, but not able to know if the other universe is one higher or one lower. If you knew, then you could make an allowance for what they knew. You have to wait the 10 days (or 100) because that's the only way to get the two universes in sync, whether you like it or not. I'll leave a comment for Silent Knight. $\endgroup$ – Bruce Jul 8 '16 at 10:11