Each person knows, from the beginning, that there are no less than 99 blue-eyed people on the island. How, then, is considering the 1 and 2-person cases relevant, if they can all rule them out immediately as possibilities?
I have found an answer to that question here:
Blue-eyed people can't see their own faces, so blue-eyed people can see one less blue-eyed face than non-blue-eyed people can. Even though I can see that there are at least 99 blue-eyed people, I don't know that they can see that, so I need to imagine people who see only 98, who would base their actions in part by imagining people who can see only 97 who would base their actions in part by imagining people who can see only 96, and so on...
This answer doesn't make sense to me. Why are they basing their logic on some hypothetical islander who can only see 96 blue-eyed people when they know such an islander cannot possibly exist and - moreover - know that everyone else knows it too?
Taking a smaller example with 6 islanders for illustrative purposes:
A (blue), B (blue), C (blue), D (brown), E (brown), F (brown)
No matter what A's eye color is, he knows:
- B will see at least one other blue (C)
- C will see at least one other blue (B)
- D, E and F will see at least two other blues (B & C)
Everyone can see at least 1 other blue-eyed islander, and everyone else knows it.
What value does Day 1 serve? Why not skip ahead to Day 2?