In the solution to the Blue Eyes logic puzzle, there is a follow-up question:
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?
Answer
After talking with you for a bit in chat, I see better what the problem is and where you're heading down the wrong path.
Here's the principal issue: knowledge is only useful if everyone on the island can make the same conclusion. Why? Let's say $C$ has a piece of knowledge that lets them skip a day, but can't verify that everyone else also has this piece of knowledge. If $C$ skips ahead a day, $C$ could easily be leaving people behind who haven't made the same conclusion.
Let's say there are three people: $A$, $B$, and $C$. They all have blue eyes.
Your premise starts off with the conclusion that $C$ sees two people with blue eyes, and knows nobody will leave the island on day 1. This conclusion is entirely correct. However, $C$ can't just go ahead and skip day 1 without first checking to make sure $A$ and $B$ will do the same. Otherwise, $C$ would skip ahead and leave $A$ and $B$ confused.
Here's where the logical trap is. While $C$ can conclude that nobody will leave the island, $C$ can't conclude that $B$ or $A$ will say the same. $C$ has two cases to explore: if they have blue eyes, and if they have brown eyes. They don't know which it is. Let's say $C$ supposes they have brown eyes (even though it's not true) - then, what does $C$ conclude about what $B$ knows?
In this thought experiment of $C$'s, $B$ sees only one person with blue eyes. But $B$ doesn't know their own eye color either! In $B$'s mind, their eye color could theoretically be brown. As a result, $A$ would leave the island on day 1!
Now, obviously, this isn't true. The point is that it's possible for $C$ to conceive of a situation in which neither $A$ nor $B$ could conclude nobody will leave the island. As a result, the knowledge that nobody will leave the island on day 1 isn't useful to $C$. $C$ may know this, but there's a chance that neither $A$ nor $B$ do, and in that case, $C$ would be skipping ahead.
To continue this thought experiment through, let's say day 1 passes. Nobody leaves the island. Now what's $C$ thinking?
Well, $C$ ponders what $B$ knows. In the case where $C$ has brown eyes, $B$ has seen nobody leave the island and has concluded that they have blue eyes. This is a correct conclusion. In the case where $C$ has blue eyes, $B$ still can't tell whether they have blue or brown eyes, but $B$ knows that everyone else is conducting the same thought experiments.
As a result, from the perspective of $C$, either $B$ and $A$ will leave the island tonight (if $C$ has brown eyes), or $A$, $B$, and $C$ will leave the island tomorrow night (if $C$ has blue eyes). The first night is critical, though, because it establishes the common knowledge that in everyone's thought experiments, there are at least two people with blue eyes.
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