Lynn

:) Good luck with that. Night!

Yeah but you're right - we could imagine a different set of parameters (like 200 people teleported to the island on day 0) where maybe you don't need the oracle, but that would probably be a different puzzle.

We could come up with all kinds of crazy scenarios. The purpose of the Oracle in the puzzle is just to give them that common reference point of "day 0".

Yeah maybe. Since everyone knows that hypothetical case isn't the case maybe they could make a leap in logic (although as purely-logical beings maybe they wouldn't or couldn't.) I think at some point we're just going outside the bounds of the puzzle parameters. We don't know that all 200 islanders arrived on the same day. What if 100 of them arrived on day 1 and started counting days, but then 1 more arrived each day for the next 100 days, mucking up all the calculations :)

OK, so I guess, in my mind, their decision to leave on day 100 relies on them all using Inductive Logic. Inductive Logic relies on the hypothetical case of n = 1. The hypothetical case of n=1 only works if you've got some external trigger to tell that hypothetical lone blue-eyed guy that he's got blue eyes. Therefore you need the Oracle.

Individually they could each be thinking they've got green eyes, couldn't they?

But aren't they thinking the same thing as me?

Hmm, let's take that case though - 100/100, and I've got blue eyes so I see 99 blue, 100 brown. What's going to make me leave? Can't I just go through life merrily thinking I've got green?

In the "skip ahead" argument - they all know that there aren't 1-97 blue-eyed people but they still need to go through that hypothetical exercise so they're on the same page. I think the same thing applies here. They know there isn't 1 blue-eyed guy but they still need the Oracle to trigger that hypothetical guy to leave.

If that first hypothetical guy doesn't have enough information to conclude that he needs to leave, I think it breaks down.

OK, but now let's say that you plop down 200 people on the same day with 100/100. Remember that our whole house of cards only works because all the islanders are using logical deduction. The logical deduction puzzle only works because it starts with that first hypotehtical case.

What would make the blue guy leave on that first day?

Well let's say that you dropped 200 people on the island on the same day, 199 with brown eyes and 1 with blue. You tell them to leave when they figure out their own eye color. What clue do they have that tells them?

Yeah we know that /isn't/ the case in our example problem, which is what makes it hard to wrap my brain around. But the whole logical analysis relies on starting with that hypothetical case and making inferences from there. You need the oracle to say "blue" to trigger the guy in that first hypothetical case to realize he's not purple.

They need some common frame of reference to start marking time, so they know what "the first night" is. And further, they need some external clue about eye color. Even if I look around and see nothing but brown eyes, without the Oracle saying: "I see someone with blue eyes" I might conclude mine are brown too, or purple or green.

But without the Oracle showing up and making the proclamation, what do you define as "the first night"? That gets into a messy question about how everyone came to be on this island and what led to these silly rules in the first place :)

@No.7892142 - Yeah basically. The upshot of this exercise is that the islanders can draw certain conclusions about what the other islanders see, but nobody can be 100% certain of their conclusions. It's only after the triggering event occurs (the Oracle) that they all have a common frame of reference to begin counting days and eventually verifying their conclusions.

@No.7892142 - Incidentally, my example shows the exercise with #6 imagining "what would #1-5 see?" to reach his conclusion. Even if you do the exercise with #6 imagining: "what would #1-5 see? Now what would #1 guess that #2-4 see? What would #2 guess that the others see?" you still end up with folks guessing 2 different numbers (I wrote a program to prove it to myself because I had such a hard time believing it :)). There just doesn't seem to be enough clues for everyone to end up on the same page.

@No.7892142 - Everyone has to use the same system for deciding how many to skip. So yes, #6 can say: "Hmm, I think it's safe to skip 4, but everyone else might conclude 3, so I'll skip 3 instead." The trouble is that #1-#5 will also think the same thing: "Hmm, I think it's safe to skip 3, but I might be wrong so I'll skip 2 instead." They still end up skipping 2 different numbers. Nobody has enough information to conclude that they're the odd man out.

Yep, makes sense.

I'm not quite sure I get there from your posted answer, so if you want to tweak it based on the final end-point, feel free, but regardless I'm going to accept it and thanks again for your patience.

Yeah. Those early days may not tell anyone anything they already didn't know, but /at some point/ it will reach a meaningful level where it's erasing uncertainty.

Even if they could rule out 95 with 100% certainty, they wouldn't be able to rule out 98 or 99... and thus (per Jon's assertion) they wouldn't know where to skip to.

But I think that's the wrong focus. The issue is not whether anyone gets down to the 95 level... it's really about "Can anyone be 100% sure that everyone else can rule out any given number"

Yeah it still seems weird that the recursive consideration model ends up with somebody saying: "But what if B can only see 95 blue people" when he knows for a fact that B must be able to see at least 98.

A can rule out Scenario 1 based on what he sees, but I'm not sure that he can be 100% certain that B can too.

Yeah.

Yeah I think you're right.

Wait...

In Scenario 1 there's only 1 blue person total. The presence of 2 blue people from anyone's perspective completely invalidates that scenario.

That is true, but they know that they see /at least 1/ and are therefore not in scenario 1.

Let's pretend that /armed with the knowledge they're not in scenario 1/, they all decide to skip day 1 and jump right into day 2. The outcome would appear to be the same. So what value came from day 1?

Scenario 3 [bbb]. Everyone can see 2 other blues. They all know they're not in scenario 1. They don't know if it's 2 or 3. They wait 1 day for.... umm.... why? What new knowledge comes from day 1 that they ALL don't already know? Then they all wait again on day 2 ... this time it's useful because if they were in scenario 2, they'd expect the other two guys to leave. When nobody leaves, they can all conclude: "Scenario 3!"

Scenario 2 [bbn]. A and B both each see 1 blue. They don't know if they're in scenario 1 or 2, but they can rule out scenario 3. C sees 2 blues. He doesn't know if he's in scenario 2 or 3, but he can rule out scenario 1. Day 1 passes and nobody leaves. A & B both realize: "Doh! Must be scenario 2" and they leave on Day 2.

New nomenclature: [bn] means blue, not-blue. There are 3 islanders total. That gives us three possible scenarios: [bnn], [bbn], [bbb]. Let's start with scenario 1. A looks around, sees no blues, realizes: "I'm the only blue", leaves on day 1.

Here's the way I did the scenario that made me ask this question... maybe that will help.

Hypothetically IF everyone could come to that conclusion, the only thing it would accomplish is reaching the end-state faster.

I definitely get that D needs to put himself in each person's shoes. I don't quite get why he then needs to recursively imagine them putting themselves in everyone's shoes :)

Kinda. I'm still not entirely sure why we need to go 3 levels deep in folks imagining what everyone else is imagining when they have so much information in front of their eyes.

You've completely lost me - I don't get how the notation tells us who's imaginging who :)

OK I think I get the notation.

So B[?B?B] means that as far as D knows, B sees C (blue) and A (blue) but knows nothing about himself or D?

C as imagined by D ... where do B and A come in?

By all means.

I'm not convinced the presence of brown-eyed folks is completely irrelevant but we can run with that idea for now.

In the 2-person example, it doesn't matter if it's A(blue) B(brown), or A(brown) B(blue) -- the net result is the same.

I agree. It's only helpful to assign letters to know who we're talking about :)

Now Jon may be right about the question of "but how do you know /which/ step to skip to." But intuitively it seems like there ought to be /some/ steps you can skip.

It's only helpful if everyone can grok that piece of information and (based on an agreed-upon algorithm) jump right to step 2.

I know they can see me - although that's not useful information because I don't know my own color. But it seems like everyone ought to be able to rule out the "there's only 1 blue person on the island" scenario.