Adding this to our box of facts:
So, given number 3 in the box, do you agree that:
If you have a yellow card, then Bob thinks it’s possible that Carl has the only green card.
I agree. Bob will think that Carl might be the only one with a green card if I have a yellow.
Adding it to the box:
Okay. Here you are sitting in the room with Bob and Carl. You know all the stuff listed above. You know something about what Bob would be thinking if you had a yellow card. Let’s continue to explore that. Let’s take up Bob’s point of view, while continuing to imagine what might be going on should we happen to have a yellow card.
Per 4, (again, IF you have a yellow card,) Bob thinks it’s possible that Carl has the only green card. But of course, Bob knows Carl doesn’t know the color of Carl’s own card. So, from Bob’s point of view, would you agree, Carl might think it possible that no one has a green card?
(If you don’t agree, we’ll take some partial steps back as necessary. But I want to see if you already agree with this.)
Here’s where it gets tricky for me…
but my final solution is yes if I as Bob know that Emoticorpse has a yellow card then Carl thinks it’s possible he has the only green card.
I left the answer to the question yes but am curious because as Bob I know that the possibility really may still be yes or no depending on what Carl actually sees that my card is and I have no idea I would know that he may very well see my card as green therefore eliminating that possibility and at the same time know I don’t have a green making it true. So truthfully If I was Bob I would say of course it’s possible so it’d be a yes but know that maybe no also.
Actually, infinite layers is always needed for common knowledge, regardless of population.
Imagine two master villains trying to outsmart each other with their schemes.
“I know you have the map”
“I know you know I have the map”
“I know you know I know you have the map”
“I know you know I know you know I have the map”…
and so on… but as long as there is any doubt on any level of knowledge about who has the map, it is not common knowledge.
Now back to the islanders. Before the guru came, there was no common knowledge whatsoever about the lower bound number of blue-eyes. Period. Think on that for a moment.
For example, everyone knows that there is at least 99 blue-eyes, but not everyone knows everyone knows that, so that knowledge is only present at the first layer. By extension, the fact that there is at least 98 blue-eyes exists as knowledge at the second level for everyone, and the fact that there is at least 1 blue-eyes exists as knowledge at the 99th level for everyone on the island.
In fact, there is no common knowledge about the upper bound of blue-eyes either, however there is a first level knowledge that there is at most 100 for all blue-eyes on the island. It turns out this is sufficient.
However, is there any common knowledge at all among the islanders? Yes there must be for the solution to make sense. Usually this clause is given as part of the problem but it is usually worded a bit differently to be tricksy, but understand that effectively this must be true:
“It is COMMON KNOWLEDGE that every islander is a perfect logician and will always obey the rules of the island”
Can you understand why, if there is doubt at any level about this fact, the inductive reasoning we use to draw our conclusion falls apart?
(If we want to be nitpicky, then given this premise it is -also- common knowledge that there is at least 0 blue-eyes. This is because when it is common knowledge everyone is a perfect logician, then any truism because common knowledge: Tautology (logic) - Wikipedia)
In comes the guru, who comes with a few special definitions about his own status:
Here is a simple axiom that you can try to prove on your own:
Everything the guru says becomes -common knowledge- IF AND ONLY IF it is -common knowledge- that the guru speaks only truth and it is -common knowledge- that everyone can hear the guru speak
So when he tells every there is at least one blue-eyes, this fact becomes common knowledge. This is a FUNDAMENTAL change that now allows a series of events to take place, which makes it become common knowledge there are at least two blue-eyes on day two, and common knowledge there are at least three blue-eyes on day three.
On day 99, it becomes COMMON KNOWLEDGE that there are at least 99 blue-eyes. Note that even now there is no change in first-level knowledge, since everyone knew there are at least 99 blue-eyes at first level before the guru ever appeared. However, all of that changes on day 100.
On day 100, it becomes COMMON KNOWLEDGE that there are at least 100 blue-eyes. But suddenly, this concept of common knowledge that we have been so obsessed about becomes irrelevant. We needed common knowledge to get to this point, but now it is enough that there is first-level knowledge that there are at least 100.
Because now for every blue-eyes there is first-level knowledge that there are at least 100 and at most 100. That means for every blue-eyes there is first level knowledge that there is exactly 100 of them, which means they now know their own eye-color.
So why does it take the brown-eyes a day later to catch on? Because for them there is first level knowledge that the upper bound is 101 blue-eyes, so even by day 100 they still couldn’t figure out exactly what eye color they were. By day 101 it becomes clear (since they are the only ones left).
This common knowledge is killing me. I understand how they all get their minimum count and their possible maximum but don’t see how they get the EXACT count. Supposedly they get it by seeing how many others already got it? but if no one got it they how any anyone else get it? I’m still stuck. Still striving to understand hope I’m trying not to bug anyone just hoping someone finds the right way to explain it to me.
I’m still stuck on how everyone gets the color of their eyes from the count. I see logic leading them as far as a POSSIBLE count not a definite one though. Because even from that Possibility they may or may not be blue eyes. I don’t see the conclusive evidence.
The brown eyes leave the island too a day later? I was wondering if they could use the same logic to escape. I guess the Guru is stuck.
No. The brown eyes cannot leave. The problem stated that there could be any number of different eye colours. Only the blue eyed islanders got the critical information.
If the Guru will make the appropriate announcement, they can leave in just the same way the blue-eyed islanders did. Otherwise, they’re stuck.
You do see how this works when the total with blue eyes is 1, 2 and 3 - right?
If so, just keep stepping up to larger numbers.
The blue eyed islanders recieved a piece of information but somehow simultaneously interpret a formula from that. A formula with all variables known to all. Even the one they have yet to find out?
Frylock was working with me one on one and I actually go the point where I may be confused but he went offline. Hopefully he gets an opportunity to tell me how I’m wrong or overthinking.
You’re definitely on the way to getting it, it seems to me. Here’s our box of facts so far:
Now, something new happens. A loudspeaker comes on. It has come on before, in past experiments, and everything it has said has always been true. You believe what it says, and you know Bob and Carl always believe what it says as well.
The loudspeaker says, “At least one person in this room has a green card.”
Strange, you think at first. Doesn’t everyone already know that?
Still, let’s follow some reasoning to see whether we can nevertheless get something new out of this event. Let’s put the event into the box of facts:
(By the way, a note concerning the original puzzle. Here’s a really simple statement of the new information the Guru brings to the table. It is this:
“The guru has stated that he sees at least one person with blue eyes.”
That is something that no one knew prior to the guru’s speaking. And it turns out that this bit of information is precisely what ends up making a difference in the reasoning involved in the solution to the puzzle.)
Notice that the box of facts is divided into two sections now. This is because we’re going to discover that after the loudspeaker speaks, some of the facts that were established before the loudspeaker spoke are no longer true.
Returning to the situation in which you’re asking, “What if I have a yellow card?” and you’re seeing things from Bob’s point of view under that hypothetical, ask yourself now, “From Bob’s point of view, can* Carl think it possible that no one has a green card?” Remember, things may have changed now that the loudspeaker has spoken.
*Though I haven’t said anything about “perfect logicians” in this scenario, let’s go ahead and affirm that it’s true in this scenario as well as in the original puzzle. So when I ask “Can carl think such and such” I’m asking whether it’s possible for a perfect logician to think it. In other words, could Carl think it without contradiction something he believes to be true?
Now, something new happens. A loudspeaker comes on. It has come on before, in past experiments, and everything it has said has always been true. You believe what it says, and you know Bob and Carl always believe what it says as well.
The loudspeaker says, “At least one person in this room has a green card.”
Returning to the situation in which you’re asking, “What if I have a yellow card?” and you’re seeing things from Bob’s point of view under that hypothetical, ask yourself now, “From Bob’s point of view, can* Carl think it possible that no one has a green card?” Remember, things may have changed now that the loudspeaker has spoken.
[/QUOTE]
No Carl can’t think that it’s possible. Carl knows that there must be at least one green card because the Loudspeaker said there is at least one.
Adding it to the box, then:
Now, still thinking about the situation in which you happen to have a yellow card, and still thinking about things from Bob’s point of view, could Carl think that Carl himself is the only person with a green card?
So for Bob options number 5 is not possible
and come to think about it option 4 is out of the question also because Bob knows that If Carl had the ONLY green card he would see that nobody else had a green card and realize that he was the one the loudspeaker was talking about.
So also now Bob knows that If Carl was the only one with a green card he would’ve known it was him and called it out knowing he was was sure and wouldn’t be executed. If he doesn’t it’s because he isn’t sure he’s the one the loudspeaker was talking about. The only way he wouldn’t be sure is if there is at LEAST one more person with green confusing Carl. Bob knows that If Emoticorpse doesn’t have a green then Emoticorpse can’t be the one confusing Carl. So the only one left that could possibly have a green that is confusing Carl is Bob himself. In that case Bob would declare his card green and not be executed and the only one left there with their card unturned or unknown and no one to base some kind of process of elimination with would be Emoticorpse.
Bob also knows that Emoticorpse is thinking this same exact logical formula because it just makes sense with the information provided he could figure it out just like Bob is doing by himself. So on the reverse side Bob knows that Emoticorpse doesn’t know he has a green but STILL sees Carl confused it’s because Emoticorpse knows that Carl sees someone else with a green and since he CAN see Bob’s card and knows that Bob’s card is not Green that He himself can be the only one confusing Carl so He now knows that his card is green and calls it out surely without being executed. leaving Bob alone if he in fact didn’t have the green. I THINK I GOT IT!!!