What Happens if a Visitor Tells Dragons Not All of Them Have Green Eyes?

[igorronaldo]

https://www.physics.harvard.edu/uploads/files/undergrad/probweek/prob2.pdf

https://www.physics.harvard.edu/uploads/files/undergrad/probweek/sol2.pdf

the solution is clear until this point:

The point is that it
is only after you make your announcement that the chain is extended the final step
to the Nth dragon. The fact that the Nth dragon heard your statement is critical
to the truth of this complete chain.
So, in the end, it turns out to be of great importance how far the chain, “A
knows that B knows that C knows that : : :” goes.
Note that if one of the dragons missed your farewell announcement (which was
“At least one the 100 dragons on this island has green eyes”), then they will all
happily remain dragons throughout the ages.

here I lost it. Does anybody has any ideas? They would be greatly appreciated.

 

[D H]

It is crucial that all of the green-eyed dragons heard the farewell announcement. It is also crucial that each of green-eyed dragon knows that all of the other green-eyed dragons heard the announcement. Since all of the dragons have green eyes, these conditions simplify to all dragons must hear the announcement and each dragon must know all of the dragons heard it.

If only one dragon has green eyes, that one dragon knows his eyes are green immediately upon hearing the announcement. That one dragon knows all other 99 dragons have non-green eyes. There's only one dragon he can't see: Himself. He must be the one with green eyes.

Things get interesting when two or more dragons have green eyes. Suppose two of the dragons have green eyes, with the remaining 98 having blue eyes. Neither of the green-eyed dragons can immediately surmise they are the one with green eyes because each can see 98 pairs of blue eyes and one pair of green eyes. On the next day, both green-eyed dragons can surmise that they have green eyes, but only if both heard the announcement and only if both are aware the other green-eyed dragon heard the announcement. Each green-eyed dragon will deduce that the one green-eyed dragon they do see would have turned into a sparrow on midnight of the day of your departure had that dragon seen 99 blue-eyed dragons. Since that other known green-eyed dragon is still a dragon on the next day, each dragon will deduce that there must be at most 98 non green-eyed dragons, and there they all are.

Suppose that only one of the two green-eyed dragons heard the announcement. The other was off doing whatever dragons do. There's no new knowledge here. That one green-eyed dragon who did hear the announcement already knows there's at least one green-eyed dragon.

Suppose that instead of a farewell announcement to all, you told both green-eyed dragons that there is at least one green-eyed dragon, but you told them this individually and in private. This too breaks the chain of reasoning that causes both green-eyed dragons to turn into sparrows on midnight of the day after your departure.

The same constraints apply if there are three green-eyed dragons, or four, or one hundred. The chain of reasoning is broken if even one of the green-eyed dragons did not hear the announcement, or if even one is not aware that all the other green-eyed dragons heard the announcement.

 

[igorronaldo]

1. The Nth dragon is you, who might have green eyes but can't see it himself

2.Note that if one of the dragons missed your farewell announcement (which was
“At least one the 100 dragons on this island has green eyes”), then they will all
happily remain dragons throughout the ages.

implies that others must be aware of the fact that he missed the announcement as well I think.

 

[D H]

implies that others must be aware of the fact that he missed the announcement as well I think.

Good point. This is a problem about perception. Suppose one of the dragons was present for the farewell but had a cold and couldn't hear the farewell announcement. The remaining 99 dragons can breathe a huge sigh of relief should this dragon let the other dragons know that he couldn't hear what was said.

On the other hand, if this one non-hearing dragon says nothing about his condition, the other dragons will still be of the opinion that every dragon heard the announcement. Those 99 dragons turn will into sparrows on the 100th day, and the last dragon may well reason that he has green eyes on the day after.

 

[arildno]

So that's why they all have disappeared, huh?
If I could, I would kill the bastard who removed magic from our world.

 

[HarrisRap]

First thing to point out is that every dragon knows that all the other dragons have green eyes.
What every dragon does not know is whether or not he/she has green eyes.
So, you can imagine, every dragon lives his/her whole life wondering whether he/she is the one lucky dragon without green eyes.

Let's start with case N=3 (3 dragons instead of 100)
Imagine you are one of those three dragons.

You want to find out if you have non-green eyes.
Here's how to find out:
Assume you do have non-green eyes.
Then watch the other two, and see if they take the logical actions consistent with you having non-green eyes.
If they don't, then you know you have green eyes.

So take the Case A: You (1) have non-green eyes.The other two dragons (2, 3) see your non-green eyes.
And they (e.g. 2) see the other dragon (e.g. 3) has green eyes.
They don't know whether they have green eyes or not, and they want to find out.
So, each of them can do the similar process of taking the case that they (e.g. 2) have non-green eyes, and see if the other dragon (e.g. 3) takes the logical actions consistent with them (e.g. 2) having non-green eyes.

So, take the case A1: One of the other dragons (2) has non-green eyes.
The third dragon (3) sees that dragon (2) has non-green eyes (and that you(1) have non-green eyes, because case A1 is a sub-case of case A).
So, the third dragon (3) now knows that he must be the only one with green eyes (because the man who left the island told them that at least one of them has to have green eyes)***
And so that night he transforms.

But, what actually happens that night: no dragon transforms.
So, that proves to dragon 2 that case A1 is false, therefore he (2) knows that he must have green eyes.
So, the second night that dragon transforms.

Ah, but what actually happens that second night: no dragon transforms.
So, that proves to you (1) that case A is false (cause if case A was true, case A1 would have to be either true or false and a dragon would have transformed on either the first or the second night), therefore you (1) must have green eyes.
So, the third night that dragon (1), you, transforms.

Well, each of them goes through the exact same thought process being "you", so on the third night, all three transform.
So, when N=3, all three dragons transform on the 3rd night. For N=100, all 100 dragons transform on the 100th night.

Once the above is really clear and you can see each step of the logic, then I suggest walk it through with N=4 to really start to see the pattern.

*** this step is the most critical. if the third dragon never heard the man's statement, then he could assume that none of them have green eyes and so possibly he doesn't have green eyes either so no need for him to transform. so, that's why if even one dragon didn't hear the statement, this logic breaks down, and no conclusions can be drawn, and they all stay dragons.

 

[HarrisRap]

And now, if you really want to test your understanding of this puzzle, answer this: What happens if you tell them all that at least TWO of them has green eyes? What about if you tell them at least THREE, FOUR, etc has green eyes?

 

[HarrisRap]

Another way to get at this is to first take a simpler case, where you say "at least 99 of you has green eyes". What happens then?
Then what happens if you say "at least 98 of you has green eyes"?

 

[HarrisRap]

BTW, as we know, if one of the dragons missed your farewell announcement, in the puzzle as stated, everyone stays dragons.
How many dragons would have to miss the announcement if you told them "at least TWO of you has green eyes" for them to all remain dragons?
And if you said "at least THREE"? FOUR?
Again, start with 99, then 98, to make it easier.

 

[HarrisRap]

the last dragon may well reason that he has green eyes on the day after.

I don't see how that dragon would come to that conclusion logically? I would think he'd just be amazed that they all finally realized they had green eyes *somehow*. And the fact that he didn't, would reinforce even more that he probably doesn't have green eyes. Regardless, logic would not tell him anything one way or the other, so he would have to go on living happily as a dragon with all his now-sparrow friends :).

Or... you may be right. Cause this last surviving dragon would think: "The only way the other dragons could know they had green eyes is if that human visitor told them something. Now, they waited 100 days, so that must mean he told them "at least one of you has green eyes." There is no other logical reason they would wait 100 days. And they didn't wait 99 days (per DH's post #2 at the top), so that must mean AAAGH I HAVE GREEN EYES." Sparrow time.

 

[Demitri]

So what do you guys think happens if the visitor tells them at least one of them does NOT have green eyes?