Cant spot flaw in nested logic in puzzle

[tonyjeffs]

This is not a puzzle question exactly. It's a question about logic that the puzzle illustrates.

There are four dragons. It is known that dragons can have brown eyes but our four all have green eyes. They can see each others eyes, but don't know their own eye colour. They will turn into sparrows if they deduce that they have green eyes. They are safe so long as they don't know their eye colour, or if their eyes are brown. Their names are Angus, Ben, Carl and Dave.
Every dragon knows that every other dragon can plainly see at least two green eyed dragons
Every dragon must therefore know that every other dragon knows that there can not be more than two brown eyed dragons, and one of them must be himself.



Angus thinks "If I have brown eyes, which is possible, I'm safe, but if that is the case, Ben will be thinking "I can see Angus has brown eyes. If I also have brown eyes, that makes two of us, but that means Carl will think "I can see two dragons with brown eyes, Angus and Ben. If I have brown eyes too which is possible, that means Dave can see three dragons with brown eyes." " "

That nested "if" seems perfectly logical in its progression. Each individual step seem (to me) valid. Why does the third level of logically nested speculation give an impossible result. Clearly something is wrong with my (alleged) logic, but I can't get my head around exactly what is wrong. Can anyone grasp and describe the error?

From Angus's perspective, Ben, Carl and Dave have identical status and are therefore interchangeable.
From Angus's perspective of Ben's perspective, Carl and Dave are interchangeable.

Thanks
Tony

 

[Quatl]

First of all you state in the beginning that all the dragons have green eyes. If that is true and each dragon can see "at least two" other dragons, then they each know that a maximum of two dragons may have brown eyes (if they see three others then only they are in question.) Anyhow the color of each dragon's eyes has nothing to do with the color of the other beasties eyes anyway. It's not really a puzzle. There is no train of logic that each dragon can employ that will let them decide their own eye color.

One flaw in your logic is the assumption you give the dragon that dragons are psychic. The result of one dragon's perceptions can't influence that of the other. ( at least not without some communication)

 

[DaveC426913]

There seems to be a rule missing. Who says how many dragons MUST be brown or green?

This statement:

"Every dragon must therefore know that every other dragon knows that there can not be more than two brown eyed dragons, and one of them must be himself."

...seems to be deduced, rather than given. I don't follow how you deduced it from what's previously given.

 

[Quatl]

It's in there:

It is known that dragons can have brown eyes but our four all have green eyes.

Still this statement does not rule out other colors as possibilities like orange. However the rest of it does imply that brown and green are the only two options.

 

[tonyjeffs]

Each dragon can see all three other dragons. They can all see each other. They all have green eyes. None of the dragons know this since they don't know their own eye colour. Each dragon can therefore see 3 pairs of green eyes. There are no possible colours besides green and brown.

Consider Dragon A. He doesn't know whether his own eyes are green or brown, so he doesn't know how many green eyed dragons the others can see. But he knows it is at least two.

I hope that's clearer.

 

[Quatl]

Each dragon can see all three other dragons. They can all see each other. They all have green eyes. None of the dragons know this since they don't know their own eye colour. Each dragon can therefore see 3 pairs of green eyes. There are no possible colours besides green and brown.

Consider Dragon A. He doesn't know whether his own eyes are green or brown, so he doesn't know how many green eyed dragons the others can see. But he knows it is at least two.

I hope that's clearer.

Much clearer :)

Angus thinks "If I have brown eyes, which is possible, I'm safe"

True

if that is the case, Ben will be thinking "I can see Angus has brown eyes.

True, but note that Angus knows that Ben's eyes are green.

If I also have brown eyes, that makes two of us,

True, remember though that this is a proposition about what Angus assumes Ben may think. Not a proposition about reality. Angus knows that if Carl holds this thought, Carl's thinking is wrong.

but that means Carl will think "I can see two dragons with brown eyes, Angus and Ben.

False, Angus knows that Ben's eyes are green. So He also knows that Carl sees Ben's Green eyes.

If I have brown eyes too which is possible, that means Dave can see three dragons with brown eyes." " "

Also false for the same reason. what Dave can see is not dependent on the other dragon's perceptions at all.

That's what I meant about the psychic comment.

 

[tonyjeffs]

Yes you got the idea :-)
And I understand what you're saying, but although it obviously is invalid, I still don't really see why "perception of perception of perception" becomes invalid.


"Perception is valid." (Angus thinks...)
"Perception of perception" is valid (Angus thinks that Ben thinks...)
"Perception of perception of perception" invalid.
(Angus thinks that ben thinks that Carl thinks...)

It just seems like it ought to provide a valid consistant result.

Maybe your last two sentences explain it. I'll think about this some more.

 

[Quatl]

(Angus thinks that ben thinks that Carl thinks...) would be fine but that's not actually what you're doing in the last two steps.

Here you do something more like "compositing" of perceptions, and at the same time ignore part of the data the original dragon knows should be shared by one it is speculating about.

It may help to keep in mind that you could start with any dragon.

 

[Rogerio]

There are four dragons. It is known that dragons can have brown eyes but our four all have green eyes. They can see each others eyes, but don't know their own eye colour. They will turn into sparrows if they deduce that they have green eyes. They are safe so long as they don't know their eye colour, or if their eyes are brown. Their names are Angus, Ben, Carl and Dave.
Every dragon knows that every other dragon can plainly see at least two green eyed dragons.

If they are equally clever, all of them will turn into sparrows.
If they are different, them the "slowest" will remain dragon.

 

[AsianSensationK]

As you transition from what Angus thinks to what Ben may think, you stay consistent with your assumptions. Angus doesn't know if he has brown eyes, and he knows that Ben doesn't know if Ben has brown eyes, so that leaves open the possibility of Ben believing that the two of them have brown eyes.

As you transition to what Angus believes that Ben believes of what Carl believes you run into trouble because you somehow believe your first order assumptions still apply. Does Angus have any reason to believe that Ben will believe the other two dragons will believe at most two dragons with Brown eyes exist? Not really. Since Carl could see two brown eyed dragons from Angus' perspective of Ben's perspective and he knows Carl doesn't know his own eye color he's got no reason to discount the proposition Ben could believe that Carl believes there exists 3 brown-eyed dragons.

But you're not actually concluding Dave can see 3 dragons with brown eyes. You're concluding Angus believes that Ben believes that Carl believes that at most 3 dragons with brown eyes can exist. That's a tricky point.

 

[Rogerio]

Angus thinks "If I have brown eyes, which is possible, I'm safe, but if that is the case, Ben will be thinking "I can see Angus has brown eyes. If I also have brown eyes, that makes two of us, but that means Carl will think "I can see two dragons with brown eyes, Angus and Ben. If I have brown eyes too which is possible, that means Dave can see three dragons with brown eyes." " "

That nested "if" seems perfectly logical in its progression. Each individual step seem (to me) valid. Why does the third level of logically nested speculation give an impossible result. Clearly something is wrong with my (alleged) logic, but I can't get my head around exactly what is wrong. Can anyone grasp and describe the error?

There is nothing wrong with your logic. But there is a substantial difference between "is possible" and "is true".

What about saying:

Angus thinks "If I have brown eyes, Ben will be thinking "I can see Angus has brown eyes. If I also have brown eyes, that makes two of us, Carl will think "I can see two dragons with brown eyes, Angus and Ben. If I have brown eyes too, that means Dave can see three dragons with brown eyes. Since this is impossible, then I can't have brown eyes, and I will turn in a sparrow." " "

This is not the whole storie, but I hope it illustrates the point.

 

[Rogerio]

Angus thinks "If I have brown eyes, which is possible, I'm safe, but if that is the case, Ben will be thinking...

Well, I'm going to try to show you how you could rewrite your sentences in a different way...

First modification:
Angus thinks "If I have brown eyes, which maybe is possible, I'm safe, but if that is the case, Ben will be thinking...

Second: You don't need "if that is the case,". The first "If" means that. So:
Angus thinks "If I have brown eyes, which maybe is possible, I'm safe, but Ben will be thinking...

Third: as you started the sentence using an "If", it is superfluous to say "which maybe is possible". So:
Angus thinks "If I have brown eyes then I'm safe and Ben will be thinking...

The last sounds better.
Hint: try to use "IF" , "THEN", "AND" , "OR" and "NOT" in your sentences.