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unreality

Three Philosophers

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this is an ancient riddle, but the answer is all mine (i'm not going to post the answer right away, i want to see how many ppl get it). I also added in the names Zab, Hab and Rab so that my answer makes more sense.

Three philosophers, Zab, Hab, and Rab, are having a quiet afternoon nap. While they're sleeping, their students come and paint each of their faces black.

When they wake up, they see that each of the others have black faces and all begin laughing at each other. But then one of them, Rab, stops laughing, because he knows that his own face is black. How did he know?

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try not to look at the answer before you've tried to solve it yourself:

If Rab had a clean face and Zab thought that himself (Zab) had a clean face, Zab would be laughing at Hab only and Hab, thinking himself had a clean face too, would be laughing at Zab. Neither would be laughing at Rab, because Rab thinks himself has a clean face also. But if Zab thought he had a clean face, and if Rab had a clean face, there would be nothing for Hab to laugh at, except... Zab. So Zab would have known he had a dirty face, and Zab didn't stop laughing. Which means there has to be something else for Hab to laugh at (Rab). Hab would have realized that too. But still both were laughing. Meaning Rab has a dirty face too.

I've tried to explain the answer as best as I could. Sorry if it doesn't make total sense.

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My real guess is something like they were sitting in a circle so that each of them could only see one face, or sommething like that

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From this solution I've created a new problem... now since all three philosophers are in the same dilemma, and Rab was the first to realize what was going on, Rab is the quickest thinker of the three. Let us say Zab is second fastest, and Hab is the slowest. However, all three think they are the fastest (only Rab is correct in this assumption). So, now it is Zab's turn to think, so to speak... what does Zab realize about his own face and why?

(It would help to know the solution to the original problem to solve this one)

Answer: "If I have a clean face, Rab will think that if he has a clean face too, Hab would have nothing to laugh at. So Rab stopped laughing, realizing he had a dirty face. Meaning I have a clean face! Or, if I have a dirty face, Rab would not have stopped laughing, since he has no way of knowing which one of us has the dirty face that Hab is laughing at, because I am a faster thinker than Rab. So I have a clean face!"

Of course if Zab knew that Rab was a faster thinker he couldn't have ruled out the fact that Rab stopped laughing for a different reason. So Zab wrongly assumes that he (himself) has a clean face. Which is wrong. They all have dirty faces.

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Just for fun, this is what Hab thinks next:

"What an idiot Zab is, laughing and thinking he has a clean face when he in fact has a dirty face. I don't want to look like an idiot, too, in case I have a dirty face, so I'm going to stop laughing now. And if I have a clean face, well I'm being nice and not laughing at people with dirty faces."

OR, more logically:

"Rab stopped laughing, realizing he had a dirty face. The only way he could have realized that is if he put himself in the shoes of Zab, thinking if Rab had a clean face, Zab would realize Hab (me) could only be laughing at Zab, thus Zab would have stopped laughing. But Zab is STILL laughing, meaning Rab has a dirty face. But if I, Hab, had a clean face, Zab would know Rab had been laughing only at him, Zab, before Rab stopped laughing, and Zab would have stopped laughing too. Thus I have a dirty face too. And Rab is a quicker thinker than Zab or me. And I am a bit slower than Zab... but I definitely think better than Zab."

So Hab and Rab aren't laughing, instead they are glaring at Zab, and Zab shuts his mouth.

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unreality, while your answer is indeed correct, it is quite confusing. Try this one:

Scenario 1 - Only 1 dirty face. 2 would be laughing and the third, with the dirty face, would have no idea why.

Scenario 2 - Only 1 clean face. The 2 with the dirty faces would know automatically because all 3 are laughing. If they saw one clean, and one dirty, only if their face was dirty would the other one with the dirty face be laughing.

Scenario 3 - All dirty faces. This MUST be the only logical way for there to be more that a moment of all 3 laughing.

The real clue is that there more than a moment of laughter.

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yep.

sorry my answer post was just to show what was going on in the mind of the philosopher because he (and the other two) automatically assumed they each had a clean face, which is why the thought process goes by the way my answer showed. But urs clears it up for anyone that didnt understand my answer, thanks!

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I will put the solution this way:

There may be three situations with different conditions.

(1) Zab's face is black, while Rab's and Hab's faces are clean.

Rab will see one black and one clean face, and he will be be laughing on Zab's black face.

Hab will also see one black and one clean face, and he will also be laughing at Zab's black face.

But Zab will see two clean and laughing faces, so he will not laugh and he will not be knowing why other two were laughing..!

Rab and Hab will be able to conclude that since Zab is not laughing, so their own faces are clean.

(2) Hab's & Zab's faces are black, and Rab's face is clean.

So Rab is able to see two black faces. He will be laughing at the black faces of Hab & Zab, rightly thinking his own face was clean.

Hab will see one black and one clean face, so he will laugh at the black face of Zab, thinking his own face was clean.

Zab will also see one black and one clean face, so he will laugh at the black face of Hab, thinking his own face was clean.

Now quicker Rab starts thinking "Hab & Zab are laughing at each other's black face and my face is clean?"

Next fastest Hab thinks, "Why Zab is laughing while Rab's face is clean. It means my face is also black as of Zab."

So realizing that his face is also black as of Zab, Hab will stop laughing.

Then slowest Zab will think, "Hab was laughing while Rab's face is clean, it means my face is also black as of Hab. Now Hab has stopped laughing because he has realized now that his own face is also black."

Thus Hab and Zab both realize that their faces are black.

(3) Faces of all three--- Rab, Hab, and Zab--- are black. In the beginning, each of them thinks like situation (2) above, that faces of other two are black, and his own face is clean. So each of them laughs at black faces of other two.

But quicker Rab starts thinking, "If my face is clean, and if Hab and Zab are laughing on each other's black faces, then why they are not able to conclude that the faces of both of them are black?" [Note that in the similar situations at Sr No. (2), Hab had been able to conclude that his own face was also black].

So Rab concludes that there could be only one reason, that he too have a black face so he stops laughing at other two.

But next fastest Hab might misunderstand here. He may think, "Rab & Zab were laughing on each other's black faces and they are seeing my face is clean. Being quicker Rab has now concluded that his own face is also black and therefore has stopped laughing. So he continues laughing at black faces of Rab & Zab.

Zab, the slowest, also thinks same way and comes to the wrong conclusion that his own face is clean. So he also contiues laughing at black faces of Rab & Hab.

Now Hab quicker than Zab further anlyzes, "After seeing that Rab has stopped laughing, why did Zab not realize that his own face is also black..?" [see Sr Number (2)].

Then Hab realizes that the conditions are different from as at Sr no. (2) above, and only possibile condition left is that his own face is also black.

Now slowest Zab is thinks "Rab stopped laughing realizing that his own face is also black like the black face of Hab, and now Hab also stopped laughing after realizing the fact that his face is also black. So my face is clean."

So we see that the slowest thinker Zab may not be able to know about his own black face.

PLEASE CORRECT ME IF WRONG.

Edited by bhramarraj

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