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Three proper ladies are traveling on a train. Each turns red within one second if they become aware of dirt on their face. They are too proper to tell the other if they have dirt, and there are no mirrors. The conductor tells them "At least one of you has dirt on your face". Within 3 seconds, all three ladies turn red. How many have dirt on their face? What is the reasoning of each of the ladies for believing they have dirt?

Assumptions: The conductor is telling the truth...the solution is either 1, 2 or 3 ladies have dirt on their face. (The answer can't be zero).

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marked as duplicate by Milo Brandt, Ross Millikan, Empty, user21820, Newb Oct 11 '15 at 6:30

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    $\begingroup$ You forgot the condition that all P.L. are P.L. (All Proper Ladies are Perfect Logicians.) ;-) $\endgroup$ – Brian Tung Oct 8 '15 at 22:16
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    $\begingroup$ I observe that this puzzle completely unnecessarily specifies genders: it works unchanged when talking about "proper people". In my opinion, the gendered versions of such puzzles/anecdotes tend to reinforce stereotypes, so I prefer the ungendered versions. $\endgroup$ – Greg Martin Oct 8 '15 at 22:26
  • $\begingroup$ @GregMartin: Now now.. don't make the common fallacies of improper generalization or false dichotomy or converse error.. Just because there are proper ladies doesn't mean all ladies are proper, nor that there aren't proper gentlemen, nor that everyone who is proper is a lady... =P $\endgroup$ – user21820 Oct 11 '15 at 6:24
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    $\begingroup$ You are applying mathematical rules in an inappropriate setting (not to mention arguing against statements I didn't make). In the actual world, stereotypes are perpetuated by non-universal statements. $\endgroup$ – Greg Martin Oct 11 '15 at 7:47
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This is a variant of the blue eyes puzzle, where "100 people on an island" has been replaced by "three ladies on a train", "blue eyes" has been replaced by "dirt on their face" and "leaving the island" by "turning red". It can be solved the exact same way.

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Assuming all P.L. are P.L., then all three must have dirt on their face.

Each reasons as follows: Let me construct a First Lemma. My First Lemma is that if only one Proper Lady has dirt on her face, she immediately turns red. The reason is simple: She must see neither of the other Proper Ladies has dirt on their faces, so since the Conductor (being a Proper Lady also) is not lying, she herself must have dirt on her face.

(Each Proper Lady actually constructs her First Lemma within the first second.)

Let me now construct a Second Lemma. My Second Lemma is that if two Proper Ladies both have dirt on their faces, they rapidly turn red. The reason is almost as simple: Each of the Proper Ladies with dirt on her face sees one Proper Lady with dirt on her face, and one without. Each of them would realize that if she herself did not have dirt on her face, the Proper Lady with dirt on her face would immediately turn red (by my First Lemma). Since that does not immediately happen, each of the Proper Ladies with dirt on their face must turn red rapidly.

(Each Proper Lady constructs her Second Lemma within the second, err, second.)

Now, I see two Proper Ladies with dirt on their faces. By my Second Lemma, if I did not have dirt on my face, both of the other Proper Ladies should have turned red rapidly. But they did not.

Oh $%&! I have dirt on my face!

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