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Who is not rich? (Posted on 2005-06-22) Difficulty: 2 of 5
Anne, Bernice and Claudia are three remarkable women, each having some remarkable characteristics.

(1) Just two are intelligent, just two are beautiful, just two are artistic and just two are rich.
(2) No one has more than three characteristics.
(3) Of Anne it is true that : if she is intelligent, she is rich.
(4) Of Bernice and Claudia it is true that: if she is beautiful, she is artistic.
(5) Of Anne and Claudia it is true that: if she is rich, she is artistic.

Who is not rich? (Show your reasoning)

See The Solution Submitted by pcbouhid    
Rating: 4.1667 (6 votes)

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Solution solution | Comment 3 of 24 |

If Anne were intelligent, she'd be rich and artistic as well (3 and 5) and not beautiful as this would account for three characteristics already. So Bernice and Claudia would be the beautiful two, which would also make them artistic (4), so Anne could not also be artistic as well--a contradiction.  So Anne is not intelligent, leaving Bernice and Claudia as the intelligent two.

If Bernice and Claudia were both beautiful, they'd both be artistic (again by 4) and together with their already established intelligence, could not also be rich. So not both Bernice and Claudia are beautiful, and thus Anne is beautiful.

By the rules, two of the ladies must have three characteristics each, and one of them two, as no one can have just one characteristic, since this would leave 7 to be shared between the other two, which can't be done.

Claudia, besides being intelligent, must have at least one other characteristic.  But both richness and beauty imply artistry (5 and 4), so she's necessarily artistic.

In the same way, we know Anne is not intelligent but is beautiful. So she must be either rich or artistic or both.  If she's rich she's also artistic (5), so she's definitely artistic.

That completes the two artistic spots, so Bernice is not artistic. Therefore she's not beautiful (contrapositive of 4).  Since she's neither artistic nor beautiful, she must be rich, to have at least two characteristics.

This also leaves Claudia as the other beautiful woman, giving her three characteristics already, so she can't be rich.  That leaves Anne as the other rich lady.

So Anne and Bernice are rich, and Claudia is not.

 


  Posted by Charlie on 2005-06-22 13:43:19
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