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)

Allright, here goes:

Each woman can have at most 3 characteristics (2), and there are a total of 8 characteristics(1), so the least chcracteristics one woman can have is the case where the other two have 3.  This leaves the third woman with 2.  Thus, the fewest characteristics one woman can have is 2.

Suppose Anne is not rich.  Then she must also not be intelligent (3).  Thus, Anne must be beautiful and artistic.  Since only one of the three women doesn't have each characteristic (1), Bernice and Claudia must both be intelligent and rich.  Each woman has at most 3 characteristics (2), and since anne has 2, the other two must have 3 each, so of Bernice and Claudia, one isn't artistic, and the other isn't beautiful.  This means the one that isn't artistic must be beautiful, and this violates (4).  Thus, Anne must be Rich.

Suppose Bernice isn't rich.  This means the other two must be rich (1), and also artistic (5).  Thus, Bernice isn't artistic (1), and also not beautiful (4).  This leaves Bernice with at most 1 characteristic, and we know she must have at least 2.  Thus, Bernice must be rich.

Since the other two must be rich, Claudia must be the one that isn't rich.

All that remains to show is that this is indeed possible under the given constraints.  We know Anne and Bernice are rich, which means Anne is also artistic (3).  This means one of the other two is not artistic (1), and therefore also not beautiful (4).  Since Claudia isn't rich and must have at least two characteristics, she can't also not be beautiful and artistic.  This means Bernice must not be beautiful or artistic.  Thus Anne must be beautiful, artistic, and rich (1); and she therefore can't be intelligent (2).  This accounts for all characteristics and satisfies all constraints. QED

 

Posted by josh on 2005-06-23 23:16:04