Not Remarkably Rich

Annette, Bernice, and Claudia are three remarkable women, each having some remarkable characteristics.

1. Just two are remarkably intelligent, just two are remarkably beautiful, just two are remarkably artistic, and just two are remarkably rich.
2. Each has no more than three remarkable characteristics.
3. Of Annette it is true that:
   if she is remarkably intelligent, she is remarkably rich.
4. Of Bernice and Claudia it is true that:
   if she is remarkably beautiful, she is remarkably artistic.
5. Of Annette and Claudia it is true that:
   if she is remarkably rich, she is remarkably artistic.

Who is not remarkably rich?

(Source: Test Your Logic: 50 Puzzles in Deductive Reasoning by George J. Summers)

This looks pretty difficult because, in themselves, implications with false antecedents don't tell us anything about the truth of their consequents. This means, for example, that if Annette is not remarkably intelligent, whether or not she is remarkably rich is still up in the air. Fortunately, there really is enough structure here to determine the truth. One can start with the assumption that Annette is remarkably intelligent. If she is remarkably intelligent then she is also remarkably rich. If she is remarkably rich, then she is also remarkably artistic. Additionally, because two of the women have to be remarkably beautiful, and none of them have more than three remarkable traits, the two remarkably beautiful women must be Bernice and Claudia. This implies in turn that they are both remarkably artistic:

	Intelligent	Beautiful	Artistic	Rich
Annette	✓		✓	✓
Bernice		✓	✓
Claudia		✓	✓

Uh-oh! We already have a problem. Fortunately, the problem stems entirely from the single assumption that Annette is remarkably intelligent; nothing further needed to be added and so this assumption can in no sense be seen as "protected", which is a good thing, because we already have some solid truth about what is the case: Annette is not remarkably intelligent and, because two of the women must be remarkably intelligent, Bernice and Claudia are. All further calculations must then start with these facts:

	Intelligent	Beautiful	Artistic	Rich
Annette	✗
Bernice	✓
Claudia	✓

After I reached this point, I had to pause for a while to figure out my next step. It turned out that it was fruitful to assume that Annette is remarkably rich, which in turn implied that she is remarkably artistic:

	Intelligent	Beautiful	Artistic	Rich
Annette	✗		✓	✓
Bernice	✓
Claudia	✓

This is a good assumption to try because it means that exactly one of Bernice and Claudia can (and must) be remarkably beautiful, but not both, because remarkable beauty in either also implies remarkable artistry for the same woman, and that would leave us with three remarkably artistic women, which is a no-no. Moreover, because only one of Bernice and Claudia can be remarkably beautiful, this assumption also means that Annette has to pick up the slack for the remaining slot, which may as well be added now:

	Intelligent	Beautiful	Artistic	Rich
Annette	✗	✓	✓	✓
Bernice	✓
Claudia	✓

Of Bernice and Claudia, which is remarkably beautiful? What if it's Bernice? If Bernice is remarkably beautiful, then she is remarkably artistic. This confers three remarkable qualities on Bernice and so she can't be remarkably rich. This means Claudia must be remarkably rich and, if Claudia is remarkably rich, then she is also remarkably artistic:

	Intelligent	Beautiful	Artistic	Rich
Annette	✗	✓	✓	✓
Bernice	✓	✓	✓	✗
Claudia	✓	✗	✓	✓

Déjà vu! We have a problem again. As before, all three women are now remarkably artistic, which is not possible. This means that, under the current assumption, Claudia must be remarkably beautiful. Can it work or do we have to go back to the drawing board? If Claudia is remarkably beautiful, then she is also remarkably artistic. That means Claudia now has three remarkable traits and cannot have a fourth of being remarkably rich. Bernice must then be remarkably rich and here is our completed tableau:

	Intelligent	Beautiful	Artistic	Rich
Annette	✗	✓	✓	✓
Bernice	✓	✗	✗	✓
Claudia	✓	✓	✓	✗

This tableau indeed fulfills all the criteria set out in the beginning. Exactly two of the women have each of the four traits and none has more than three. Remembering that only implications with true antecedents need to be checked, it is equally true that, if Claudia is remarkably beautiful, she is remarkably artistic and that, if Annette is remarkably rich, she is remarkably artistic. Which leaves Claudia without remarkable wealth, but I suppose if she really wanted it she could go into consulting or modeling or the like with her other assets.

Final answer: Claudia is not remarkably rich.