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For example, "John was asked to solve a complex challenge."

I find this to read very awkwardly, as if someone told me they solved a banana. Is this a case of verb-noun mismatch?

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up vote 4 down vote accepted

So long as the noun is something solvable, this would be a valid construction. Thus puzzles, Rubik's cubes and equations are all nouns which can be the object of the verb "to solve". So if the challenge was a puzzle, it could be solved. If the challenge was, however, physical in nature, it would be more natural to say that the challenge was completed. The answer to your question is thus "it depends on the nature of the challenge in question - if the challenge can be said to have a solution, it is acceptable to use the verb 'solve'".

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Even if the noun is not something solvable, it would be a valid construction, though nonsensical. One can certainly write that ?"John was asked to solve a complex banana". There is no verb-noun mismatch. An example of verb-noun mismatch is *"John was asked to solve a complex bananas". Incidentally, as "solve" can mean "dissolve" there is actually nothing nonsensical about solving a banana. The problem is more practical: how to coagulate the banana. –  MετάEd Jun 14 '12 at 23:46
I assume that usage is either technical or archaic. Most people would not understand what you meant if you said that you were going to solve a banana. The phrase may be grammatically correct, but our goal on EL+U is as much to explain how to communicate as it is to understand grammar. In context a banana could be solvable (a combination banana?), but the point is that some challenges are solvable in a sense that people would commonly understand, both grammatically and logically. –  Christi Jun 14 '12 at 23:51
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