Tolkien's subtle sarcasm in the last line of that quote may give a hint as to the reason:
Quote:
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It did not seem possible to Faramir that any one in Gondor could rival Boromir, heir of Denethor, Captain of the White Tower, and of like mind was Boromir. (emphasis mine)
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If Boromir thought of himself as unrivalled, it is very possible that he did not consider Faramir capable of finding the hidden vale of Imladris and thus the answer to the riddle. But the original question (which I find very interesting!) of this thread remains unanswered - why would the leaders of Gondor have chosen to send away the leader of their army?!
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'Mercy!' cried Gandalf. 'If the giving of information is to be the cure of your inquisitiveness, I shall spend all the rest of my days in answering you. What more do you want to know?' 'The whole history of Middle-earth..
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