Like Dupin, Minister D, who steals the Queen's compromising letter, is a daring and "ingenious" person who is able to put himself in other people's shoes, figure out how they think, and therefore outsmart them. Like Dupin, he is a "poet," one who can think creatively and outside of conventional boundaries through concentrating on an understanding of human psychology.
The Prefect, unlike either Dupin or Minister D, is a "mathematician" who will always apply the same precise and thorough methodology to finding a stolen letter. He would always assume that a stolen letter was carefully hidden from view and therefore would search for it hidden places. Because the letter is so important, he applies his method with extra minuteness, looking everywhere hidden for the letter.
However, as Dupin would have done, Minister D uses his knowledge of human behavior to outwit the Prefect. Anticipating how the Prefect would conduct his search, the Minister left the letter out in plain sight, after replacing its envelope with one very dissimilar to the envelope that was described to the Prefect and that he was looking for.
Dupin realizes that Minister D is very intelligent, just as he himself is, and so manages to think like him by asking himself how a highly intelligent person would hide a letter. He even mimics the way the minister stole the letter from the Queen when he takes the purloined letter from the minister:
It was the minister who gave me the idea of the facsimile, when he stole the original letter. To get the letter back I just did the same thing he did when he stole it. You could even say that between our letters there is a perfect, ahem ... correspondence.