Focus on the relationship between the amateur detective and Minister D.
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Dupin explains to the narrator that the police were very skilled but that the case was not suited to the unimaginative. He provides the example of a schoolboy who was particularly skilled at a guessing game in which he was to guess whether his opponent had an odd or even number of marbles and in which he bet one marble per game. The schoolboy won because he was able to emulate his opponent's logic by imitating the other boy's face in order to see how the expression made him think. The police only think about what they believe to be the best course and fail to consider the thoughts of the Minister.
Dupin notes that the Prefect believes that D. is a fool. However, D. is also a mathematician and can thus combine creativity and logic. According to Dupin, while normal mathematicians lack imagination and would have hidden the letter away in exactly the type of place where a policeman would search, the Minister foresaw the probable avenue of investigation and chose an alternate route. Dupin offers the example of a game in which one attempts to guess the point on a globe of which the other is thinking. A novice will choose an obscure name, but a skilled player will choose a very prominent name, knowing that the other person will discard such names as possibilities because they are too obvious. The Prefect does not understand this reasoning, but Dupin places himself into the mind of the Minister and realizes that the Minister would have decided to hide the letter in the most obvious place possible.
After coming to this conclusion about the letter, Dupin visits D.'s apartment while wearing green glasses that conceal the fact that he is looking around the apartment. At length, he discovers several visiting cards and a letter that has been torn and altered in appearance hanging carelessly from a rack on the mantelpiece. D., it appears, placed the letter in full view after turning it inside out, readdressing it, and making it appear useless. Dupin memorizes the appearance of the letter while talking with the Minister and leaves a gold snuff box at the apartment. The next morning, he comes back on the pretense of having forgotten his snuff box, and when D. rushes to his window to observe a disturbance involving gunshots that Dupin previously orchestrated, Dupin substitutes the letter with a fake that he created the night before and soon returns home.