please discuss in you own words a summary of the story the purloined letter?
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Reprising their roles from "The Murders in the Rue Morgue," C. Auguste Dupin and his friend the unnamed narrator appear in a small library room in Paris, silently smoking and, in the case of the narrator, contemplating two of Dupin's previous cases involving the Rue Morgue murders and the death of Marie Rôget. Monsieur G., the Prefect of the Parisian police, enters the apartment to ask Dupin's opinion of a case, although he refuses to do so in the dark because the idea is "beyond his comprehension" and thus an "oddity." He describes the case as simple but puzzling, but ignores Dupin's suggestion that perhaps its simplicity and self-evidence is what confuses the police.
According to G., a letter has been stolen from the royal apartments that the police know the thief will use for blackmail. The letter belongs to a lady who was forced to hastily place it on a table when the person from whom she wished to conceal the secret entered the room. The Minister D., who also entered, saw and interpreted the contents of the letter correctly. He then placed a letter of similar appearance beside it before retrieving the incorrect paper prior to leaving. The lady saw the substitution but was unable to point it out because of the presence of the third person, who noticed nothing. Since then, D. has used his possession of the letter for political blackmail, and because the lady is unable to publicly reclaim the letter, she has asked the police to retrieve it for her.
The narrator notes that the minister must still have the letter, since to relinquish it would be to lose his power of blackmail, but the police have been unable to locate it, despite having thoroughly searched D.'s apartment. D. cannot be keeping the letter on his person, since the police have already searched him twice. Dupin remarks that the minister cannot be much of a fool, although the Prefect disparages the man for being a poet and therefore, in the Prefect's view, unintelligent. The narrator asks the Prefect about the police's method of search, and the Prefect explains how thoroughly they have searched the apartment, particularly since the reward for the retrieval of the letter is so great. The narrator agrees with the Prefect that the letter must not be in the apartment, but Dupin asks G. to search it again before asking for a complete description of the letter.
A month later the Prefect returns, having found nothing on a second search, and mentions that he will offer a reward of fifty thousand francs, since the retrieval of the letter has become increasingly important. Dupin tells the Prefect to write the check; the astonished Prefect does so, takes the letter from Dupin, and rushes away from the apartment. 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.
The narrator asks why Dupin did not simply steal the letter. Dupin answers that D. might have been desperate enough to have his attendants kill Dupin. In addition, he notes that after a year and a half of being subjected to the Minister's blackmail, the lady will now have the upper hand. He predicts that D. will soon embarrass himself and cause his political downfall, but he has no pity for the man because D. is "an unprincipled man of genius" who once did Dupin a wrong, which Dupin good-naturedly promised to return. Dupin admits that he would like to know the man's thoughts when he opens the letter to read a quote from Crebillon's Atrée et Thyeste which translates to "If such a grievous plan is not worthy of Atreus, then it is worthy of Thyestes." Dupin knows that D. will recognize Dupin as having gotten his revenge.