Final answer:
The coincidence of the Prefect's visit is not a random occurrence but is tied to Dupin's accurate anticipation of both the Prefect's and the Minister's actions, revealing his superior detective skills.
Step-by-step explanation:
The coincidence of the Prefect's visit lies in the fact that it matches Dupin's expectations about the Prefect's inability to locate the missing letter due to his underestimation of the Minister's intellect. The Prefect's conventional methods were predictable and based on the assumption that the Minister, being a poet and supposedly a fool, would not anticipate typical police tacticts. Dupin, on the other hand, recognizes that the Minister is clever and adaptable, realizing that a more nuanced approach was needed to conceal the letter from the Prefect's thorough, yet straightforward, searching methods.
Dupin uses his understanding of the Minister's character as both a mathematician and a poet to adapt his measures accordingly. The Prefect's conventional search methods are ineffective because they underestimate the Minister's ingenuity, illustrated by Dupin's ability to predict and outmaneuver the Prefect's actions. This misjudgment leads to the Prefect's surprise and reveals Dupin's superior deductive skills when he presents the recovered letter. Hence, what seemed like a coincidence of the Prefect's visit is actually a result of Dupin's calculated anticipation of the Minister's and the Prefect's moves.
The simplicity with which the Minister conceals the letter ultimately trumps the Prefect's exhaustive, yet ironically myopic, search—a notion that Dupin finds humorously self-evident, much to the Prefect's bafflement. Dupin's understanding that genius lies in simplicity, which is overlooked by the Prefect, demonstrates the central paradox that the most elusive truth might be hiding in plain sight.