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在读 具体数学(英文版第2版)
P2 In fact, we'll see repeatedly in this book that it's advantageous to look at small cases first. The next step in solving the problem is to introduce appropriate notation: name and conquer. Smart mathematicians are not ashamed to think small, because general patterns are easier to perceive when the extreme cases are well understood (even when they are trivial). P3 Our experience with small cases has not only helped us to discover a general formula, it has also provided a convenient way to check that we haven't made a foolish error. Such checks will be especially valuable when we get into more complicated maneuvers in later chapters. The recurrence only gives indirect, local information. With a closed form, we can understand what Tn really is. P4 we go through three stages: 1 Look at small cases. This gives us insight into the problem and helps us in stages 2 and 3. 2 Find and prove a mathematical expression for the quantity of interest. For the Tower of Hanoi, this is the recurrence (1.1) that allows us, given the inclination, to compute Tn for any n. 3 Find and prove a closed form for our mathematical expression. For the Tower of Hanoi, this is the recurrence solution (1.2). Our analysis of the Tower of Hanoi led to the correct answer, but it required an "inductive leap"; we relied on a lucky guess about the answer. One of the main objectives of this book is to explain how a person can solve recurrences without being clairvoyant. 引自 C1 RECURRENT PROBLEMS
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