数学解题法：算法、理解与应用

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"《算法导论：问题求解》是一部深入探讨数学的本质、应用与解决问题方法的经典之作。它强调数学不仅仅是关于数字的加减乘除，或者复杂的算法，而是涵盖了如何做（The How）、何时做（The When）以及为何这样做（The Why）。作者关注的核心是培养学生的理解能力，使他们不仅能够正确执行算法，而且能理解其背后的逻辑，以及何时选择何种策略。乔治·波利亚的问题求解四步骤在这个教程中起着关键作用。这四个步骤包括： 1. 理解问题：首先要明确问题的定义，解析问题中的关键要素，确保对题目有全面的理解。 2. 明确目标：确定解决这个问题所需达成的目标，知道你正在寻找什么样的答案或证明。 3. 计划策略：设计一个通用的方法或算法，考虑可能的解决方案，并评估它们的有效性和可行性。 4. 实施和验证：执行你的策略，检查结果是否符合预期，如果必要，进行调整并重复尝试。书中提供的问题和练习旨在帮助读者在实践中掌握这些步骤。初次接触证明理论的学生可能会面临挑战，因为这不仅涉及到解题技巧，还包括如何清晰地表述和展示推理过程。每一章节都会提供必要的定义和背景知识，以及解决问题的基本框架。学生需要投入时间阅读、尝试不同方法，反复思考，逐步提升自己的问题求解能力。通过这本书的学习，读者不仅能提升技术技能，还能培养批判性思维和逻辑推理能力，这对于未来无论是专业数学研究还是实际工作中的问题解决都至关重要。《算法导论：问题求解》是一本旨在引导读者掌握解决问题核心技巧的实用教程，适合所有希望深入理解数学精髓的读者。"

1. The How, When, and Why of Mathematics

they don’t understand a word you are saying, sometimes saying it

out loud in a coherent manner will allow you to recognize an error

you can’t spot when you are reading. If you are permitted to work

together, switch proofs and ask your partner for criticism of your

proof.

When you are convinced that your argument is correct, it is time

to write up a correct and neat solution to the problem.

Here is an example of the P

olya method at work in mathematics;

we will decipher a message. A cipher is a system that is used to hide

the meaning of a message by replacing the letters of the alphabet by

other letters or symbols.

Exercise 1.1.

The following message is encoded by a shift of the alphabet; that

is, every letter is replaced by another one that has been shifted n

places further down the alphabet. Once we reach the end of the

alphabet, we start over. For instance, if n were 7, we would make the

replacements a

→ h, b → i, ...,s → z, t → a, ....Now the exercise:

What does the message below say?

PDEO AJYKZEJC WHCKNEPDI EO YWHHAZ W YWAOWN

YELDAN. EP EO RANU AWOU PK XNAWG, NECDP?

Let’s use the ideas from P

olya’s list to solve this. If you have

solved problems like this before, it might be a better exercise for you

to try on your own to see how this ﬁts P

olya’s method before you

read on.

1. “Understanding the problem.” Each sequence of letters with no

blank space between the letters represents one word. Each let-

ter is shifted by the same number of places: namely n.Son is the

unknown in this problem and it is what we need to ﬁnd. Once

we know the value of n, we can decipher the whole message. In

addition, once we know the meaning of one letter, we can ﬁnd

the value for n.

2. “Devising a plan.” A cipher text may have weak points. What are

these? How about the short words? Looking at the short words, in

some sense, substitutes an easier problem for the one we have.

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数学解题法：算法、理解与应用

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