作者：Terence Tao
出版社：Oxford University Press
副标题：A Personal Perspective
发行时间：October 5th 2006
来源：下载的 PDF 版本
Goodreads：4.2 (93 Ratings)
豆瓣：9.0（61人评价）

摘录：

Proclus, an ancient Greek philosopher, said:

This therefore, is mathematics: she reminds you of the invisible forms of the soul; she gives life to her own discoveries; she awakens the mind and purifies the intellect; she brings to light our intrinsic ideas; she abolishes oblivion and ignorance which are ours by birth ...

But I just like mathematics because it is fun.
Mathematical problems, or puzzles, are important to real mathematics (like solving real-life problems), just as fables, stories, and anecdotes are important to the young in understanding real life. Mathematical problems are ‘sanitized’ mathematics, where an elegant solution has already been found (by someone else, of course), the question is stripped of all superfluousness and posed in an interesting and (hopefully) thought-provoking way. If mathematics is likened to prospecting for gold, solving a good mathematical problem is akin to a ‘hide-and-seek’ course in gold-prospecting: you are given a nugget to find, and you know what it looks like, that it is out there somewhere, that it is not too hard to reach, that it is unearthing within your capabilities, and you have conveniently been given the right equipment (i.e. data) to get it. It may be hidden in a cunning place, but it will require ingenuity rather than digging to reach it.
In this book I shall solve selected problems from various levels and branches of mathematics. Starred problems () indicate an additional level of difficulty, either because some higher mathematics or some clever thinking are required; double-starred questions (*) are similar, but to a greater degree. Some problems have additional exercises at the end that can be solved in a similar manner or involve a similar piece of mathematics. While solving these problems, I will try to demonstrate some tricks of the trade when problem-solving. Two of the main weapons—experience and knowledge—are not easy to put into a book: they have to be acquired over time. But there are many simpler tricks that take less time to learn. There are ways of looking at a problem that make it easier to find a feasible attack plan. There are systematic ways of reducing a problem into successively simpler sub-problems. But, on the other hand, solving the problem is not everything. To return to the gold nugget analogy, strip-mining the neighborhood with bulldozers is clumsier than doing a careful survey, a bit of geology, and a small amount of digging. A solution should be relatively short, understandable, and hopefully have a touch of elegance. It should also be fun to discover. Transforming a nice, short little geometry question into a ravening monster of an equation by textbook coordinate geometry does not have the same taste of victory as a two-line vector solution.

This book was written 15 years ago; literally half a lifetime ago, for me. In the intervening years, I have left home, moved to a different country, gone to graduate school, taught classes, written research papers, advised graduate students, married my wife, and had a son. Clearly, my perspective on life and on mathematics is different now than it was when I was 15. I have not been involved in problem-solving competitions for a very long time now, and if I were to write a book now on the subject it would be very different from the one you are reading here.
Mathematics is a multifaceted subject, and our experience and appreciation of it changes with time and experience. As a primary school student, I was drawn to mathematics by the abstract beauty of formal manipulation, and the remarkable ability to repeatedly use simple rules to achieve non-trivial answers. As a high-school student, competing in mathematics competitions, I enjoyed mathematics as a sport, taking cleverly designed mathematical puzzle problems (such as those in this book) and searching for the right ‘trick’ that would unlock each one. As an undergraduate, I was awed by my first glimpses of the rich, deep, and fascinating theories and structures which lie at the core of modern mathematics today. As a graduate student, I learnt the pride of having one’s own research project, and the unique satisfaction that comes from creating an original argument that resolved a previously open question. Upon starting my career as a professional research mathematician, I began to see the intuition and motivation that lay behind the theories and problems of modern mathematics, and was delighted when realizing how even very complex and deep results are often at heart be guided by very simple, even common-sensical, principles. The ‘Aha!’ experience of grasping one of these principles, and suddenly seeing how it illuminates and informs a large body of mathematics, is a truly remarkable one. And there are yet more aspects of mathematics to discover; it is only recently for me that I have grasped enough fields of mathematics to begin to get a sense of the endeavor of modern mathematics as a unified subject, and how it connects to the sciences and other disciplines.

Understand the problem. What kind of problem is it? There are three main types of problems:

• ‘Show that ...’ or ‘Evaluate ...’ questions, in which a certain statement has to be proved true, or a certain expression has to be worked out;
• ‘Find a ...’ or ‘Find all ...’ questions, which requires one to find something (or everything) that satisfies certain requirements;
• ‘Is there a ...’ questions, which either require you to prove a statement or provide a counterexample (and thus is one of the previous two types of problem).

The type of problem is important because it determines the basic method of approach. ‘Show that ...’ or ‘Evaluate ...’ problems start with given data and the objective is to deduce some statement or find the value of an expression; this type of problem is generally easier than the other two types because there is a clearly visible objective, one that can be deliberately approached. ‘Find a ...’ questions are more hit-and-miss; generally one has to guess one answer that nearly works, and then tweak it a bit to make it more correct; or alternatively one can alter the requirements that the object-to-find must satisfy, so that they are easier to satisfy. ‘Is there a ...’ problems are typically the hardest, because one must first make a decision on whether an object exists or not, and provide a proof on one hand, or a counter-example on the other.

Write down what you know in the notation selected; draw a diagram. Putting everything down on paper helps in three ways:

• (a) you have an easy reference later on;
• (b) the paper is a good thing to stare at when you are stuck;
• (c) the physical act of writing down of what you know can trigger new inspirations and connections.

Other problems, though, are not as deep. Here are some simple examples, all involving a natural number n:

• (a) n always has the same last digit as its fifth power n5.
• (b) n is a multiple of 9 if and only if the sum of its digits is a multiple of 9.
• (c) (Wilson’s theorem) (n − 1)! + 1 is a multiple of n if and only if n is a prime number.
• (d) If k is a positive odd number, then 1k+2k+···+nk is divisible by n+1.
• (e) There are exactly four numbers that are n digits long (allowing for padding by zeroes) and which are exactly the same last digits as their square. For instance, the four three-digit numbers with this property are 000, 001, 625, and 876.

Basic number theory is a pleasant backwater of mathematics. But the applications that stem from the basic concepts of integers and divisibility are amazingly diverse and powerful. The concept of divisibility leads naturally to that of primes, which moves into the detailed nature of factorization and then to one of the jewels of mathematics in the last part of the previous century: the prime number theorem, which can predict the number of primes less than a given number to a good degree of accuracy. Meanwhile, the concept of integer operations lends itself to modular arithmetic, which can be generalized from a subset of the integers to the algebra of finite groups, rings, and fields, and leads to algebraic number theory, when the concept of ‘number’ is expanded into irrational surds, elements of splitting fields, and complex numbers. Number theory is a fundamental cornerstone which supports a sizeable chunk of mathematics. And, of course, it is fun too.

The notation ‘x = y (mod n)’, which we read as ‘x equals y modulo n’, means that x and y differ by a multiple of n, thus for instance 15 = 65 (mod 10). The notation ‘(mod n)’ signifies that we are working in a modular arithmetic where the modulus n has been identified with 0; thus for instance modular arithmetic (mod 10) is the arithmetic in which 10 = 0. Thus, for instance, we have 65 = 15 + 10 + 10 + 10 + 10 + 10 = 15 + 0 + 0 + 0 + 0 + 0 = 15 (mod 10). Modular arithmetic also differs from standard arithmetic in that inequalities do not exist, and that all numbers are integers. For example, 7/2 �= 3.5 (mod 5), but rather 7/2 = 12/2 = 6 (mod 5) because 7 = 12 (mod 5). It may seem strange to divide in this round-about way, but in fact one can find that there is no real contradiction, although some divisions are illegal, just as division-by-zero is illegal within the traditional field of real numbers. As a general rule, division is OK if the denominator is coprime with the modulus n.