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Solving Mathematical Problems: A Personal Perspective
Authored by a leading name in mathematics, this engaging and clearly presented text leads the reader through the various tactics involved in solving mathematical problems at the Mathematical Olympiad level. Covering number theory, algebra, analysis, Euclidean geometry, and analytic geometry, Solving Mathematical Problems includes numerous exercises and model solutions throughout. Assuming only a basic level of mathematics, the text is ideal for students of 14 years and above in pure mathematics.
128 pages, Paperback
First published January 1, 2006
112 people are currently reading
2067 people want to read
About the author
Terence Tao
48 books217 followersTerence "Terry" Tao FAA FRS (simplified Chinese: 陶哲轩; traditional Chinese: 陶哲軒; pinyin: Táo Zhéxuān) is an Australian-American mathematician who has worked in various areas of mathematics. He currently focuses on harmonic analysis, partial differential equations, algebraic combinatorics, arithmetic combinatorics, geometric combinatorics, compressed sensing and analytic number theory. As of 2015, he holds the James and Carol Collins chair in mathematics at the University of California, Los Angeles. Tao was a co-recipient of the 2006 Fields Medal and the 2014 Breakthrough Prize in Mathematics.
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Displaying 1 - 23 of 23 reviews
December 19, 2014
I was quite disappointed after reading this book. There is not much to learn from it, as it has been written by Tao in his (mathematical) youth, and by that time he didn't have a solid writing style yet.
Very easy to read, probably in one day you can finish it. Around 100 pages, it contains only a few chapters on main olympiad topics. After each solved problem Tao proposes a few related (or not) ones to repeat the technique suggested. There are no answers to the problems, but they are in general fairly easy. Main ideas I got from the book:
1. Strategies in problem solving
A bit like How to solve it, from Polya. Heuristics to approach problems, with main ideas:
Understand the problem
Understand the data
Understand the objective
Select good notation
Write down what you know, draw a diagram.
Modify the problem ( slightly and significantly )
Prove results
Simplify, exploit data and reach tactical goals.
Basically, you should do this using a low risk approach . Do not apply ideas blindly, but rather think ahead if it can attain the goal.
2. Number Theory
Try to relate the problem to things you know, e.g. sum of digits and divisibility by 9.
Guess the answer ( e.g. yes/no). Guess the easy options first, in order to save time.
Tao modifies the problems till one he can solve, following a logical path when taking decisions.
Try small cases.
Use the known facts you wrote down.
3. Examples in algebra and analysis
Always try to use tactics that get you closer to the objective, unless all available direct approaches have been exhausted - In this case go sideways or backwards !
Use induction !
4. Euclidean Geometry
Draw a picture!
reason by absurd
use contradiction
5. Analytic Geometry
Use vectors
Exploit symmetry
Make guesses
Split into cases
Quantify things when possible.
6. Sundry examples
Choose a good notation!
Look for symmetries
Very easy to read, probably in one day you can finish it. Around 100 pages, it contains only a few chapters on main olympiad topics. After each solved problem Tao proposes a few related (or not) ones to repeat the technique suggested. There are no answers to the problems, but they are in general fairly easy. Main ideas I got from the book:
1. Strategies in problem solving
A bit like How to solve it, from Polya. Heuristics to approach problems, with main ideas:
Understand the problem
Understand the data
Understand the objective
Select good notation
Write down what you know, draw a diagram.
Modify the problem ( slightly and significantly )
Prove results
Simplify, exploit data and reach tactical goals.
Basically, you should do this using a low risk approach . Do not apply ideas blindly, but rather think ahead if it can attain the goal.
2. Number Theory
Try to relate the problem to things you know, e.g. sum of digits and divisibility by 9.
Guess the answer ( e.g. yes/no). Guess the easy options first, in order to save time.
Tao modifies the problems till one he can solve, following a logical path when taking decisions.
Try small cases.
Use the known facts you wrote down.
3. Examples in algebra and analysis
Always try to use tactics that get you closer to the objective, unless all available direct approaches have been exhausted - In this case go sideways or backwards !
Use induction !
4. Euclidean Geometry
Draw a picture!
reason by absurd
use contradiction
5. Analytic Geometry
Use vectors
Exploit symmetry
Make guesses
Split into cases
Quantify things when possible.
6. Sundry examples
Choose a good notation!
Look for symmetries
September 21, 2023
I started reading this thinking it would be similar in content and arousal of interest as the other Maths texts I have read. This was not the case and I felt slightly idiotic when I realised I had purchased a textbook rather than I book on interesting Maths concepts and the history of the subject. But, nonetheless, this was coherent, a fairly good introduction to Problems from Mathematics Olympiads, and at times really quite interesting. This book was aimed at the 14+ age category, I am 17 currently studying Advanced Higher Maths and a fair few areas were completely unknown to me, so I am not be sure how 14 year old me would have coped with this, I'm pretty sure I would not have gotten a single shred of it. I started this book hoping to get a deep insight into basic mathematical problem solving, and was doing all of the exercises, focusing on the logic of his solutions, and paying a great deal of attention to how Terence Tao worked through each problem. By the end of the book I was just reading to get an incredibly general impression of the essence of solving a mathematical problems, and found myself not really bothering to understand every statement and every step. This was certainly a personal issue, as like I said, I went into this book, not really expecting what I got, and believing it would have been the usual, fascinating and enriching Mathematical literary composition, when in reality it was closer to a textbook. My copy was second hand, and the previous owner had highlighted quite a prodigious number of mistakes, I would say there were two or three minor mistakes per chapter, which honestly felt a bit shoddy to me, even huge textbooks covering a whole years syllabus does not seem to have nearly as many mistakes as this book did. Overall I would say that if you are really interested in learning to solve and approach problems akin to those in High School level Mathematical challenges, read this book, otherwise you might be better off with something else.
My opinion is yet to hold any real value. And that is the only part of this review you can trust.
My opinion is yet to hold any real value. And that is the only part of this review you can trust.
July 31, 2014
Terence Tao is one of the most famous and early mathematicians nowadays. He was the younger medal-winner in the IMO (International Mathematical Olympiad), he won medals in three consecutive years, finally achieving the gold prize.
In this book Tao aims to show the reader various tactics involved in solving mathematical problems at the IMO level, for which he assumes a basic level of mathematics, trying to avoid difficult or less known results or theorems.
A book much more famous in solving problems is How to Solve It: A New Aspect of Mathematical Method, by George Pólya, but these two are very different texts. The latter is more theoretical, providing ways of thinking, meditation about problems and how to face them or some clues to solve them. But he offers a few examples in which you can apply them. In fact, he repeats himself continuously.
However, the point of view of Tao in this book is entirely opposite, for he write a short foreword in the way of Pólya and then almost all the book is a set of examples in different branches of maths (number theory, algebra, analysis, Euclidean geometry and analytic geometry) showing various tactics to face problems. In each of them he explains how to approach the solution, discarding unsuitable ways (and why) and choosing the best one.
Besides, he proposes a lot of problems the reader can try to solve, some of them similar to the yet solved ones, others simply a more general one. A good chance for fun, trying to apply the strategies he shows previously.
In this book Tao aims to show the reader various tactics involved in solving mathematical problems at the IMO level, for which he assumes a basic level of mathematics, trying to avoid difficult or less known results or theorems.
A book much more famous in solving problems is How to Solve It: A New Aspect of Mathematical Method, by George Pólya, but these two are very different texts. The latter is more theoretical, providing ways of thinking, meditation about problems and how to face them or some clues to solve them. But he offers a few examples in which you can apply them. In fact, he repeats himself continuously.
However, the point of view of Tao in this book is entirely opposite, for he write a short foreword in the way of Pólya and then almost all the book is a set of examples in different branches of maths (number theory, algebra, analysis, Euclidean geometry and analytic geometry) showing various tactics to face problems. In each of them he explains how to approach the solution, discarding unsuitable ways (and why) and choosing the best one.
Besides, he proposes a lot of problems the reader can try to solve, some of them similar to the yet solved ones, others simply a more general one. A good chance for fun, trying to apply the strategies he shows previously.
February 16, 2022
I quite liked this book and would have found it extremely useful if I had managed to find it when I first started studying math and physics in university. The book is, as promised by the title, focused on methods for solving problems rather than a more rigorous treatment of any particular problem. It acts as a nice survey of topics in mathematics (including number theory, analytic geometry, functional analysis, etc) and includes a legitimately fun "sundry" chapter at the end. The personal tone of the author also reminds me of Div, Grad, Curl and all that which I think channels the goal perfectly -- it is relatively easy for the reader to follow the thought process of Terry Tao.
My only criticisms of this book are that it is especially geared towards solving abstract mathematics problems (with a clear bias towards mathematics competitions) as opposed to applied problems that require abstract mathematics (as in much if physics). The other, more personal, criticism is that many of the solutions depend heavily on modulo arithmetic, which again tends to lend itself towards abstract problems.
All that said, I enjoyed reading it and will probably refer back to the individual chapters if I find myself working/struggling in one of those domains.
My only criticisms of this book are that it is especially geared towards solving abstract mathematics problems (with a clear bias towards mathematics competitions) as opposed to applied problems that require abstract mathematics (as in much if physics). The other, more personal, criticism is that many of the solutions depend heavily on modulo arithmetic, which again tends to lend itself towards abstract problems.
All that said, I enjoyed reading it and will probably refer back to the individual chapters if I find myself working/struggling in one of those domains.
July 3, 2023
Alto king. +20 y god
Read
March 18, 2020Understand 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)
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)
January 21, 2021
It won't take much of your time, only a bit more than 100 pages. But it's still sort of hard to read if you have no math background like me. In general I'd still recommend to give it a go if you have some background and want to understand the way Terrence Tao approaching math's problem.
February 27, 2021
C’est une curiosité que ce livre écrit par un jeune Terence Tao, adepte des olympiades mathématiques. Rien que la couverture : un jeune Terence Tao, penché sur un problème avec le célèbre Erdös. Ce livre détaille uniquement des techniques pour résoudre des exercices "habituels" pour ce genre de concours. La résolution utilisant donc parfois des informations de contexte pour trouver la solution.
Il est assez facile de lire le livre ”comme un roman”, profitant des explications claires (la plupart du temps), du jeune Terence. C’est ce que j’ai fait, gardant à plus tard le soin de revenir en détail sur les exercices proposés en prolongement/approfondissement.
4 étoiles c’est peut-être un peu généreux, mais ce livre n’est pas une mauvaise affaire après tout : il se lit vite pour les pressés, mais il faut passer du temps pour tout assimiler et faire les exercices...
Il est assez facile de lire le livre ”comme un roman”, profitant des explications claires (la plupart du temps), du jeune Terence. C’est ce que j’ai fait, gardant à plus tard le soin de revenir en détail sur les exercices proposés en prolongement/approfondissement.
4 étoiles c’est peut-être un peu généreux, mais ce livre n’est pas une mauvaise affaire après tout : il se lit vite pour les pressés, mais il faut passer du temps pour tout assimiler et faire les exercices...
April 5, 2024
It feels weird to say I read this--I didn't "read" it in the same way I would normally "read" a math paper, as in I didn't go through the proofs carefully, confirm results, etc. I read it because I wanted to get into the head of a mathematical genius (Tao wrote this at age 15) at a young age to see how he thought. It was gratifying to confirm what I often thought was a key to being a good mathematician... Tao was not afraid of error. He was willing to "play" with problems and examine them from directions that eventually didn't work, yet did not see that as failure. By playing with problems in the same way you'd manipulate a Rubiks Cube (randomly, using a formula, or even with a hammer), you learn things and become a better problem solver. Very interesting and enlightening read.
August 10, 2025
This book doesn't aim to teach the reader how to solve math problems - there are better books on this topic. Instead, Terence Tao welcomes the reader to his thought process. He selects problems from various math competitions and walks you through his personal approach to solving each one. Reading the book feels like a detective story, with Tao meticulously eliminating possibilities and uncovering the most elegant solution. The sheer beauty of the problems and their solutions makes this a 5-star read.
February 17, 2020
Tao's elucidations on how to approach mathematical problems contain valuable ideas and make puzzling over the solutions to the examples discussed in the book an entertaining pastime while also being (mostly) understandable even for readers not too familiar with mathematics.
Note: I did not work out the additional exercises provided in the book for a lack of time (and mathematical prowess)
Note: I did not work out the additional exercises provided in the book for a lack of time (and mathematical prowess)
June 11, 2022
This app is just getting on my nerves. I wish there was a way I couldn't rate it for 1 star. I tried reading the book Solving Mathematical Problems by Terence Tao but I couldn't read it. It just kept showing me reviews. Please can someone guide me on how to read it and fast.
Want to read
November 19, 2019Read
April 7, 2020Abandoned this subject.
September 27, 2021
Bien mais dur pour mon petit cerveau
Read
May 3, 2022This book has wisdom in it. It will help me improve my mathematical problem solving and more.
August 12, 2022
Great book to develop interest in Mathematics for the young bunch of enthusiasts who want to begin exploring it.
January 7, 2023
This book is great. If you are struggling with math then get this book and read it. It is better than many live teachers out there.
February 20, 2023
It is really focused at underage mathematicians who are going through the Olympiad phase of a mathematician. I could salvage some of his younger thoughts though.
August 25, 2025
I understood barely anything, it seemed well put together and reminded me of my days in highschool.
July 17, 2019
Was about Terence Tao's approach to mathematical games.
Displaying 1 - 23 of 23 reviews