Chaque vérité que je trouvois étant une règle qui me servoit après à en trouver d’autres [Each truth that I discovered became a rule which then served to discover other truths].(René Descartes, “Discours de la Méthode“)

Problem solving, from homework problems to unsolved problems, is certainly an important aspect of mathematics, though definitely not the only one. Later in your research career, you will find that problems are mainly solved by knowledge (of your own field and of other fields), experience, patience and hard work; but for the type of problems one sees in school, college or in mathematics competitions one needs a slightly different set of problem solving skills. I do have a book on how to solve mathematical problems at this level; in particular, the first chapter discusses general problem-solving strategies. There are of course several other problem-solving books, such as Polya’s classic “How to solve it“, which I myself learnt from while competing at the Mathematics Olympiads.

Solving homework problems is an essential component of *really* learning a mathematical subject – it shows that you can “walk the walk” and not just “talk the talk”, and in particular identifies any specific weaknesses you have with the material. It’s worth persisting in trying to understand how to do these problems, and not just for the immediate goal of getting a good grade; if you have a difficulty with the homework which is not resolved, it is likely to cause you further difficulties later in the course, or in subsequent courses.

I find that “playing” with a problem, even after you have solved it, is very helpful for understanding the underlying mechanism of the solution better. For instance, one can try removing some hypotheses, or trying to prove a stronger conclusion. See “ask yourself dumb questions“.

It’s also best to keep in mind that obtaining a solution is only the short-term goal of solving a mathematical problem. The long-term goal is to increase your understanding of a subject. A good rule of thumb is that if you cannot adequately explain the solution of a problem to a classmate, then you haven’t really understood the solution yourself, and you may need to think about the problem more (for instance, by covering up the solution and trying it again). For related reasons, one should value partial progress on a problem as being a stepping stone to a complete solution (and also as an important way to deepen one’s understanding of the subject).

See also Eric Schechter’s “Common errors in undergraduate mathematics“. I also have a post on problem solving strategies in real analysis.

## 44 comments

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29 October, 2008 at 3:31 am

fnasimDear Terry

I am Farhan Nasim from Bangladesh. Thanks for your advice on

Solving mathematical problems.I have solved a STEP problem. The solution is here. Please make a comment on the solution.

Note: My English is quite poor, you may experience this in the solution.

31 December, 2008 at 6:43 am

AnonymousHi,

Not to be rude, but a translation of Descartes that captures the original poetry of his phrase better might be:

Each truth I discovered was a rule that then served to discover other truths.

[Corrected, thanks – T.]1 January, 2009 at 3:46 pm

thomasteepeDear Professor Tao,

here are two articles on the benefits of clever note-taking for math problem solving:

http://www.artofproblemsolving.com/Resources/AoPS_R_A_Mistakes.php

with a strong emphasis on math competitions

and

with a number of ideas on how to use mind maps in math.

Best regards,

Thomas

25 January, 2011 at 2:19 pm

thomasteepeI’ve outlined some more ideas math problem solving:

The 8 page paper contains a problem solving layout I find very useful personally (and indeed m u c h better than the previous mind map approach), and a densely packed one-page collection of essential problem solving tools.

6 January, 2015 at 6:05 am

thomasteepeI’ve spent more time thinking about combinations of math problem solving and note-making.

Here’s the current state of thought:

18 January, 2009 at 4:14 am

fnasimHi Thomas

Thanks for your useful links. Would you mind reviewing a problem solved by me. Please go here.

Best Regards

F Nasim

12 March, 2009 at 8:04 pm

analgeomaticaHi dear Professor Tao,

I am very interested in elementary geometry and higher dimension Euclidean geometry, could you please upload chapter 4 in your problem book (I see it is about geometry), thank you very much.

I hope you are interested in elementary geometry, too, nice to meet you here!

Best regards,

Tran Quang Hung.

23 March, 2009 at 4:10 am

UndergradHi Prof Tao,

As an undergraduate student I often face the problem of deciding how many textbooks problems I should do before moving on, for example, Is ten questions per chapter of Rudin’s Principles of Math Analysis adequate? The more problems I do on a specific topic the slower it takes to reach graduate level mathematics. On the other hand, if I just do the homework problems I feel I won’t be fast enough for answering questions during exams.

Is there any way of deciding this question?

many many thanks

23 March, 2009 at 9:52 am

DajHi Professor Tao,

http://www.mathlinks.ro/viewtopic.php?t=266042

12 October, 2010 at 8:09 am

Depression and problem solving in mathematics: the art of staying upbeat « Republic of Mathematics[…] attitude to problem solving and to mathematics in general. Terry wrote a helpful blog post “Solving mathematical problems“. He recommends playing with a […]

8 January, 2011 at 9:43 pm

hezhigangDear Prof. Tao:

I hava translated this essay into chinese, I’m sorry I couldn’t translated it well enough, as my ability in english is as poor as mathematics. Your blog is Gospel to those who are interested in Professional mathematics.

译文url: http://article.yeeyan.org/view/99083/121600

2 March, 2011 at 7:44 pm

LeifMy name is Leif and I wanted to ask you on your opinion on something-forgive me if this seems strange or random. As someone who has apparently fit the profile of “child prodigy” and “genius” (e.g. the fact that you graduated high school and university and earned your doctorate so young), I’m wondering if you think you possess something that only a few others have in terms of intellectual ability or not. I have always believed that if someone applies himself and puts in enough time, effort, concentration, and perseverance they can accomplish whatever they set their mind to. And I greatly appreciated the article on your page where you say that someone does NOT have to be a genius to be a mathematician. Does that mean you don’t consider yourself a genius or you don’t really see a distinction between yourself and others who apply themselves and are ambitious? I am not saying I don’t believe some people are naturally more gifted at certain things or develop stronger skills or have stronger talents than others. I just think it’s possible to achieve levels of excellence and brilliance even in your weak areas through consistent practice. That’s why I disagree with this post by astronomer Julianne Dalcanton http://blogs.discovermagazine.com/cosmicvariance/2007/02/25/the-cult-of-genius/ which i found linked from your page where she doesn’t believe most people can reach the level of Feynman-Einstein-Hawking smart. I’m wondering why not? With enough time and effort. After all, they made their share of mistakes and setbacks but did not give up and they were not superhuman or genetically enhanced as far as I know.. And isn’t part of their fame due to circumstance and perhaps even chance-not their intellectual ability but I mean their status and the fact that their discoveries happened to be earth-shattering or were given more attention by the public at-large beyond the scientific community? Surely some of the scientists working today will make equally groundbreaking or insightful discoveries or develop innovative theories and thus can fairly be labeled “genius” or as having the same level of smarts?.

I think I have developed a stronger aptitude for language than for math and due to suffering from depression in high school and middle school I didn’t push myself nearly as much as I could and lost much of my motivation. However, I don’t see why it’s not possible for me to develop mathematical abilities as strong as my linguistic abilities or even pursue a career in astronomy (which I love) or physics or even pure mathematics. And I know you say similar things on your career advice blog, and I know it’s important to be realistic and plan for graduate school and beyond I just really don’t like how people put this label of genius or prodigy on certain people to (in my opinion) make them seem able to achieve things that most other people cannot–even the levels of Einstein or Mozart. And I don’t think that’s arrogant or unrealistic..but I wanted to get your honest opinion. Thanks in advance for your time.

15 March, 2011 at 7:35 pm

AmateurHey Leif,

This book might be useful in pursuing the answer for your question:

http://www.amazon.com/Talent-Overrated-Separates-World-Class-Performers/dp/1591842247

Disclaimer: I just read the summary and reviews of that book. I plan to read it and a few related books later. I’m, at the moment, too busy with studying Maths stuff.

11 April, 2011 at 3:20 pm

AnonymousThanks Amateur for the link I appreciate that you replied to me since Mr. Tao still hasn’t..

23 April, 2011 at 3:01 am

OneminusDear Prof. Tao,

I am a high school student,I loved math got good grades in my middle school years.But I find math hard and i often make many mistakes now.In fact ,i think i can work out many problems while doing my homework .But i am very nervous during my math exams and i almost forget everything i have learnt.

Now i still love math but i am afraid of exams.What should i do?

3 June, 2011 at 8:11 am

How to Solve It and some proposed analogies for problem-solving « Boundless Rationality[…] Terence Tao’s take on problem-solving […]

27 September, 2011 at 9:21 pm

Sabysachi MukherjeeDear Professor Tao,

I am a fifteen year old student currently in high school.I am currently self-studying some non-rigorous calculus. (I have 2 more years in high school).

I am curious to know what I should study next.Should I study some analysis or is it group theory that you recommend?I am confused.

30 October, 2011 at 10:46 am

AnonymousWell, after calculus, one usually studies multivariable calculus. Try http://ocw.mit.edu/courses/mathematics/18-01sc-single-variable-calculus-fall-2010/ to see if you’ve learned everything in regular calculus, and then go onto http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/ .

24 December, 2011 at 7:27 pm

Learning Chinese « tummykung[…] original: https://terrytao.wordpress.com/career-advice/solving-mathematical-problems/ […]

11 December, 2013 at 5:23 pm

Tomas KojarThe link for the book doesn’t work.

[Fixed, thanks – T.]2 February, 2014 at 1:17 pm

AnonymousOn a somewhat unrelated note: do you engage in light to moderate physical activity in order to concentrate over extended periods of time?

16 April, 2014 at 4:15 am

palaki want more and more suggetions

10 October, 2014 at 7:04 am

Deepak kunar shuklaBut i dont get time to read all the theories how can i directly solbe the problems

30 July, 2020 at 10:11 pm

Rajdeep SindhuIf you only want to ‘solve the problems’ without understanding what you’re doing, I wouldn’t exactly call it ‘solving’ a problem. To quote John von Neumann, “There’s no sense in being precise when you don’t even know what you’re talking about”.

20 October, 2014 at 6:07 pm

Solving mathematical problems | RAMY's Technical Blog[…] Tao did a good job on Solving mathematical problems. This helped me a lot on my math […]

6 February, 2015 at 5:26 pm

Career Advice by Prof Terence Tao, Mozart of Mathematics | MScMathematics[…] advice on how tosolve mathematical problems, you can try my book on the […]

14 May, 2015 at 5:15 am

AnonmathDear Dr. Tao,I am an undergraduate studying math.While trying to solve problems from my text books (like Stein’s Complex Analysis ), I notice that very often I cannot solve the hardest problems from them. Since research is about hard problems, does that mean I don’t have what it takes to be a mathematician?

14 May, 2015 at 10:37 am

Terence TaoAs you are still several years away from having to attack research-level mathematics problems, your

currentskill in solving such problems is not particularly relevant (much as the calculus-solving skill of, say, a seventh-grader, has much bearing on how good that seventh-grader will be at calculus when he or she encounters it at the college level). The more important consideration is the extent to which your problem-solving skills areimprovingover time. For instance, if after failing to solve a problem, you receive the solution and study it carefully, you may discover an insight or problem-solving technique that eluded you before, and will now be able to solve similar problems that were previously out of reach. One should also bear in mind that being able to partially solve a problem (e.g. to expand out the definitions, solve some special cases, and isolate key difficulties) is also a very important measure of progress (see this previous post of mine on this topic), as is the practice of constantly asking yourself “dumb” questions in the subject (as discussed in this post). One should also not focus on the most difficult questions, but rather on those just outside your current range. (More generally, you may find other links on my career advice page to be of use to you.)14 May, 2015 at 7:12 pm

AnonmathDear Dr. Tao, how should I select those problems which are just outside my range? Thank you.

11 July, 2015 at 5:56 am

译：解决数学问题 by 陶哲轩 | 万里风云[…] 按：此篇是对陶哲轩的一篇博客 Solving mathematical problems的翻译。翻译仅仅为我的个人兴趣，水平有限，如有纰漏，烦请指正。您不得将其用于商业目的。 […]

27 July, 2015 at 5:36 am

nabilDear sir ,

I learn mathematics as a hobby because I did not have a chance to study college-level mathematics at any good educational institutions.

Do one best learn mathematics which one does not know anything about , say symplectic geometry , best by starting from page 1 of a book on symplectic geometry , and work out all (or most) the proofs and problems as he read. Or is it much more efficient to start from a problem in , say , classical mechanics and work out the mathematical structure of hamiltonian systems on his own from scratch using symplectic topology books as a guide ?

13 February, 2016 at 2:32 am

subodhI am fond of mathematics and want to solve problems

18 June, 2016 at 5:24 am

Sneha Datalu HSHave doubts regarding msc mathematics

12 December, 2016 at 1:27 pm

AnonymousI love this

3 January, 2017 at 9:13 am

For Auld Lang Syne and Cosyne / Global Math Department[…] on the reasoning than on the procedure for answering a word problem. Terry Tao’s quote from this post adds some light to my thinking: “Finding a solution is a short term goal and increasing […]

20 February, 2017 at 9:54 pm

Learning in a better way – My Hilbert Space[…] https://terrytao.wordpress.com/career-advice/solving-mathematical-problems/ […]

12 March, 2017 at 6:54 am

Maths studentOne could also try to use a slightly different notation than the usual one, and solve the problem in this radically new notation. Then perhaps one will work with the pseudo-visuals in the brain, instead of the symbols.

24 June, 2018 at 7:47 pm

Tanner StrunkHi Dr. Tao,

I am a (as of this summer) third year graduate student in a mathematics PhD program, and I was wondering if you also have a post on something like “Where to Look for New Problems.” Certainly faculty in any decent mathematics department are a great source of open problems, but I have also heard that it is good to go outside of the expertise of, say, your advisor, and to be working on multiple research projects simultaneously (the latter being advice from you in another blog post, if I recall correctly).

What might you suggest as an avenue for finding new problems for an aspiring mathematician?

24 August, 2018 at 6:32 am

Jonathan SpencerThe link is broken on your link to “There’s more to mathematics than grades…” Your link here has an apostrophe, which should be removed.

Cheers!

[Corrected – T.]13 November, 2018 at 8:30 am

AnonymousThanks prof terry tao

14 December, 2018 at 10:16 am

Gohar Ali shahSir!

Terrence tao, Asalamalikm!

Dear, sir I am need your great book of “ANALYSIS TWO”

SIR PLZ send my in my email address.

goharali.shah99@gmail.com

A great tnx. From PAKISTAN.

8 April, 2019 at 4:37 am

Bertha AurelliaThanks for sharing The routine seems to map nicely first principle for solving problems. .

Apply Maths to Real World Problems

As much as possible, try to apply real-world problems when approaching maths. Maths can be very abstract sometimes so looking for a practical application can help change your perspective and assimilate ideas differently.

Probability, for example, can be used in everyday life to predict the outcome of something happening and determine whether you want to take a risk such as if you should buy a lottery ticket or gamble.

23 June, 2019 at 4:01 am

Problem Solving : 3 general suggestions – Mathematics Scouts (μαθηματικά ανιχνευτές)[…] Before I begin providing suggestions about problem solving in mathematics, I would like to spend a few words about the general topic of problem solving. Problem solving is among the favorite exercises of a high school mathematics enthusiast. But not everyone excels equally well in the practice of solving problems. That is to say, there are some students who can solve a problem quickly and neatly, while some students have a lot of difficulty to solve them. This often leads to some students getting demotivated, and hence his or her knack towards the subject of mathematics observes a notable decline. But mathematics is much more than just solving problems. The interested reader is suggested to read more from this page of Prof. T. Tao’s blog. […]

12 September, 2019 at 1:20 am

Aden PowerOne of the first links, that says “not the only one”, seems to go to a 404.

[Corrected, thanks – T.]