Studying Math: Want a Challenge?

Many students who write to us are involved in math competitions. They don’t always say that explicitly, but we can tell when the problems they ask about may be far beyond ordinary homework, requiring deeper problem-solving skills. The three questions I’ll look at today are from students asking how to prepare for these competitions, or just to develop their own abilities beyond the normal.

The Math Olympiad

Here is a question from Ryan, in 2000:

Preparing for a Math Olympiad

This question may be a little weird.

I'm fifteen years old and I'm in grade 10. I really like math and I do relatively well in school. But I want to be a candidate for the International Math Olympiad in grade 11 and/or grade 12. In order to do so, I have to push way past the subjects covered in class. I'm ready to work really hard and make sacrifices in order to reach my goal.

My question for you is: what is the best way to learn senior level math and beyond, and know how to do well on math contests? I can't just look over grade 12 math contests and learn the math. Do you have any ideas on how I could efficiently "reach the top"?

Doctor Ian first stated the obvious:

I once had a neighbor once who was always trying different diets. We were talking about it one day, when she blurted out that she'd do anything to lose weight... except eat less, and exercise more.

Anyone who has become really good at math did it by reading books and practicing. I know that sounds too simple to be true, but there it is.

He then added general advice about finding useful books (back then, there probably weren’t many websites, if any, dedicated to this sort of preparation) and using them to self-teach:

Here's how to choose a book: Look for one where the stuff in the first chapter is too easy, and the stuff in the last chapter is too hard. That means you need to learn the stuff that's in between.

When you've found a few like that, open each one up to a section that tries to explain something you don't understand, and see how well the author's style of teaching meshes with your style of learning. Choose the one that seems best suited for you.

As you go through the book you've selected, work the problems in each section until you can do them easily. If you get stuck, go back to the previous chapter, or the previous book, until you find what you failed to learn earlier.

When you reach the end of one book, select another one, using the same process described above. Keep selecting and reading books until you know everything.  :^D

If you haven't read anything by or about Richard Feynman, you should remedy that deficiency as soon as possible. (_Surely You're Joking, Mr. Feynman_ would be the natural place to start.) He was one of the great problem-solvers of all time, and his insights in that direction will be of great value to you.

You should also look at some of George Polya's classic works on problem-solving.

How about practicing? Math competitions commonly involve problems that go beyond the ordinary: that is, puzzles.

And work as many puzzles as you can. The value of puzzles is that they teach you alternative ways of thinking about problem solving by forcing you to learn to 'think outside the box', so to speak. As with textbooks, start with easy puzzles and work your way up to harder ones. When you've solved a puzzle, take some time to stop and think about what assumptions or habits of thought you had to work past in order to find the solution. (Why didn't you see it immediately? Where else might you be blinded by the same prejudices?) Write them down, and look over your notes from time to time.

One thing that you should keep in mind is that the people you'll be competing against have not, by and large, learned a lot of math in order to compete. They compete because they have learned a lot of math, and they have learned a lot of math because they love learning it.

He closes with a story reminding us to be patient when learning, which I have to omit here.

Challenging yourself

Now let’s turn to a 2014 question from Stephanie, who had been asking a number of difficult questions:

Giving Myself a Challenge

In my younger years, I participated in a lot of maths competitions, which put me ahead of many of my classmates.

Now I'm 14 and approaching IGCSEs. Can I do those contest puzzles alongside my textbooks? Which ones do you think will help me most?

Doctor Ian took this question, too:

I got a good chuckle from reading you say, "In my younger years ..." :^D

Yes: I think competition problems and textbooks are complementary. 

In a textbook, you have information being presented to you in an orderly way, and you have problems that are created to let you practice using that information. So most of the time, when you're solving a problem from a textbook, there's not much question about what you're supposed to do. Sometimes you have to decide which variant of a technique should be applied, or there's some small wrinkle that arises (for example, "What if the local maximum is less than the value at an endpoint?"), but it's pretty straightforward. It's *designed* to be straightforward.

By contrast, a competition problem is often designed so that the "straightforward" approach is more difficult than some other, easier approach. But to find that easier approach, you have to come at it from a different angle; and it's figuring out that angle -- by altering the problem so that it more closely resembles something you've seen before -- that provides the challenge, and the benefit.

They are two different kinds of training, and the good “mathlete” needs both.

For example, you might have a problem where the straightforward approach would require you to create and then factor some third-degree polynomial. But if you recast it as a particular kind of problem in geometry, then you can use symmetry to get the answer quickly.

But there won't be anything in the problem to tell you that! It's something you have to arrive at yourself, by allowing your mind to wander through almost everything you've ever learned, looking for interesting connections that you haven't noticed before.

You're not going to see problems like that in your textbooks (unless they're really unusual!). Competition problems are the best source that I know of for this kind of experience.

And in my opinion, at least, this skill -- this kind of alchemical changing of the problem you have into one you'd rather have -- is probably the most important "problem solving skill" that there is, inside and outside of mathematics.

So I would encourage you to use textbooks to incrementally increase the range of your knowledge; and competition problems to challenge you to take the new things you're learning and tie them to what you've already learned, in ways that can surprise and delight you.

Stephanie had several follow-up questions, to which I will show only Doctor Ian’s answers:

If you want to use the "mathematics as language" analogy, then math textbooks are like language textbooks, and competition programs are like actually going out and having conversations with people.

The thing about conversations is that they are largely uncontrolled, which is to say, just about any topic can lead to any topic; and often the connections can be somewhat obscure ... just like with competition problems!
The best advice I can give you ... is to develop the habit of trying to solve any problem you come across in as many ways as possible.

Got one answer? What's a second way to attack the problem? What's a third way? A fourth way? If you switch around the knowns and unknowns, which of those solution methods still work? What new ways of approaching it would be required?

A second good habit to develop is generalizing the solutions you learn. If this technique will solve this problem, what other seemingly unrelated problems could it be used to solve? Is this technique a special case of some more general idea? Is it a generalization of some more specific ideas that I've learned in the past, and can now subsume under this one? Can I abstract this idea so that it applies to things in other fields of mathematics? To areas outside of mathematics?

In this way, just about any problem serves the role of a "competition problem," even if it's not expressly constructed for that.

Turn your textbook into a competition lesson, by taking everything farther! Polya has some similar comments in How to Solve It.

Stephanie responded:

Thanks for your advice, Dr. Ian! I will remember your two ways of solving problems.

Given the vast range of subjects in "conversations" (I do love that extended analogy -- thank you!), do you think there are any groups of competition questions that will be at my current IGCSE level? or that will help with my IGCSEs in some way?

Doctor Ian concluded:

To be honest, I don't have a lot of experience with various competitions, or with IGCSE for that matter. So I'm not able to tell you which ones would be most beneficial for you.

But that's why I discussed how to turn *any* problem into a "competition" problem. That is, if you get good at that, you won't have to spend time looking around for problems that other people have made up. You can make up your own.

It's worth noting, by the way, that this is very close to the heart of how mathematics gets extended. Mathematicians solve one problem, then think about how they could change it (e.g., by adding or removing constraints), until they find that they have a problem that no one has solved before. Then they solve that new problem, re-visit it -- and the process continues.

Also, I realize that in my previous messages, I forgot to mention one other habit that you might want to develop. Feynman talks a lot about this one. That is the habit of trying to "see" ahead of time what a solution should be, before you sit down to try to calculate it. Sometimes this looks like finding estimates for upper or lower bounds, or noticing where symmetry can rule out certain kinds of solutions, and so on.

There are at least two reasons for doing this. One is that if you know approximately what the solution has to be, then you can catch errors if you arrive at something far away from that. The second is that it can give you a better understanding of the problem, and in so doing, may help you make a connection to something you might not otherwise have thought about.

Of course, for any given competition, today there will be sample questions and tests from the source, and probably other sites dedicated to helping students prepare, so one can find out what sorts of problems to expect; but developing good habits while doing any kind of work is perhaps more valuable.

By the way, some of the Math Doctors have had considerable experience working with students on math competitions, and they have often answered questions about particular competition-style problems, giving additional general hints as well. These are much harder to search for!

The Putnam Exam

Let’s look at one more question, this one about an extremely challenging college-level competition.

Preparing for the Putnam Exam

I was wondering if you could give me some specific advice on how to study for the Putnam Exam because I will be taking it for my first time.  I know the test is very hard, and I would like some helpful hints from people who are more experienced than me at mathematics.

I am reading a book called Techniques of Problem Solving and I will probably finish it before the Putnam Exam is given.

Two of us answered this one. First, Doctor Pete described the exam, and gave general advice. Then  Doctor Vogler got more specific:

Thanks for writing to Dr. Math.  I competed in the Putnam exam three times, and I thoroughly enjoy math puzzles, so I still like to work on the problems even though I am no longer an undergraduate and therefore no longer eligible to compete.

If you look at some old exams, I think you'll notice that in each of the two sections (A and B), the six questions are approximately in order of difficulty.  So the first question is generally within reach of anyone who's taken calculus and maybe linear algebra, although most such students will still find the question very challenging. (Remember that the median score on any question is zero.)  The second question is harder, but still feasible.  I found question three usually doable but very challenging.  Then question four was only sometimes doable, and five and six rarely.  The first four or five problems usually only require exposure to normal undergraduate mathematics (like Doctor Pete listed off for you) and a lot of creativity or cleverness.  The sixth (and sometimes the fifth) question will usually require some advanced math that most undergraduates haven't learned yet.  That might seem unfair, but it's supposed to be the most challenging problem on (arguably) the most challenging math test in the world.  And, even if not all students learned that subject, they all had the chance.  (You have a university library with a math section that you can use, right?)

So it is probably better to spend more time on earlier problems until you get those than wasting time trying to work the sixth problem, which you're not likely to get.  (Unless you're really *really* smart.)  In other words, a half hour per problem is not very realistic.  If you want to finish all six problems, then you had better get the first one or two done very quickly so that you have more time on the sixth.  A more realistic approach might be:  a half hour for the first problem, an hour for the second, an hour and a half for the third, and never mind the other three.  Or perhaps even: three hours for the first problem, and never mind the rest.

Note that your goal has to be to do as well as you can do! This is not for those who can’t stand failure. (If you like challenges, then you have to accept failure as a step in learning.)

I would agree with Doctor Pete that you should practice by working problems on old exams, but I would not say that the ability to do well on this exam is not related to the ability to do math in general.  In trying to work old problems, and in reading solutions to the ones you can't get, you will learn mathematical techniques that are very useful in many fields of mathematics, and which will come up in your courses later.  You can learn some really neat things.  Better yet, you learn these techniques with a use in mind, so that makes them seem very worthwhile.  If you learned the same thing in a class, you might think, "Why do we have to learn this?  When will I ever use this?"

By the way, you can find the questions for previous Putnam exams, along with complete solutions for each question, in a fall issue of the American Mathematical Monthly, around the October or November issue of the next year.  (Does it take professional mathematicians that long to get good solutions to all of the problems?)  You can probably find that math journal in the periodicals section of your university library.  Ask the librarian if you need assistance.

Doing (trying!) old exam problems can be a great learning experience, even apart from taking the exam.

Be familiar with writing clear proofs, because this is the style of writing that they will be looking for, even when it isn't a proof they ask for.  Be familiar with mathematical rigor, such as using definitions as they are given to you, and not making incorrect assumptions.  You can use other theorems that you've learned in your courses, but you usually won't need obscure theorems, just the ones normally taught in undergraduate courses.

And one other thing:  They will often ask a question with a fairly large number, like "What is the 30th term in this sequence?"  Almost always it will require *way* too much time to generate 30 terms in the sequence, and that's not what they want you to do.  You should generate a few terms, find a pattern, write out a formula for the n'th term, and then prove it by induction.  They only say "30'th" to disguise the fact that it's easier to get the general n'th term than it is to get 30 terms.  Then you just substitute n=30 into the formula.  (Notice that this type of question is an example of a question that doesn't ask for a proof, but you should give one anyway: a proof by induction of your formula.)

We have previously discussed the process of finding patterns and formulas; this takes it to a whole new level. I will be getting back to that topic soon, looking at this more advanced type of problem.

1 thought on “Studying Math: Want a Challenge?”

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