On Writing Problems

Reflections about composing satisfying mathematical challenges

Problems and Exercises

To most people, the expression “math problem” brings to mind a teacher assigning something like “section 5.2, problems 11 through 37 odd” from a ten-pound textbook.

But these are not what a mathematician would recognize as “math problems”. In fact, there is a very good chance that in the textbook itself, the heading preceding the numbered list of calculations-to-be-done is not “Problems” but “Exercises”. Far from being a meaningless terminological choice, many people who work in mathematics and related fields draw a definite distinction between these two superficially similar tasks. The difference can be defined in terms of what demands the task makes on the people to whom it is posed.¹

To complete an exercise, the student must merely apply facts and techniques that he or she already knows. Exercises serve an important pedagogical role: they help the student to cement the knowledge presented in lecture or the text, and help develop fluency with and understanding of the techniques or algorithms being learned. But a student who only works exercises is only doing mathematics in the most superficial sense.

By contrast, to solve a problem requires creativity and original thinking. When one begins work on a problem, this is done without a complete idea of how the solution might proceed. A true problem does not reduce immediately to rote calculation or the application of known techniques. It demands attention, thought, exploration, and perseverance through uncertainty and against obstacles that seem to bar the path to an answer.

While writing exercises is relatively straightforward — just create a setup that demands the use of the technique to be exercised — inventing problems is substantially trickier. How does one author a problem that demands creativity and insight on the part of the solver, and which is intellectually and aesthetically satisfying to solve? Creating a good problem requires orders of magnitude more time and creativity than writing an exercise.

One Person’s Problem is Another’s Exercise

One immediate difficulty with this distinction is that the amount of creative thinking required to answer a question can vary greatly depending on the background of the person solving it. For someone who does not know calculus, the problem of finding the area enclosed by a parabola and a line is a difficult one indeed (just ask Archimedes). But for a student in a calculus class, it is a straightforward exercise.

This implies that a key consideration when composing a problem is “for whom?” A good problem for a middle school student may be a trivial exercise for an undergraduate math major, while a good, challenging problem for the undergraduate might be incomprehensible to the middle-schooler and a straightforward exercise for a more knowledgeable mathematician.

But creating a good problem is not merely a matter of posing it to the right audience. Though a good problem for Alice might be merely an exercise for a more knowledgeable Beth, the converse is not true. A question which is an exercise for Beth, but relies on ideas not known to Alice, is not at all likely to be a good problem for Alice. It might, for instance, be simply unapproachable by Alice without background that she doesn’t have, or it might just be uninventive and uninspiring and so not a good problem for anyone.

How, then, does one write a good problem?

Principles for Good Problems

I’ve been solving and creating problems for the majority of my life.² I’ve written problems just to share for fun; I’ve composed problems (both for pay and as a volunteer) for competitions for middle school, high school, and college students; and I’m currently on the problem development team for Project Euler.³ I’m hardly the world’s foremost expert on writing problems, but I’ve given a lot of thought to the topic over the years. I like to think that I have good taste in problems by now.

So what makes a good problem, and how does one go about creating one?

A Good Problem Should Be Natural

The vast majority of good problems feel natural or well motivated to their target audience. By this I don’t mean that they have an entertaining (or worse, “realistic”) story that somehow provides a justification for doing the problem. In fact, an elaborate story is often a “smell”⁴ for a badly-written problem. A good mathematics problem should leave the solver feeling like it is the sort of problem he or she should want to solve: perhaps it has an elegant statement, or it is related to something already known to be interesting.

If you are writing a problem and have what you think is an interesting idea, it’s often worth pondering whether it can be expressed in a clean and natural way, or whether it is easily motivated by relating it to something the potential solver already considers interesting or important. Sometimes you need a wall of text anyway, just to specify all the details well enough to make the problem statement totally unambiguous. But ideally the core idea is simple enough that it can be expressed in just a few sentences to someone in the target audience, and should have a natural “hook” to grab the solver’s interest.

A Good Problem Should Have Depth

Most good problems have some sort of depth, not just a single trick. This can mean multiple parts or steps are needed to solve, or it can mean that the key idea itself has depth and independent interest (even if some of that depth is not strictly necessary to solve the problem). It’s often tempting to compose problems that are merely a layer of obfuscation over a one-off, adhoc trick such as a particular algebraic manipulation. This is usually a bad idea. Such a problem is probably going to be unsatisfying to solve, and doesn’t really lead the solver to learn anything interesting by solving it.

Instead, a great problem will have hidden depths to it: ideas that shed light on other problems, tendrils of connection that worm their way into other areas of mathematics, emergent structure that is more intricate than the initial statement. This sort of depth is hard to deliberately engineer, which is one of the reasons that problem creation is an experimental and artistic endeavor as much as a technical one.

A Good Problem Should Stretch the Solver

Good problems live near the edge of the ability of their target audience. They should require the solver to do something they have never done before to solve, and yet be accessible enough that the solver can actually discover and do that thing. A problem might be too hard or too easy⁵ for the target audience. A problem which is too hard is simply a brick wall. But a problem which is too easy is of less value as well. It won’t be as satisfying to solve (because there is no real challenge to overcome), it may feel less fun (since there’s less of a discovery process), and it won’t teach the solver much (since they already know what they need to know to solve it).

This means that it’s hard to write good problems for a general audience. Many contests rely on the fact that they contain multiple problems at different levels to make up for this: the easier problems on a contest may be at the right level for inexperienced competitors, but trivial for experienced ones, while the hardest problems may remain real challenges for the best competitors at the cost of being brick walls for the weaker ones.

One common pitfall for excellent problem solvers who are new problem authors is hard-problem-syndrome. Whether through ignorance of the capabilities of their audience, or through a kind of intellectual machismo, they often write problems that are hard for them, not problems that are appropriate to their target audience. Sometimes this results in contests with problems that are uniformly too difficult⁶; other times, it results in underdevelopment of the easier problems, with the result that many of these are substituted with problems which are mere exercises even to their target audience.

A Good Problem Should Be Unambiguous and Clear

This one is more about the presentation of the problem than the underlying problem itself, but is worth talking about because it’s such a common issue with inexperienced problem writers. It’s easy to write a problem statement that seems clear to you, but is not actually clear to anyone else, and may even be ambiguous or underspecified.⁷ This is strictly a bad thing, and can completely ruin what otherwise might be an interesting problem. Composing statements that are all of clear, concise, and unambiguous is a sine qua non of good problem writing.

Note, though, that “ambiguous and underspecified” is not the same thing as “open-ended”. Unless you are writing for a contest, open-ended problems can be great! But what makes the problem open-ended is not uncertainty about the setting, but expansiveness in the question. A good open-ended problem is an invitation to exploration, but it must have a clear starting point.

A Good Problem Should Be Novel

Problems should be new to the solver. Of course, what “new” means and how strictly novel the problem needs to be can vary. Certainly, a problem must be new enough to not be a mere exercise for the intended audience. Ideally, it should also feel new to the solver: it should include some element that they’ve never considered before.

Whether a problem should be new in an absolute sense is another matter, and depends greatly on the goals of the place where the problem is to appear. In private lessons or in coursework, a problem which is a minor twist on, or which is even identical to, some folklore⁸ result or an existing published problem, is perfectly fine. Many of the problems which appear as challenges in advanced mathematics textbooks are excellent and well fit for purpose, despite (obviously) no longer being original after (or even before) the first time the course is taught. There is also a place for presenting and expanding on existing problems in informative publications (such as “problem of the week” style articles), though in that case it’s arguable that the work being done here is purely expository, not problem-writing.

The calculus changes when considering problems written for competitions, or for outlets where novelty is the point. At Project Euler we generally try to take care not to publish problems that have appeared before, and if we can find a duplicate or near-duplicate of the problem elsewhere, that is usually sufficient reason not to publish.⁹ It’s my opinion that contests should also strive for novelty in this sense. As a general rule, if a problem is functionally equivalent to one which a contestant could reasonably have seen before, but is comparatively difficult if they haven’t, it’s probably a bad problem for a contest (though it might be a great problem in another context!). A recent example of this is Problem 10 from the 2025 AMC 10A. The problem is very obviously equivalent to a well-known bit of folklore and will take about ten seconds for anyone who is familiar with the folklore problem. (Thankfully, the problem is not that hard even if you don’t know the folklore, but its inclusion was still a bad choice in my opinion.)

A Good Problem Should Be Fun to Solve

Last and most importantly, good problems are fun.¹⁰ There should be a sense of joy, in discovery if not in execution, when tackling a problem. While doing exercises to practice skills isn’t usually exciting, solving problems involves creativity, exploration, and learning: all things that most humans find fun. The ultimate test of a good problem is whether the people you wrote it for enjoyed working on it, and feel like they got something valuable out of the experience. By extension, if you didn’t have fun coming up with or solving the problem yourself, you should be suspicious that the problem might not actually be very good.

This rule of thumb is harder to apply when writing problems for a less experienced audience. The people best positioned to write good problems are very experienced, but this very experience means that many problems that will be an enjoyable challenge for less experienced solvers will be trivial for them. Still, a well-trained sense of “taste” for what makes a problem fun can overcome this obstacle.

How to Invent Problems

How does one go about actually coming up with problems? In broad strokes, there are two main approaches. You can start with the setting, or you can start with the solution.

Starting with the setting uses your own curiosity as a guide. You try to generate interesting setups for a problem, without a clear idea yourself of what the solution will look like, or even if the problem is solvable. What if we examined this situation? What are the consequences of these definitions? Essentially, you begin by posing partial problems to yourself, and exploring them to see if they lead anywhere good. If you find something interesting along the way, then you try to develop that into your problem.

Starting with the solution, on the other hand, is to begin with a goal already in mind. You know the place that you want to lead the solver to, and you set out to find a trailhead from which they can find their way there. This is the natural way to create a problem if you want to use it to teach your audience something in particular. In this method, the problem statement is almost the last thing you come up with.

Both these methods have their advantages and potential pitfalls. Starting with the setting has two obvious possible failure modes (and one less obvious).

First, you might have to spend a long time, and throw away many ideas, before you come up with a good problem. I’ve spent hours analyzing some promising setting, only to discover that the interesting bit is simply too difficult to lead to a good problem. In this way, it’s like real research, albeit on a smaller scale.

Second, you can’t make any guarantees about the nature of the solution to the problem you come up with. If you’re trying to create problems for the primary purpose of teaching a particular topic, this approach is very hit-or-miss, and may very well be unsuitable for that reason.

The less obvious problem with starting with the setting is that you can get so invested in the setting that you fail to notice that the problem you ultimately come up with is just not all that interesting to solve. You have to be willing to take a step back and, as many authors say about writing, “kill your darlings”.

But in compensation, starting with the setting can generate the most elegant and motivating problem statements, and leads to many of the most inventive problems. When you, the problem writer, don’t know at first where the solution will go, then it is very likely that your audience won’t either. You may get into truly novel and uncharted territory. It’s even possible you’ll open up some avenue for a bit of your own research, if the setting proves interesting and deep enough!

Starting with the solution has a different set of problems. The main one is that the resulting problem can feel forced, because the setup was contorted in order to make the intended solution materialize. This can cash out as an unnatural setting and problem statement, or as one where the intended solution feels either telegraphed or artificially obfuscated. You can also end up with problems whose solution mostly amounts to seeing the trick, where the problem reduces to seeing the right transformation that reduces the problem to one where its intended solution is clear, or where solving the problem is a skill check that asks “do you know this one technique?” That is, it’s possible to set out to write a problem, and end up with an obfuscated exercise.

Nevertheless, there are clear benefits to this method as well. Each problem-writing attempt is more likely to succeed, and you are able to target a particular pedagogical goal with your problems much more easily than if you start with the setting.

If you’re learning to write problems, I recommend attempting both methods. Try writing a problem by tweaking the setting of a problem you liked (start with the setting), or by choosing a strategy you enjoy using and try working backwards to come up with a problem to which you can apply it (start with the solution). Your first attempts probably won’t be very good, and probably won’t be very original, but eventually you can increase your hit rate and the quality of your problems. The key is to be aware of the possible failure modes of your chosen strategy.

I personally use a mix of these strategies when composing problems. Arguably, I sometimes use a mix of strategies on the same composition: start by looking for a problem to which I can apply a given idea, then tweak it to make it more interesting in some way, then solve the tweaked version and see where that leads. Both methods have led me to create well-received problems.

Have Fun!

Just like problems should be fun to solve, they should also be fun to invent. If you’re not having fun, you can stop. Your problems will be better if you’re having a good time when devising them anyway. I write problems because I like writing problems. Why bother, otherwise?

1. Not all mathematicians make this terminological distinction, and textbooks do not always honor it either. Certainly there are books that label as “problems” things which most would call “exercises”, and some which do the reverse. But even though the terminology is not always consistently applied, the conceptual distinction is important.

2. One might argue that I’ve been creating problems in a different sense since I was an infant.

3. Project Euler releases weekly mathematics problems that are intended to be solved by a combination of pure mathematical reasoning and inventing and programming appropriate computational algorithms. Though some of the early entries are easier and less interesting than the majority on the site, most of the approximately 1000 problems in the archive are excellent problems for a large proportion of the site’s intended audience. See my post about why you should participate in Project Euler for more about Project Euler.

4. Analogous to a “code smell”: it’s something that’s not itself inherently bad, but its presence often indicates that something is wrong. In this case, the elaborate story often means that the problem author felt like the story was necessary to motivate an un-natural problem statement, and was having a hard time doing so.

5. But note that a more advanced audience than the direct target, to whom the problem might be too easy, or even more of an exercise, could still find solving it to be enjoyable if it is high quality in other ways.

6. For instance, “Mock AMC“ contests where two thirds of the problems are at the difficulty of the hardest 20% of problems on the real competition.

7. Yes, in real life, many situations are ambiguous or underspecified. With few exceptions, these are very bad characteristics for posed problems. (The exceptions are essentially limited to problems that are explicitly about developing models for actual real-world situations. If you aren’t composing problems for a large project in a class or competition that’s explicitly about such modeling, you don’t want ambiguous or underspecified statements.)

8. “Mathematical Folklore” consists of problems, theorems, or particular proofs which are well-known within a community (maybe contest problem afficionados, mathematicians generally, or mathematicians working in a particular field), but don’t have a particular concrete source.

9. There are some exceptions. Of course many very early problems were not original; there are also a handful of problems which are primarily intended to be ways of introducing the solver to a particular subject matter, and for which that requirement of novelty is relaxed or ignored. But for the most part, if a particular problem has appeared before it appeared on Project Euler, it’s because someone has overlooked something.

10. The astute reader may ask: type 1 fun (fun in the moment), or type 2 fun (fun only after the experience is over)? I think that the best problems are generally fun while you are solving them, and not merely afterwards when you reflect on your struggle and accomplishment and what you’ve learned. But if a problem provides enough type 2 fun, that’s sometimes good enough.