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How Do You Come Up With Good Problems?

One of the biggest feature of Brilliant is that you can write your own problems and share them to people who are interested. Even though this feature has been around for quite a while, I didn't check it out until recently. The reason behind this is that I know that I am a lousy problem writer. I can't come up with problems. But I'm giving it a go. So I've written a couple of problems recently. But I don't feel much excited with the problems I'm coming up with. So, I need your help.

What advice would you give to someone who has recently started making problems? How do you come up with problems? How do you improve yourself? Please share your thoughts.

Better yet, you could describe the experience you had coming up with a problem you're really proud of. I'd love to hear your stories.

Thanks in advance!

Note by Mursalin Habib
2 years, 9 months ago

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I am always "playing around" with mathematical ideas, and indulging my curiosity, for my own entertainment. So questions tend to spring up naturally in my head as I do so. Some are more interesting than others, and when I have (what I think is) a good one, I set about writing it up "formally" and sharing it on Brilliant.

So my advice would just be, what math ideas are you curious about? Find other problem sources, and see if any problems spark your interest and lead you to ask other questions (of yourself): Will that always be the case? Does that trick always work? What if you consider rational numbers, instead of just integers? Etc., etc. The more you get in the habit of asking yourself questions, the more naturally question-asking will come in general. Matt Enlow · 2 years, 9 months ago

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@Matt Enlow Your last line is absolutely true! Jatin Yadav · 2 years, 9 months ago

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@Matt Enlow That is some really good advice. Thanks! I'll keep asking myself more questions then! Mursalin Habib · 2 years, 9 months ago

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@Matt Enlow Thanks for these inspiring lines,Sir. Soham Dibyachintan · 2 years, 8 months ago

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@Matt Enlow Quite true. Researching both mathematics/physics ideas (infinite summations, electricity + magnetism, etc) and problems (here in Brazil almost every university had its own entrance paper, giving us students a huge database of questions) helps the brain to create similar or even very different questions and interesting approaches on its theory.

Mixed theories (geometry + geography, history + chemistry, etc) is also a pattern that appears on our universities' papers. This has led me to find and also develop many, many cool questions.

A personal funny story is that I solved UNICAMP's 2014 Objective Math Paper in about 5 minutes, because I had already created and solved very similar questions.

PS: Here are the questions:

If \(\cos x = \tan x\), evaluate \(\sin x\).

In a right angled triangle with perimeter \(6\), its sides form an arithmetic progression. Evaluate its area.

Evaluate he modulus of the complex number \(z = i^{2014} - i^{1987.}\) Guilherme Dela Corte · 2 years, 9 months ago

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There are many deep facets to posing good questions, and composing incisive problems. In my opinion, most of Matt's advice applies not only to writing good problems, but living a fulfilling life as well.

We have learned a lot about problem writing from writing our own problems and observing all of your behavior.

Here is some shallow advice on problem publishing(not as hard as problem writing, but almost as important):

  • Avoid the ALLCAPS button at all costs.

  • Give your problem a cool title. A cool title is one that both describes what you fill find inside the problem, but leaves a sense of curiosity that makes you want to click it and find out. Naming math problems may seem trivial, but is actually one of the key factors in whether or not someone clicks on a math problem and sustains the interest to actually read it, solve it, and hopefully talk about it with you.

  • Though good problems are beautiful in and of themselves, humans are cognitively very visual. Attach an image to your problem that is either useful, beautiful, or hilarious. In our experience, hitting all 3 of those qualities is miserably difficult. For instance, I have no great ideas about what kind of image would ideally accompany Labib's BdMO headache

In other words:



That said, just as the hobbits did in fact march into Mordor, it is actually possible to make problem solving visually more friendly. Graphic design is very hard to do well, but pretty easy to do just good enough at. Peter Taylor Staff · 2 years, 9 months ago

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@Peter Taylor You should just put this comment under guidelines for posting problems on Brilliant and don't forget to add the meme :) Mursalin Habib · 2 years, 9 months ago

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@Peter Taylor I think you saw the lords of the ring Avisek Agarwal · 2 years, 9 months ago

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It's definitely not easy asking questions, and much harder to ask the right question. It is an important skill that should be developed, but is often ignored in school.

One way to start, is to look at common everyday things that you do, and then ask if there are alternative ways of approaching them in a logical / rational way. For example, in SF, there are bus-stops on almost every corner, and my house is located between 2 bus-stops. The one that is slightly nearer is uphill and will be reached first by the bus. Which one should I walk (or bike) towards? Does it matter if I can spot the bus as it rounds a corner? What other factors are there to consider? As you work out the different parts, it can turn into a nice algebra or calculus problem, and not to mention help you decide what to do :)

If you prefer theoretical questions, you can start with a fact / theorem / another problem, and then modify the scenario from there. An example would be Dividing a set of lines, do you know which famous question it was adapted from?
If you remove restrictions, can you reach similar conclusions? If you add more restrictions, what else can you say? This was how I stumbled upon my Economics thesis - I was studying auction behavior, and failed to theoretically replicate certain empirically observed behavior, and eventually showed that one of the (main) assumption was invalid. Calvin Lin Staff · 2 years, 9 months ago

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Now that I think about it, I hardly ever create my own questions from scratch. I usually just use others' problems or adapt them myself. This is because coming up with good problems is hard. But it is especially hard when you have to come up with them on the spot. You shouldn't search for good problems; they should come to you.

For instance, one day I was packing up a tent from a recent camping visit. I notices that the tent required 7 stakes but came with 10 stakes. I also noticed that the stakes that we used got really muddy. So an idea just popped into my head: what is the expected number of times I can go camping before all the stakes are dirty? Why? I don't really know. It just came to me, and I remembered it because I knew I could solve it with the math I knew.

Other times, I'm just playing around with random expressions and symbols, like the summation symbol. And like Matt says, after playing around with them I can start to formalize my question and share it on Brilliant. This is that question.

Best of luck to you. Bob Krueger · 2 years, 9 months ago

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Though I am also a newbie at problem making, but still I can share what I have learnt:

  1. Choose a subject.

  2. Choose a topic.

  3. Choose a situation.

  4. Try to extract the most you can out of this situation. Never leave it after making only 1 problem. Think of all possibilities. More than 1/2 of times, you discover that a better problem could have been made from the same situation. For example, I made 4 problems related to JEE-Mains and the one posted was the best I could think.

  5. Recheck the calculations, check for a typo, and give a final touch to the problem.

If you are unable to think of a situation, try other good problems, and there is a good chance that you would come up with a better problem based on the same trick as deployed in a problem you tried. I firmly believe that trying more and more problems makes you a better problem creator. Jatin Yadav · 2 years, 9 months ago

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I try to think of a problem that is interesting in its own right (like TSP or the knapsack problem as two examples) and then put them into real world situations.

I don't think it always makes a problem more interesting to embed it in a real situation, but it often can. Chung Gene Keun · 2 years, 9 months ago

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I think you should look around for problems in your daily life. For example, if you are travelling in a bus, you could think about the number of people in the bus and maybe come up with a problem like , number of women is half the number of men and the number of seats on the bus is 48. Assuming that all seats are occupied and no one is standing, how many women are there? This was just a basic example, and you could maybe develop on this idea. NOTE- I am not a good problem writer myself, but I am trying to be and I am trying this method. Shabarish Ch · 2 years, 9 months ago

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I like to take boring math concepts and spruce it up a little bit. Most of the times when I feel great, I make it more challenging and require more thought than the concept requires, other times I just think of a fun little story to go along with the problem. :)

Also, like @Peter said, attach an image if relevant. I should start finding some images online... Kevin Mo · 2 years, 9 months ago

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Luckily, I have just the post for you. Finn Hulse · 2 years, 7 months ago

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@Finn Hulse I have read it before. Nice post! It nicely deals with what you should do to turn an inspiration into a well-written problem. I was actually trying to know more about how good-problem writers get their inspiration. That's why I was looking for other people's problem-creation stories.

Thanks anyway! Mursalin Habib · 2 years, 7 months ago

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@Mursalin Habib Okay... well, most of my inspiration comes to me in math class. My teacher gives us crappy problems (of course), and I like to take those principles, and turn them into something cool. My article does address how to get inspiration, I'm pretty sure, though. Finn Hulse · 2 years, 7 months ago

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According to me I think that for making problems you need a good imagination power. Maths basically deals with problems, if you include Mathematics in your daily life routine , you will be able to find different varities of problems arising to your mind. Not only by coming across those problems , you also have to go deep through it, to completely have the idea about the matter.. Nashita Rahman · 2 years, 9 months ago

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