What Makes A Good Problem?

When you publish a problem on Brilliant, it's only natural to hope that the community falls in love with it and wants to work on your problem. It feels good when thousands of people have viewed your problem, and you bask in their admiration. However, only a few problems have that special distinction. This wiki explains how you can greatly improve the quality of your problems by

• identifying your target audience and
• choosing the best presentation.

1) What makes the problem stand out?

• Represents a daily life scenario: people love seeing how math and science are applied to the real world.
• Clarifies a misconception: people are curious when they come across a counter-intuitive result, and want to understand how to think about it.
• Thought-provoking: it encourages people to apply their knowledge instead of just relying on memorization of formulas.
• Involves multiple concepts: a problem becomes more challenging if it involves multiple concepts.

2) What level is the problem?

• Easy: It should neither involve a lot of tedious calculations nor contain unexplained technical jargon.
• Medium: It may involve a single application of a difficult concept but doesn't require a lot of knowledge.
• Hard: Showcase a complex situation which requires the use of multiple concepts. It requires a lot of thought processes to completely solve the problem.

This gives you insight into the ability of your audience and determines the amount of information that you should provide in the problem.

Everything on the internet is competing for your limited attention. If the presentation and content do not feel outstanding, then we have been conditioned to hit the backspace and move on. To avoid losing your audience, be aware of the following:

1) Phrasing: People can only read what we have written down, and not what we were thinking or intending.

• Keep it short and uncomplicated. In general, remove unnecessary information that is not relevant to the problem.
• Organize the information. Keep similar information together so that it is quicker to understand.
• Identify the crux of the problem. Make it the focus of the problem.

2) Theme/Motivation: People will not engage if they find the problem boring.

• Make the problem engaging with a relevant theme. Real-life applications tend to work well, but do not overdo it.
• Consider providing a hint to encourage answering the problem.
• Keep your target audience in mind as you write up the problem.

3) Contains imagery: People will only engage with the problem if they can easily comprehend its meaning.

• A picture says a thousand words. When relevant, it helps to draw the reader in and quickly provides the necessary context.
• For geometry problems, having a picture often makes it easier to understand what is being described.

4) Directive/Answer options: Meaningful options makes a sensible question.

• Keep it simple. Avoid making the reader do unnecessary work in order to submit an answer.
• Consider using multiple-choice options to encourage people to give the problem a try, even if they are not fully certain.

How can we improve the following problem?

My 2 favorite positive integers satisfy the property that if I take any one of these numbers and multiply by itself by a total number of times, where this number is numerically equal to the other number, then the resultant product is 16. What is the sum of my 2 favorite positive integers?

Answer: 6.

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First, let's identify our target audience.

• What makes the problem stand out? It involves both multiplication and indices.
• What level is the problem? This is likely of easy difficulty because the reader is required to know how multiplication and indices are related to each other.

Second, let's choose the best presentation.

• Phrasing: The problem seems convoluted. For example, to verbally describe the second sentence without any variables will make it extremely difficult to digest. It's much easier to just use mathematical notations to describe it. Plus, it isn't necessary to state that they are my 2 favorite numbers, simply stating that there are 2 specific positive integers in question is sufficient.
• Theme / Motivation: Readers can easily see the symmetry behind the second sentence once we've written out the math expressions: \(x^y = 16, y^x = 16\).
• Imagery: In this case, the math expressions speak for themselves.
• Options: A numerical answer would work best in this case. There's little to no benefit by adding multiple choices because it doesn't show how one can arrive at these numbers.

Based on the above, we can improve the problem by

1. rewriting the question by providing math notations and
2. removing the context of "favorite numbers."

As such, this leads us to create the following problem:

Let \(x\) and \(y\) are positive integers such that \(x^y = 16\) and \(y^x = 16\). What is the sum \(x+y?\)

How can we improve the following problem?

There is a point that is 5 away from the edge of a circle that is towards the center of the circle. If you draw all of the chords through this point, you will find that the shortest chord has length 30. What is the radius of the circle?

Answer: 25.

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First, let's identify our target audience.

• What makes the problem stand out? It is thought-provoking because there doesn't seem to be enough information in the problem to proceed.
• What level is the problem? This is likely of medium difficulty because it is a simple application of a concept.

Second, let's choose the best presentation.

• Phrasing: The problem seems convoluted. For example, the second sentence can be made explicit/immediate by phrasing it as "The shortest chord of the circle that passes through this point has length 30."
• Theme/Motivation: This is actually a really interesting geometry problem, but that has not been expressed as yet. Too much time is spent trying to figure out what the problem is, instead of appreciating it. In addition, a hint might be helpful, because there doesn't seem to be enough information at the start to proceed.
• Imagery: This will strongly benefit from having an attached image.
• Options: A numerical answer would work best in this case. Having multiple-choice options doesn't make it more tempting to answer, because it is not clear how we can arrive at those values.

Based on the above, we can improve the problem by

1. providing a pictorial image for people to understand what the description is and
2. providing a hint for people to get started.

As such, this leads us to create the following problem:

How can we improve the following problem?

At \(t=0\), a particle \(A\) is located at the origin \((0,0)\) and a particle \(B\) is located on the \(y\)-axis at \((0,-d)\). Then, \(A\) starts traveling along the \(x\)-axis at a constant velocity of \(u.\) \(B\), on the other hand, travels with a constant speed of \(u\) such that, at every instant, its velocity vector is oriented towards \(A\)'s current location. Let \(r(t)\) denote the distance between the particles at time \(t\). Find \(\lim_{t \to \infty} r(t)\).

Options:
(A) \(\frac{d}{2}\)
(B) \(d\)
(C) \(2d\)
(D) Zero
(E) Infinite

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First, let's identify our target audience.

• What makes the problem stand out? It represents a daily life scenario where we want to chase an item directly. As they work through the problem, it might showcase a misconception.
• What level is the problem? This is likely of hard difficulty because we have to relate several concepts and solve a differential equation.

Second, let's choose the best presentation.

• Phrasing: Let's be honest, most people would not make it through the entire paragraph, because it appears boring.
• Theme/Motivation: Being a classic "chasing problem," we could add some characters to make the problem more interesting.
• Imagery: Since it can be a real-world application, having an image that illustrates all the information would be very helpful.
• Options: While we could convert this into a numerical answer by setting \( d = 1 \), having the options not only makes it easier for people to guess, it also showcases that there is a "nice" answer to this seemingly complicated problem.

Based on the above, we can improve the problem by

1. figuring out how best to present the problem as a real-world/familiar scenario and
2. providing a pictorial image for people to immediately grasp what is happening.

As such, this leads us to create the following problem:

Tom and Jerry both have equal top running speeds \(u\) and are initially at points \(A\) and \(B,\) respectively, separated by a distance of \(d\). They both spot each other and immediately start running at their top speeds. Jerry runs on a straight line perpendicular to the line \(AB\) and Tom runs in such a way that its velocity always points towards the current location of Jerry. Let \(r(t)\) denote the distance between Tom and Jerry at time \(t\). Find \(\displaystyle \lim_{t \to \infty} r(t)\).

Options:

(A) \(\frac{d}{2}\)
(B) \(d\)
(C) \(2d\)
(D) Zero
(E) Infinite

Now that you've seen these ideas in play, take a problem of yours and see how you can make it great. If you would like to help going through a problem, comment in the feedback box below!