# 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. Discover how to increase the likelihood that your problem becomes a favorite that the community enjoys and engages with.

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## What makes a good problem

- Simply stated - do not try to be tricky.
- Clear and concise
- Complete - all the necessary information is provided or explained
- Correct - respond to reports and help clarify the problem
- Has proper formatting - if you need help, just ask
- NOT trivial, nor trolling, nor tedious

\[ \begin{array} { | l | l | } \hline \text{Consider a sequence of 10 numbers.} &\text{ Consider an arithmetic progression (AP) of 10 terms.} \\ \text{Total of the first, third, fourth, eighth and ninth terms is 100.} &\text{The sum of the terms in odd position is 100.} \\ \text{Sum of the second, fifth, sixth, seventh and tenth terms is 200.} &\text{The sum of the terms in even position is 200.} \\ \text{The common difference of this AP?}& \text{What is the common difference of this AP?} \\ \hline \end{array} \]

- Which problem highlights a meaningful property of arithmetic progression?
- Which problem is a tedious laborious uninspiring calculation?

\[ \begin{array} { | l | l | } \hline \text{ What is } & \text{ What is } \\

23512 \times 91251 \times 5438 \times 45161 \times 113 \times 74242 ? & 2015 \times 2017 - 2016 \times 2016 ? \\ & \text{ Hint: Let } x = 2016.\\ \hline \end{array} \]

- Which problem inspires a sense of accomplishment when completed?
- Which problem would you prefer to work on?

## What makes a great problem

- Interesting learning experience for the audience
- Grabs people's attention and encourages them to try
- Tempting to get started, meaningful to spend time on
- Targeted at the right audience
- NOT long winded with unnecessary directives
- Has a relevant image / visual presentation

- Which problem are you more likely to get started on?
- Which problem is easier to understand the setup?

Which problem statement is easier to understand?

Which directive is more engaging?

## How to get start writing popular problems

As you are working on a problem, did you learn some cool fact while solving it? Were you inspired to think about something similar and discovered something new? If you, you can post it as a follow-up to the problem!

If you just learned a theorem, what are some ways that applying it can simplify that steps that you had to do in the past? Others would be equally excited about seeing a better and faster way to approaching the problem.

While trying to solve a problem, you discovered that if you frame it in another way or give other relevant information while also removing some existing information, then the questions become much interesting!

I) In a similar problem that I posted earlier, I discovered that I can obtain the answer by L'Hôpital's Rule. However, after some digging, I discovered that we could generalize this question to a more powerful and interesting question. Try to solve the following question as well!

\[ \begin{array} { | l | l | } \hline \text{ Old question } & \text{ Your new question } \\

\text{What is } \displaystyle\lim_{y\to0} \dfrac{\sin y}y & \text{True or false: } \lim_{z\to0} \left( \dfrac{\tan z}z\right)^{10} = 1 \\ \hline \end{array} \]II) In a similar problem, I've written up a long tedious solution involving coordinate geometry. Now's here's my follow-up question, but try to solve it by applying this cool theorem that I just learned: Ptolemy's theorem!

\[ \begin{array} { | l | l | } \hline \text{ Old question } & \text{ Your new question } \\

\text{Cyclic quadrilateral has side lengths} & \text{If the product of the measures of diagonals is 10,} \\ \text{ 1, 2, 3 and n in that order.} & \text{ what is the sum of the product } \\ \text{What is the possible value of n?} & \text{of the measures of the pairs of the opposite sides?} \\ \hline \end{array} \]III) In my previous problem, I gave a standard constraint which makes the problem-solving technique pretty straightforward. However, if I were to change/remove some of these constraints, the problem becomes much tougher to solve. Come and give this one a try as well!

\[ \begin{array} { | l | l | } \hline \text{ Old question } & \text{ Your new question } \\

\text{If } x+y=1, x^2+y^2=2 & \text{If } x^2+y^2=2 , x^3+y^3=3 \\ \text{then what is } x^3+y^3 ? & \text{ Find all possible values of } x+y .\\ \hline \end{array} \]

**Cite as:**What Makes A Good Problem?.

*Brilliant.org*. Retrieved from https://brilliant.org/wiki/what-makes-a-good-problem/