# Andy Sandbox

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

#### Contents

## Identify the audience

At Brilliant, we have different **levels** (1 through 5) of problems which indicate how difficult the problem is. Although it's good to be aware of how difficult your problem is, it's more important to think of a **target audience**. We organize our problems into three target audiences: **Basic**, **Intermediate**, and **Advanced**.

BasicBasic users want to learn and challenge themselves with new concepts in math and science, but they don't want to have to learn obscure terms, theorems, and definitions before they get started. Basic problems should neither involve tedious calculations nor contain unexplained technical jargon. It should be easy to understand what is asked for and easy to get started on the problem without any special knowledge. It's often best to provide multiple choice responses so that the Basic user has a general idea of what the answer could be.

[[Explain why basic is not about facts or simple calculations. It's about problem solving]]

The following problem is great for Basic users because it does not require any special math knowledge to get started and it is easy to understand what the goal of the problem is.

[[Suggestions on how to modify the problem for a basic audience]]

IntermediateIntermediate users want to use their prior knowledge of math and science to challenge themselves. Intermediate problems may involve a single application of a challenging concept (up to courses offered in secondary school), or they may combine easier concepts in creative and challenging ways. Multiple choice responses are less prevalent in Intermediate, although sometimes they are appropriate.

The following problem is great for Intermediate users because it combines several concepts (probability and geometry) in a creative and challenging way.

[[Suggestions on how to modify the problem for an intermediate audience]]

AdvancedAdvanced users want to test the limits of their already extensive knowledge of math and science. Advanced problems showcase a complex situation which requires the use of advanced (from courses offered in undergraduate school and above) concepts. Just as with intermediate problems, multiple choice responses are not typically used, but they are sometimes appropriate. Just because a problem is Advanced does not mean you should require several steps of tedious calculations. It's often best to keep the concepts challenging but the arithmetic simple!

The following problem is great for Advanced users because it takes a practical situation and challenges users to apply Calculus concepts to formulate a solution.

Two villages (points B and C) are to receive a train connection to their neighboring city (point A), as shown in the diagram.

In order to save money when building the train route, only the connection with the shortest rail route is selected.

How many kilometers of rails must be laid? Round the result to the nearest integer.

Note: The railroad does not necessarily have to be built on the 30 km segments indicated in the diagram. A railroad switch can be built to allow a railroad to split into different paths.[[Suggestions on how to modify the problem for a advanced audience]]

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

## Make the problem stand out

There are a number of different ways to make your problem stand out, and these are just a few of them.

Represent a daily life scenarioPeople love seeing how math and science are applied to the real world. The following problem is about the forces that hold up magnets on a refrigerator.

Gravity becomes weak in the presence of a magnetic field, so there is no force in vertical direction Friction acts upwards and gravity acts downwards The magnetic force acts upwards and gravity acts downwardsPeople use magnets to hang notes on refrigerators. Despite the downward pull of gravity, magnets do not fall, even though the magnetic force doesn't pull up.

How is this possible?

Clarify a misconceptionPeople are curious when they come across a counter-intuitive result, and want to understand how to think about it. The following problem clarifies a misconception about computing the number of cubes that could fit inside a box.

Yes NoSuppose you want to fill a \(150\times60\times37\) cuboid with \(5 \times 5 \times 5 \) cubes. To calculate how many fit, you divide the two volumes:

\[ \dfrac{37\times60\times150}{5\times5\times5} = 2664. \]

You conclude that given optimal placement, 2664 of the cubes fit inside the cuboid. Is this correct?

Assume that all the faces of the cubes are in parallel with the faces of the cuboid.

Provoke thoughtProblems are fun when they encourage users to apply their knowledge instead of just relying on memorization of formulas. The following problem starts off with a seemingly impossible equation puzzle and then challenges users to come up with a way to make the equation true.

Yes No\[\large \begin{array} & 0 & & 0 & & 0 & = 6 \end{array}\]

Using basic arithmetic operations \((+,\ -,\ \times,\ \div),\) parentheses, and

any other operationson the left side of the equal sign, can you make this equality hold true?

Note: Some of the other operations you might try are the trigonometric functions \(\big(\sin(\cdot)\), \(\cos(\cdot)\), and \(\tan(\cdot)\big)\), factorial \(!\), floor \(\lfloor \cdot \rfloor\), and ceiling \(\lceil \cdot \rceil\). You can put in as many operations as you like, but you can't put in any additional numbers or digits!

Challenge with multiple conceptsA problem can become more challenging if it involves multiple concepts. However, be careful with putting

toomany different steps into a problem because tedious extra steps aren't very much fun. The following problem requires logical thinking, algebra, and a little bit of combinatorics to arrive at the solution.

Garry and Fabiano Garry and Magnus Fabiano and Magnus There isn't enough information to know for certainMagnus, Garry, and Fabiano are playing in an exhibition badminton event. In this exhibition, after two players face off for a set, the winner of the set stays on the court to play the player who was sitting out.

At the end of the event, Magnus has played 9 sets, Garry has played 14 sets, and Fabiano has played 15 sets.

What pair played

set number 13?

## Improve the presentation

[[Rename this section and put it first]]

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:

## Phrasing

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

\(\color{darkgreen} \text{Do}\) \(\color{darkred} \text{Don't}\) Keep it short and uncomplicated. Include unnecessary information that is not relevant to the problem. Organize the information so that the important details and assumptions are easy to find Make it difficult to find relevant information for solving the problem. Identify the crux of the problem and make it the focus. Divide the focus of the problem into different things.

## Theme / Motivation

People will engage with a problem if they find it interesting and relevant.

\(\color{darkgreen} \text{Do}\) \(\color{darkred} \text{Don't}\) Give your problem a relevant theme or realistic situation. Become so focused on thematic elements that you neglect the underlying math or science. Consider providing a hint or example to provide context. Intentionally mislead your audience without fair warning. Keep your target audience in mind as you write up the problem. Force a solution that is tedious or beyond the level of your target audience.

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 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.

## Applying these ideas to improve a problem

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.

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

- Rewrite the question by providing math notations.
- Remove 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.

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

- Providing a pictorial image for people to understand what the description is
- 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 \(u\). \(B\), on the other hand, travels with a constant speed \(u\) 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) \(\dfrac{d}{2}\)

(B) d

(C) 2d

(D) Zero

(E) Infinite

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

- FIguring out how best to present the problem as a real-world/familiar scenario.
- 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) \(\dfrac{d}{2}\)

(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 help going through a problem, comment in the feedback box below!