# Calvin Testing 6

<< Calvin's Version>>

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

## Identifying your target audience

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.
- It 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: 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.

## Choosing the best presentation

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.

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.
- Multiple Choice options 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:

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!

<<**Rohit's Version**>>

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 elaborates on what makes a problem great.

There are two aspects of a great problem 1) The problem content 2) The presentation

Let us look at them in details

## The problem content

The problem content contains the core idea of the problem, and how you want to ask it. There are two pillars which make a great content.

**1) Potential idea:** <<Need some elaboration>>

Counter intuitive/ astonishing result/ Clarifying a misconception: People grow more curious if they come across a problem whose result is counter intuitive. These problems are generally more liked by the community.

A daily life scenario or real world application: It is always great to see the science/maths working in our day to day life.

Thought provoking: The problem should not be a direct formula or memorization but should force the audience to apply their knowledge or thoughts.

**2) Fitting directive:** Ask the problem best suited for the target audience. Brilliant has categorized the problems in three categories. 1) Easy 2) Medium 3) Hard. Each category has specific needs to meet.

Easy: The easy problems should not involve a lot of calculations and scrap work. They should not contain complicated technical jargons that basic level users might not know.

Intermediate: These problems may involve applications of slightly difficult concepts and some scrap works.

Hard: The problems in this category are the ones which show a complex situation which may require the use of multiple concepts. These problems are not easy to crack and require a lot of thought process to solve the problem.

## The presentation

The presentation is how you can make your problem more appealing and stand out in the crowd of problems.

**1) Phrasing:** People will read what we finally write in the problem and not what we think or intend with the problem.

Short and uncomplicated: Remove unnecessary content that does not relate to the problem.

Split paragraphs: Writing a lot of information in one paragraph is difficult to comprehend. One should split the paragraphs having distinct information keeping the similar information in one paragraph.

**2) Theme/motivation:** A problem can be greatly enhanced if a motivation is provided behind the series of events that occur in the problem. A problem becomes more engaging and fun to read with a theme. Themes like Tom chasing Jerry or Knight telling truth and knaves telling lies greatly enhances the problem.

**3) Imagery** A descriptive image which aids the problem statement helps the reader to quickly understand the problem.

**4) Options**

Short and uncomplicated: Lengthy options are generally a turn off for many readers. They find it very cumbersome to comprehend and most likely not engage with the problem.

Meaningful: Random options sometimes can degrade the beauty of the problem. The dimensionally incorrect or completely vague options should not be added to the problem. It is better to have two or three meaningful options than to have four or five vague options.

## Example 1

Let us now go through an example and figure out if this is a great problem, if not, then how can we improve it.

The problems with this write up

Convoluted phrasing: <EXPLANATION NEEDED>

Not imagery: <EXPLANATION NEEDED>

Now let us see how we can improve this problem. Consider the following write up.

Benefits of this write up

Imagery: <EXPLANATION NEEDED>

Short and to the point: <EXPLANATION NEEDED>

## Example 2

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 \(\lim_{t \to \infty} r(t)\).

The problems with this set up is

Boring and Unrelatable: <EXPLANATION NEEDED>

Difficult to comprehend: <EXPLANATION NEEDED>

Now consider the following set up.

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 \(\displaystyle \lim_{t \to \infty} r(t)\).

Benefits of this write up

Engaging and Curious: <EXPLANATION NEEDED>

Imagery: <EXPLANATION NEEDED>

<<**Pihan's Version**>>
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.

## Attributes of a great problem

While there are many factors in figuring out what make a great problem, we have categories them into 4 types:

#### 1) Problem quality: Do you have a innovative idea that you want to share?

**(1.1) Explicit:** All the necessary information has been provided. There's no trickery involved.

It wouldn't nice to state a term that is not properly defined, or have multiple interpretation. Otherwise, people might get the wrong meaning.

**(1.2) Learned:** What do you gain out of trying the problem? You actually learned something out of it / sense of accomplishment.

We want to show that this is not an ordinary memorization with no purpose. But by actually engaging with it, the readers are able to learn something beneficial out of it.

# (2) Problem format: How can you ask that question?

**(2.1) Directive:** Choose the right kind of directive you want to ask. There's a lot of similar looking directives to convey your point.

What's the most exciting question that you can ask so that it stirs their imagination the most?

**(2.2) Suggestion:** Does adding a hint help steers the readers to the right direction?

You have already asked a question. But there could be multiple ways to solve it, maybe it's still worthwhile to hint that there's a creative way to solve it?

# (3) Phrasing: How make your problem more readable?

**(3.1) Neat:** All the information provided are presented clearly in a proper formatting.

It's easy to grasp all the information if it was presented a clean manner. Maybe join/partition some of the information ,or keep the similar information together?

**(3.2) Candid:** Message is conveyed straight to the point.

There's no need to beat around the bush. No need for redundancies.

# (4) Imagery: How to make it grabs people's attention?

**(4.1) Relatable:** Symbols alone can look very abstract. Can this be made to be a more relatable scenario?

While it can be daunting to look at all the math symbols, how can we rephrase them in such a way that it's immediately more understandable in layman's point of view / daily life.

**(4.2) Illustrative:** Does an image/example helps elucidate your message?

Words might explain what's going on, but an image/example helps reinforces/clarifies the message.

## Example of average versus great problem

Let's compare the 2 write-up of the same setup below. By checking these attributes listed above, we are able to figure out which version better presented.

Version A | Version B | |

Write-up | ||

Explicit? | Yes | Yes |

Learned? | Yes | Yes |

Directive? | Yes | Yes |

Suggestion? | No | Provided, but not compulsory |

Neat? | No | Yes |

Candid? | No | Yes |

Relatable? | No | Yes |

Illustrative? | No | Yes |

As you can see, by comparing these 2 versions, we managed to check off (much) more attributes for Version B, this suggests that Version B's write-up is better (and an improvement of) than Version A.

**Cite as:**Calvin Testing 6.

*Brilliant.org*. Retrieved from https://brilliant.org/wiki/calvin-testing-6/