**The third stage of writing a solution : Adept**

Review the guidelines for a Journeyman.

Now that you have written several solutions, you are aware that there are many different ways of saying the same thing. Just like writing poetry, some people write haikus, limericks or allegories, while others write ballads, epics or Shakespearean sonnets. There are many different ways to express yourself, and to make the best choice, it will help to keep your audience in mind. Sometimes a short haiku suits the bill, while other times an epic ballad is called for.

Let's face it, we all want others to "Vote up!" our deserving solutions and admire the marvelous thoughts we have shared with them. In order for that to happen, we have to understand who will be reading your solution and what would motivate them to vote on our solutions. This will help us adapt our writing in a manner that is pleasing to them, to keep them engaged and involved.

## Guidelines

Here are some guidelines for an Adept:

**1) Identify your audience.**

This is easy. On Brilliant, your audience is (often) other members who are of a similar mathematical ability as you are. Include enough detail so that you are clearly convinced about the truth of the statement, since your audience members are not mind-readers. You should not assume that they have already solved the problem, though they may be able to fill in small gaps in your solution. If you write solutions to problems that are not in your level, please tailor your solutions accordingly.

\[ \boxed{ \begin{array} { l} \text{All the world's a stage} \\ \text{And all the men and women merely players.} \\ \end{array} } \]

**2) Explain your thinking step by step.**

Your proofs will now have to link several different ideas together, and you have to ensure that each statement is well supported. Place yourself in the shoes of your reader and ask: “Why is this true? Is there any reason why it can be false?” This can help you ensure that each argument is justified. Often, the order in which your ideas are presented is extremely important, and crucial to helping someone understand how it all fits together.

\[ \boxed{ \begin{array} { l} \text{You put your right hand in,}\\ \text{You put your right hand out.}\\ \text{You put your right hand in,}\\ \text{And you shake it all about.}\\ \text{You do the Hokey Pokey,}\\ \text{And you turn yourself around.}\\ \text{That's what it's all about!} \\ \end{array} } \]

**3) State the approaches that you are using.**

Simply stating the type of proof you are thinking of can help your reader identify the checkpoints. If you tell them that you are "Proving by contradiction", they will keep an eye out for the contradictory statement, which can reduce the confusion that they might have. Likewise, if you tell the reader that you are "proving by induction", they will watch out for the base case and the induction hypothesis.

Just as stating the technique tells your reader what is happening soon, stating the approaches gives your reader a broader sense of what will be done, and how they will arrive at their destination.

\[ \boxed{ \begin{array} { l} \text{ One does not simply walk into Mordor. } \end{array} } \]

**4) Define your variables, use consistent notation.**

By stating your definitions at the start, you are making it clear to the reader what the terms mean, which they can easily reference in future. Rather than “Consider the number which when added to anything doesn’t change it. When we add this number to the number \(n\), since it doesn’t change it, hence we get \(n\)”, instead say “\(n+0 = n \)”.

\[ \boxed{ \begin{array} { l } \text{ Imagine if Harry Potter was played by a different actor in all the movies... } \\ \text{ It would be hairy. } \end{array} } \]

## Examples of solutions

Having outlined these guidelines, let's look at a few examples. The following solutions were written up for this question:

Arpit's awesome averages:

Nine numbers are written in increasing order. The average of the nine numbers is the middle number, the average of the five largest values is 68, and the average of the five smallest values is 44. What is the sum of the nine numbers?

Let’s look at the following solution:

Solution 1:We find that the middle number is \( \frac{ 68 \times 5 + 44 \times 5 } { 10} = 56 \) since it is the average of all the numbers. Hence 56 is the middle number.

It is the average of the largest and smallest number, so we have to add it back eight times and get \( 56 \times 9 = 504 \).

How could the above guidelines help us to improve this?

**Guideline 1:** The extreme brevity of the solution definitely doesn't provide enough detail for another person who hasn't figured out the solution to understand the solution easily. A short solution that skips most of the steps is worse than no solution at all.

**Guideline 2:** While the author understood what he was doing, most of it was lost in the writeup. Each of the paragraphs doesn't seem true at first glance, and there isn't enough supporting evidence to explain why this is so. The reader needs to dig around and understand the problem more, in order to read the solution.

How would we fill in the gaps? The first paragraph could have been written as:

If we add the first 5 numbers and the last 5 numbers, we will get all 9 numbers, with the middle number repeated. Since the average of all nine numbers is the middle number, hence this sum is equal to 9 times of the middle number plus itself, or 10 times of it. Thus the middle number is \( \frac{ 68 \times 5 + 44 \times 5 } { 10 } = 56 \). I leave it to you to tease out the meaning of the second paragraph.

**Guideline 3:** Stating that the main goal is to find the middle number, will help explain why we care about the algebraic manipulations that are being done.

**Guideline 4:** If we started off by naming the nine numbers \( a, b, c, d, e, f, g, h, i, \) then we can easily refer to these variables. The first paragraph could be clearly written as:
\[ \begin{align}
44 \times 5 + 68 \times 5 & = (a+b+c+d+e) + (e+f+g+h+i) \\
& = (a + b + c + d + e + f + g + h + i ) + e \\
& = 9e + e = 10 e. \end{align} \] Hence, \( e = \frac{ 68 \times 5 + 44 \times 5 } { 10 } = 56 \).

Remember to use the previous guidelines that you have learnt! For example, it is very helpful to explain why \( 44 \times 5 = a+b+c+d+e \), by reminding your reader about the given conditions.

Don't forget to use the guidelines that you learnt as a Journeyman!

Now, let’s compare it to this solution:

Solution 2:By Raphaël N.

I tried to make it as detailed and simple as I could

1. Let the nine numbers in increasing order bea, b, c, d, e, f, g, h, i.

2. The middle number ise.

3. Since the average of theninenumbers equals their middle numbere, then:

\[\frac{a+b+c+d+e+f+g+h+i}{9} = e\] \[a+b+c+d+e+f+g+h+i=9e\] 4. The average of thefivelargest values(e, f, g, h, i)equals68, which means that \[\frac{e+f+g+h+i}{5} = 68\] \[e+f+g+h+i = 68 \times 5 = 340\] 5. By following the same steps with the five smallest values, we get : \[a+b+c+d+e = 220\] 6. By adding the last two equations : \[a+b+c+d+2e+f+g+h+i=560\] 7. By combining this equation with the first one : \[9e+e=560\] \[10e=560\] \[e=56\] 8. The sum of the nine numbers equals 9e, so we can get it by multiplying 9 by 56. Or, to make things easier, we can just subtract 56 from 560. The answer is504.

I like this solution because it clearly lays out what is happening, and shows us how to get there. The individual steps are laid out, and there is no chance of misunderstanding the solution. We see how much easier giving names to these 9 integers helps us in reading the statements. It explains how, and where, we use the facts about averages, and gently guides us towards the conclusion.

You can view this solution (and the problem) by clicking on the hyperlink Solution 2. If you enjoyed this solution, remember to vote it up!

Aspire to be better. Proceed on and be a Magus.

## Comments

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TopNewesthow do you attach a diagram to your solution? – Krishna Ramesh · 2 years, 5 months ago

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@Krishna Ramesh @Niranjan Khanderia To attach an image, you first upload it to an image hosting site like Imgur or Flickr, and then use the following code:

For example,

gives

Calvin

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– Himanshu Arora · 2 years, 4 months ago

And how do you include the code blocks? Like the one in your comment after "For example," ..Log in to reply

– Niranjan Khanderia · 2 years, 5 months ago

Thanks. Can there be any other Microsoft, Google site instead of " image hosting site like Imgur or Flickr " ?Log in to reply

I used Imgur and Flick as examples as they are the most common / easy to use. – Calvin Lin Staff · 2 years, 5 months ago

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– Niranjan Khanderia · 2 years, 5 months ago

Thank you so much.Log in to reply

– Krishna Ramesh · 2 years, 5 months ago

all right thanks a lot!!Log in to reply

How many levels of solution writing are there? How do I identify what level I am on? – Nanayaranaraknas Vahdam · 2 years ago

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Each "level" represents being aware of a certain aspect, and provides a way for you to improve your solution writing ability by letting you know what else to consider / include in your solution. – Calvin Lin Staff · 2 years ago

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– Nanayaranaraknas Vahdam · 2 years ago

What are the names of the five levels?Log in to reply

Solution Writing Notes. – Calvin Lin Staff · 2 years ago

Check outLog in to reply

I have not understood that how to attach diagram to a soln. – Sargam Yadav · 3 months, 1 week ago

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How can I write "angle ABC"?

Calvin edit: \( \angle ABC \) – Debarya Das · 5 months, 2 weeks ago

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`\angle ABC`

. – Calvin Lin Staff · 5 months, 2 weeks agoLog in to reply

How to upload a photo by clicking it??? – Ayush Pattnayak · 7 months, 1 week ago

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When you click the image button in the formatting toolbar, you will be asked to select an image to upload. – Calvin Lin Staff · 7 months ago

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– Ayush Pattnayak · 7 months ago

Thank you,sir.Log in to reply

whenever i raise some thing to a number it appears in the next line...

for example ; when i want to write in a single line \ [ (a+b) ^ {2} \ ] = \ [ a ^ {2} \ ] +2ab+ \ [ b ^ {2} \ ] .................(I HAVE ADDED SPACE IN B/W TO SHOW HOW I WRITE) it appears as : \[(a+b)^{2}\] = \[a^{2}\] + 2ab + \[b^{2}\]

What to DO now??

Edit: \[ ( a + b)^2 = a^2 + 2ab + b^2 \] – Sahil Jindal · 1 year, 9 months ago

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I've edited your equation, and removed all the intermediate brackets, and shown you how to get it to display together. You can edit your comment to see the latex, or hover your mouse over the image. – Calvin Lin Staff · 1 year, 9 months ago

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– Sahil Jindal · 1 year, 9 months ago

One more query Sir : I want to write something in combinatorial form ( for instance I want to write n on top of C and 2 on bottom).... How to do...?Log in to reply

– Calvin Lin Staff · 1 year, 9 months ago

{ 2n \choose r + 1 } gives us \( { 2n \choose r + 1 } \).Log in to reply

– Sahil Jindal · 1 year, 9 months ago

Thank You so much Calvin Sir :)Log in to reply

In some of my solutions I have tried to write square roots, but have had no success. I even followed the formatting guide, but I don't see what I'm doing wrong. For example: \sqrt{2). I've also had a similar problem with exponents.

Edit: \sqrt{2} produces \( \sqrt{2} \) – Curtis Clement · 1 year, 11 months ago

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– Niranjan Khanderia · 7 months ago

If there is only one letter, there is no need of { }.However if it is an alphabet, keep a space. Keeping a space always is a better practice. \(\sqrt{2}\) and \(\sqrt2\) appear the same. Just see how I have written in Latex.Log in to reply

I have edited your comment to show you how to type \( \sqrt{2} \) by using \sqrt{2} – Calvin Lin Staff · 1 year, 11 months ago

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