Adept Solution Writer

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: $\begin{aligned} 44 \times 5 + 68 \times 5 & = (a+b+c+d+e) + (e+f+g+h+i) \\ & = (a + b + c + d + e + f + g + h + i ) + e \\ & = 9e + e = 10 e. \end{aligned}$ 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 be a, b, c, d, e, f, g, h, i.
2. The middle number is e.
3. Since the average of the nine numbers equals their middle number e, 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 the five largest values (e, f, g, h, i) equals 68, 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 is 504.

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.