**The 1st stage of writing a solution : Apprentice**

Writing up a solution can be scary. You stare at a blank screen and wonder: What do I need to write? How much do I need to write? Where do I start? Who is reading this? Why am I writing this?

There is no "correct" answer to these questions. By writing up your solution, you are explaining to someone else the thought process which you used to solve the problem. Just as there are many ways of approaching a problem, there are also many ways of writing up a solution. As you practice writing solutions, the experience you gain helps you get a better sense of how to write a good solution. There isn't an "absolutely the #1 best solution", but there are numerous great solutions that are waiting to be written.

## Guidelines

Here are some guidelines for an Apprentice:

**1. Use words to explain. Use complete sentences, correct grammar, punctuation and phrasing.**

Remember that you are trying to express your ideas to others, and it is very hard to do so through interpretative dance. Use complete sentences and proper phrasing to explain what you are doing.

\[ \boxed{ \text{ Understand you, if Yoda-speak you do. Done easily not. } } \]

**2. Check for careless mistakes.**

Always double-check your math, and reduce errors. Mistakes happen all the time, and are only a problem if you do not catch them. If arithmetic mistakes carry through your work, it could likely result in no one believing or understanding your solution.

\[ \boxed{ \text{ Do not be the boy who cried } 2+2=5. } \]

**3. Check that your mathematical statements display correctly.**

In the preview box, look at your equations again and check that they appear as you want them to. If an equation is too long, you might want to split it up.

Often, using the \(*\) sign for multiplication instead italicizes the surrounding text. For example, "\( 2 ^*2 = 4 \)" appears as "2*2 = 4*". Instead, use a capital X to denote multiplication, as in \( 2 X 2 = 4 \). You can avoid this by placing your equations with using brackets, as in \( \backslash ( 2 ^* 2 = 4 \backslash) \), which would appear as \( 2*2=4\). This is known as LaTeX, which allows you to write beautiful equations.

\[ \boxed{ \text{Taxi is different from } + \alpha \times i. } \]

**4. State the numerical answer clearly.**

Remember to include the final numerical answer, and not just the steps that you took. This makes it much easier for your audience to understand your proof as a whole, and indicates to them how you arrived at your solution.

\[ \boxed{ \begin{array} {l l } \text{When telling a joke, remember to deliver the punchline.} \\ \text{Do not leave mid - } \\ \end{array} } \]

**5. Ensure that all necessary parts are included.**

If your solution used multiple steps, remember to show all of your working. Skipping steps to 'save time' will result in others not understanding what you did, and waste all of their time.

\[ \boxed{ \begin{array} { l l } \text{ Did you hear about the wooden car with the wooden wheels and the wooden engine?} \\ \text{ It wooden go. } \\ \end{array} } \]

## Examples of solutions

Now that we have outlined these guidelines, let's look at a few examples. Consider the following problem:

The quadratic expression \( x^2 - 10x + 50 \) can be rewritten as \( (x-a)^2 + b \). What is the value of \( a +b \)?

Let's look at the following solution:

Solution 1:\( x^2 - 10 x + 50 = x^2 - 2.5x + 5^2 + 20 = (x-5)^2 + 25 \).

\( a = 25, b = 5 \).

How can we use the above guidelines to help us improve this solution?

**Guideline 1:** While we may (eventually) figure out the exact thought process of the equations, adding some words to explain what you are doing is immensely helpful. We are only able to read what you wrote, and not what you are thinking in your head.

**Guideline 2:** There is an arithmetic mistake made in the first line, though it was corrected at the end. Can you spot it?

At the last step, the wrong values of \(a\) and \(b\) are given. Remember to check that you wrote down what you were thinking.

**Guideline 3:** Saying \( 2.5 x \) can be confusing, since \(2.5 \) often refers to the decimal value of \( \frac{5}{2} \), while the author most likely meant \( 2 X 5 \).

**Guideline 4:** What is the final answer?

**Guideline 5:** You should explain how we arrived at the values of \(a\) and \(b\) (namely by comparison). It is also helpful to explain that the initial line was obtained by the technique of Completing The Square.

Now, let's compare this to the following solution (of the same problem):

Completing the square is the technique.

\(x^2-10x+50\) can be expressed as \(x^2-10x+25-25+50\), or \(x^2-10x+25+25.\)

(The \(25\) came from \((\frac{-10}{2})^2\), which is the desired \(c\) term of the perfect square.)

Simplifying, \((x-5)^2+25\) is the desired result. \(a\) and \(b\) are \(5\) and \(25\), so the answer is \(30\).

This is a great solution. It is easy to understand and follow, and provides explanations for each of the steps taken. There are no mathematical errors, the equations display nicely due to using LaTeX, and the answer is clearly stated.

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 Journeyman.

Note: You can now view Latex codes by hovering over the equation. Read Seeing actual \(\LaTeX\) for more details!

## Comments

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TopNewestThanks a lot. . But I find it difficult to write solutions.. Considering I'm using my android phone 😄 but really. . Its a great experience... Thanks a lot with the guides..

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Use the desktop version :) Think about how you want to present the solution, and then type it up at home.

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So how does one upload images from one's android phone on the app? I know the legendary one says something about it but i did not really follow that.... can someone please explain it to me??

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A great guide! I will be using these guidelines with my students. Thank you Calvin!

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Mr. Calvin LIn thank you very much for guidig me .

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I was so happy that I was going to write my first solution today but I never knew what it will turn out to be... It was so complex and more than what you call "scary".

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It's an experience that everyone goes through. It's like swimming, where you have to take the plunge into the pool. It can initially be scary, but you will soon be able to take it in stride and enjoy being able to swim.

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Thanks for your advise, sir. Today I wrote another solution and it went well. I guess I'll be writing good solutions to even more complex problems soon.

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Thank a lot sir. I will try to follow ur guidelines

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I wrote the most complex answer I had ever written in my life (it's not too long, I'm still 13) in the 100th problem of the Summer Challenge. I wonder @Calvin Lin or anyone can you check if what I wrote is considered a "good solution"?

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How do I ask questions sir??

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How do i ask questione??Like even I have some doubts in science

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Just realized how hard writing an answer is! Anyways will keep doing :P

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Keep at it!

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Thanks a lot for the information.It's really help me to write the answer.

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thank you very much...

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Thanks sir but their are some bugs which has to be fixed. I use brilliant uncompfortable in browser by default somemistake it cannot understand...you may look their... This website or organizer should have their public suggestion or report box to fix the bug or issue concerne with website and other managing.

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Answering

fromoptions:Suppose, the older one is a boy i. e. he has told a lie that means he has no younger sister.

So, the truth may be 1. He has an older sister 2. He has a younger brother

Now, let the later one is a girl i. e. she speaks the truth that means she has no brother. It's contradictory.

Hence, let the later one is a boy i. e. he tells a lie. It means they are same in gender.

It fulfills the all conditions.

So the two are brothers.

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Thanks a lot.

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Dear sir I am thankful to your comments. I was travelling that much I could that moment. Also I am not gadget saVvy.so it may take some time get used to this. .thanks for comments .

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It’s really wonderful that you prepared these notes, Calvin. I’m at the pre-Apprentice level right now, but I will now launch a quest to achieve Legendary status. I know what you are talking about: I have seen some Legendary solutions posted, with brilliant LaTeX formatting as well. [Fortunately I already have LaTeX skills, just not demonstrated yet on Brilliant.org.]

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That's great! Solution writing (or more generally, clear communication) is an important skill to develop, and takes time and effort to improve.

I look forward to seeing your Apprentice solutions.

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Thanks!!!!!!!:-)

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Mr. Calvin i am very thankful to for guiding me.

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Thank you very much, Mr. calvin, for your advice. I would surely remember these. Thank you very much once again. :)

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Its really scary, I might commit mistakes😅 I am just an ordinary math teacher

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Don't worry! The community will help you out, and offer advice for improvement.

You miss 100% of the shots that you do not take :)

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i have given a solution . Please inform whether the way of explanation is correct.

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The community will interact with the solution that you wrote, and post comments if they are uncertain, and vote up your solution if they like it.

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