# Short and simple guidelines to writing a good solution

Just writing up this quick set of guidelines to aid people (including myself) in writing good solutions and proofs.

1) Keep your audience in mind. Show every essential step of your working, so that the reader isn't puzzled by your solution and doesn't have to guess at your motivations; at the same time, don't show every single step, as it may come off as condescending. A good balance makes a great solution.

2) Don't write all of it in equations. Include prose which outlines what theorems/lemmas/results you are invoking. A solution chock full of line after line of equations is hardly a pretty sight. Keep it clean.

3) Still, don't go rambling on. It's very annoying. Leave that for the comments section, where you can append a footnote or two.

4) Use LaTeX. It makes everything easier to read and understand, and clarity is paramount.

5) But don't wrap everything in LaTeX! It's extremely difficult to read. I've seen this a lot on Brilliant. Far too much, actually. It lessens your load, too, so hey! Win-win for both the writer and the reader.

6) Use connectors such as "therefore", "hence", "since", "because", etc to keep the solution cohesive as a whole. They provide a good sense of flow within any proof, which makes it easier for the reader to process.

Other tips:

Keep a good balance of \ [ \ ] and \ ( \ ). As a rule of thumb, a good ratio lies between 2:1 and 1:2. This helps to keep yourself in check, to avoid verbosity by including only important equations/expressions/etc worth wrapping in \ [ \ ].

$\ln, \cos, \sin$ and other special functions are given to you, so use them. I don't know about others, but it's not pleasant to see a stray $tan(x)$ in a solution, personally.

Provide useful links to theorems and results. Good websites for reference include MathWorld and Wikipedia.

Don't squeeze a fraction into a tiny space, eg $e^{\frac{A}{B}}$. It doesn't look good, and it's hard to read. Try $e^{A/B}$ or $\exp(\frac{A}{B})$ instead!

Use lemmas sparingly. Easier questions (mostly) don't require lemmas, but it will be apparent to the advanced problem solver when (and when not) to use lemmas.

I hope that at least somebody out there will be inspired to increase the quality of their proof-writing! I strongly feel that lucidity in writing is always quintessential for any sort of mathematical writing, and so these guidelines serve to help one improve upon that important aspect.

Anything you would like to add? Comment and I will add on to these guidelines.

Note by Jake Lai
3 years, 11 months ago

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

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*italics* or _italics_ italics
**bold** or __bold__ bold
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1. numbered2. list
1. numbered
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Note: you must add a full line of space before and after lists for them to show up correctly
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[example link](https://brilliant.org)example link
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# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in $$ ... $$ or $ ... $ to ensure proper formatting.
2 \times 3 $2 \times 3$
2^{34} $2^{34}$
a_{i-1} $a_{i-1}$
\frac{2}{3} $\frac{2}{3}$
\sqrt{2} $\sqrt{2}$
\sum_{i=1}^3 $\sum_{i=1}^3$
\sin \theta $\sin \theta$
\boxed{123} $\boxed{123}$

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Also, see the Solution guides by Calvin

Staff - 3 years, 11 months ago

@Jake Lai, what is a lemma?

- 1 year ago

A lemma is a micro-theorem that one uses to prove a more interesting theorem. If you are familiar with programming, it is like a subroutine.

Staff - 1 year ago

This is really a good solution. It is a pity that I did not find him when I was at university. It would also helped me save my time, although my solution is as good as this. My friends advised me reliable and without any scam essayexplorer.com, where thanks to a lot of reviews I found freshessays. In fact, I can’t imagine how to live without it, because it saves time for more useful things.

- 3 months, 3 weeks ago