The fourth stage of writing a solution : Magus
Review the guidelines for an Adept.
Have you ever been trapped in the world of a good book, where page after page weaves vivid imagery before your eyes? And when you finally put the book down, you realize that it is morning and you have school (or work) in an hour? There are many different literary genres that capture our attention, ranging from drama to fantasy, horror to mythology, sci-fi to autobiographies. Similarly, good solutions come in many different flavors, but all share the ability to capture and hold the attention of the audience.
If your solution is convoluted and involves multiple parts, you are more likely to lose your audience along the way. By providing a clear path to follow, it is much more likely that your reader will become immersed in your writeup and emerge with an understanding of your insights.
Here are some guidelines for a Magus:
1) State your plan.
Stating the plan to the reader helps them understand your overall approach, especially for problems involving several parts. This way, if they have already done the initial work, they may skip ahead and read the relevant ideas.
Using conjunction words like "First", "Next" and "Finally" can help the reader identify the various checkpoints of your argument. Numbering your points also provides more structure to your writeup, just like in these guidelines.
2) State the important element / aspect of your proof.
Sometimes, clearly stating a Theorem (or Lemma, or Proposition) that you will subsequently prove, can help the reader easily compartmentalize your proof. Even if they may not understand the specific proof to the Theorem, they can still understand the rest of your argument.
Note that unless it is indeed a well-known theorem, preferably with a name given to it, you should include a proof (or at least a quick sketch of one). It is also a good time to check that the conditions of the Theorem are indeed satisfied.
3) Start at the start, end at the end.
It will be clearer for you to start your explanation from the beginning, instead of working with the end result. Think about how to provide instructions from the cinema to the library. If you say that “The library is on Cottage Street, which you can get to by making a left turn on Bay street after you have travelled on it for half a mile. To get there, you would want to make a right turn at the third intersection”, it can be confusing even though your directions are correct. Instead, it is better to say “From here, walk down to the third intersection and turn right onto Bay street. After half a mile, make a left turn onto Cottage Street, and the library is on your right”.
There are rare cases in which presenting a solution from the end is a better approach. (This may arise when the motivation behind the solution becomes much more important than the solution itself.) In general, I would suggest that you avoid doing so.
4) Use proper spacing.
Use lines and paragraphs to help separate the various ideas. Label your equations and formulas, especially those you will be using repeatedly. As a rough guide, a complete argument should take one paragraph.
It is sometimes helpful to place equations on their own line. Avoid having several lines of equations within a paragraph, especially if they involve multiple steps of reasoning.
5) Check that your LaTeX is properly typeset.
LaTeX is extremely sensitive to the commands that you input, and how you decide to use your brackets. If you forget or misplace a bracket (especially for exponents and indices), your equations will display incorrectly. Worse still, forgetting to close a bracket may result in an error, and your beautiful equation will not display. Instead, the raw code will be shown in a box, like so:
If you have to spend 10 seconds trying to interpret the latex syntax, you are likely to skip over understanding the proof. The same is true for your audience.
Having outlined these guidelines, let's look at a few examples. The following solutions were written up for this question:
Aram’s real solutions: The sum of all real solutions of the equation can be expressed as , where and are positive integers. What is the value of ?
Details and assumptions: For this question, the domain of the square root function is non-negative numbers.
Let’s look at the following solution:
The answer is , which is the sum of 9, 4 and 1. The sum of the roots is 6, which can be written as . The roots are and . The equation is equivalent to which follows by manipulating the given equation.
How could the above guidelines help us to improve this?
Guideline 1: It is unclear what the initial plan is. Why is the equation with roots equivalent to a nice polynomial equation? Even if it is, how was this obtained? There are too many steps that are skipped, and even if the statement is true, it would look highly suspicious.
Guideline 2: If the solution stated that the equation was squared, the writer would realize that other roots could have been introduced (remember to check the conditions!) during this procedure. Stating the important elements of your proof reminds you of what you are doing, and why it works (or doesn't).
Guideline 3: This solution is completely written in reverse, and even if the statements are valid, the lack of initial justification makes it hard to believe.
Guideline 4: The are two main ideas involved here. First, we want to manipulate the equation to a form that we understand. Second, we want to determine the roots which are valid and find their sum. It is best to separate these into different paragraphs.
Guideline 5: As you can see, when LaTeX isn't properly typeset, it can hinder you from reading the proof easily. Did you realize that one of the roots is incorrectly stated? The first root should have been . The final equation is wrong, and should have been .
Don't forget to use the guidelines you learnt as an Adept!
Now, let’s compare it to this solution:
Solution 2: By Anqi L.
Let us rewrite the following way:
Step 2 : We square both sides to get -
Step 3: We open up the brackets to get -
Step 4: The idea here is to factorise the above equation. Well to do this, we can remark that , and (coincidentally) . Now this motivates us to write:
Step 5: Now we need to solve:
A reasonable prediction would be 4 roots. However due to the details and assumption, we can in fact directly obtain from the question the important condition that . This implies that for:
(a) We need We get or We can reject the first root since it is less than 2.
(b) Similar analysis as the above case shows that we can reject the second root since: . So for this case we get one root namely: .
In conclusion the answer is : . Hence _9+2+3 = 14 _
This solution indirectly states that we want to rewrite the solution into a reasonable form, and then understand the given roots. It explains the important aspects in both parts, especially in the latter where it states "A reasonable prediction would be 4 roots. However due to the details and assumption, we can in fact directly obtain from the question the important condition that " . Each main idea occupies it's own line, and the case-by-case argument 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 Legendary.