The final stage of writing a solution: Legendary
Review the guidelines for a Magus.
Congrats on making it thus far. You should be able to engage your audience with a well presented solution. In this post, we explore various ways of adding that additional oomph to help you stand out from the crowd and gain additional glory. By taking time to consider different ways of writing up your solution, you can ensure that your solutions are clear, yet brief.
It is interesting to present different facets of the problem, especially if they can yield greater insights into more general cases. Be warned: This can easily overpower your solution and make it hard to decipher what you are actually saying. Do not try this until you are extremely confident of your solution writing ability.
Here are some guidelines for a Legendary Solution Writer:
1) Take 5 minutes to consider how to present your solution.
Once you have solved the problem, spend some time to look over your work. You are likely overwhelmed, and can benefit from clearing your head. Set your pen aside and review what you have come up with.
What is the best way to approach the solution? While a plan is important, planning the plan can be more valuable. The solution should provide a clear description of the steps taken, and your plan should be a walk in the park, and not a walk in the maze.
Are there various parts that can be improved? If you have a long ugly algebraic expression, is there a way to simplify the calculations? Using a good substitution of variables might make it easier to wade through the mathematical muck that others have to read.
2) Optimize for clarity first and brevity second.
This is obvious, but needs to be stated. If your solution is unclear, you should spend a few lines to tidy up the presentation. However, do not be too long-winded and belabor every minute detail.
3) Include motivation.
If your ideas are complicated or unexpected, you should have an initial discussion that explains what led you to think about these ideas, and why they are natural in the setting of the problem. This will make it easier for your reader to understand how they themselves can come up with a similar solution in future, and they are much more likely to thank you for enlightening them.
If you are using a complicated algebraic expression, explain why you thought of it in the first place, and state the important features. If it is similar to another well-known situation, you can mention that it provided the motivation for you.
4) Generalize the problem, if possible.
For problems which are a special case of a more general class, it can be useful to present the proof to the generalized problem, and state that we are looking at a certain special case. This will often allow you to illuminate more insight into the underlying theory and structure presented in the problem.
Even if you suspect that a generalization exists, you should mention it even if you are unable to prove it. This will provide some food for thought for those who have completely digested your solution.
Examples of solutions
I will present several proofs which have inspired the Brilliant community, and point out how they will benefit generations to come. You can view their solutions by clicking on the hyperlinks. As always, if you enjoyed the solution, remember to vote it up! One day, I hope to add your solution to this list.
1) Anqi L. - Solution to Sequential Roots.
Great explanation of your first instinct. I knew that was what most people would attempt too, and get stuck there. Glad to see that you can push through, and even provide a reasonable explanation for the choice of 'strange' coefficients used later.
2) Sreejato B. - Solution to Irrationally irrational irrationals.
He identifies the key result needed in the problem and stated it as a separate initial claim. Even if you are unable to understand the proof of this claim, you can jump to just after QED and see how the claim helps us approach the rest of the problem. The diagram is pictorially helpful too :)
Read the explanation of the motivation in the comments. This strips away some of the mystique of the (seemingly random) equations, and demonstrates how you can also come up with a similar awe-inspiring solution.
Read the comments for the generalized solution and understand why the number 20 isn't special. This comment presents a higher level of thinking, and explains why \( m^2 + m - 1 \) is so crucial.
Effective use of illustrations and places the solution in a larger context of physics, rather than just a list of equations. Explaining about work done, etc. provides the little extra details that make a solution worth reading to understand the general concepts involved, instead of just the specific steps to solve the problem.
Now that you've seems some excellent solutions, go forth and multiply!