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When I'm thinking about a question, I often find that I've done a lot of roundabout work only to find a neat direct shortcut solution, especially when I'm into a new concept. It's kind of frustrating, why haven't I found out that shortcut at the beginning?

How common is this phenomenon (among working mathematicians)? What is some advice to reduce this phenomenon, I mean, the unnecessary work?

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    $\begingroup$ The answers to your question are roughly. 1) Very. 2) Become more familiar with that concept. Meaning that, in the end, the work is not unnecessary, because now you can see those patterns. $\endgroup$ Commented Nov 6, 2022 at 13:35
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    $\begingroup$ It means that you are learning things by your own efforts. On this site, I see tons of students approaching everything as contest problems, meaning anything taking more than 3 minutes is discarded. Which is frustrating for me. $\endgroup$
    – Will Jagy
    Commented Nov 6, 2022 at 13:35
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    $\begingroup$ But this question, while "meta" in some sense, is not really a good fit for the meta site, because here we should be discussing the use of the main site. $\endgroup$ Commented Nov 6, 2022 at 13:36
  • $\begingroup$ @WillJagy What about taking more than 3 days (spare time, that is)? I'm not asking about those problem in a test or something. I'm asking about reading a monograph and trying to reach a better understanding. $\endgroup$
    – Michael
    Commented Nov 6, 2022 at 13:48
  • $\begingroup$ MSE Meta is for asking questions about the site, but to answer your question, I would say it is pretty common. I've had a professor who worked on finding some lower bound in a physics problem for months, but when his colleagues said they could've done so-and-so, he realized he did a lot of roundabout work. It's not a waste, though. You just have to struggle through the problem. That's how you learn. $\endgroup$ Commented Nov 6, 2022 at 18:01
  • $\begingroup$ @JyrkiLahtonen I consider this posting an exceptional case. In my opinion, this is clearly a question for MathSE reviewers. Further, this question would be (perhaps wrongly, that is a separate issue) more than likely downvoted, closed, deleted on MathSE. That leaves metaMathSE. Further, as far as I am concerned, this is an outstanding question to ask. $\endgroup$ Commented Nov 6, 2022 at 19:20
  • $\begingroup$ Something that can alleviate the struggle (and has done so for me) is (1) to have a good peer system, and (2) to read math freely from just about anywhere. (2*) Reading mathematics in the night , just before you go to sleep, might sound shockingly unprofessional, but I've found it helpful to sleep over problems! Nothing helps like a peer system : it's like a mini PolyMath project of sorts with them. If you read math freely, then anything you read upon will not seem alien, in the least, to you : you will always have an easier concept to fall back , build on, and return to the harder topic. $\endgroup$ Commented Nov 7, 2022 at 4:41
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    $\begingroup$ @SarveshRavichandranIyer Thanks for your advice! Sadly, the only peer-system-like stuff I got is online forums like this one. :( $\endgroup$
    – Michael
    Commented Dec 3, 2022 at 3:03
  • $\begingroup$ That is unfortunate, because this site will not be forum like in many ways.you know what, good luck though. I sincerely hope that this site can be of help to you in some way. $\endgroup$ Commented Dec 6, 2022 at 16:23

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Expressing ideas similar to those already posted in the comments, but with different wording:

I consider Elbow Grease to be an extremely powerful weapon in the solving of Math problems. This typically means taking a piece of scratch paper, manually examining specific cases/data with respect to an assertion or problem, and stretching your intuition to grasp the patterns resulting from your Elbow Grease effort.

What I am terming Elbow Grease seems somewhat (perhaps largely) synonomous with your use of the word roundabout.

In a situation where I do not immediately see an elegant path to a solution, my first step is almost always Elbow Grease. If this results in my intuition evolving to intuitively grasp an answer at the end of my labors, fine.

This means that the next time I am confronted with such a Math problem, I will be able to bring my stretched intuition immediately to bear on such a problem. So, the next time, I may or may not be able to avoid Elbow Grease.


Your goal should be to stretch your intuition whenever possible. This will minimize the subsequent Elbow Grease situations.

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    $\begingroup$ elbow grease noun Strenuous physical labor and effort. (American Heritage Dictionary) $\endgroup$
    – GEdgar
    Commented Nov 7, 2022 at 13:21
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    $\begingroup$ I wasn't familiar with this phrase. I have been in the habit of calling it getting some dirt on my hands. $\endgroup$ Commented Nov 7, 2022 at 20:08

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