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Proofs, not just answers

Brilliant only allows questions with a natural number as an answer. This is very understandable, because it allows for automatic checking of one's work. It turns out to work brilliantly.

However, what I think is more interesting about math than calculating an answer, is proving something. Brilliant was set up for olympiad mathematics originally, but olympiad math is usually not about giving the right answer, but about proving something.

There are many olympiad problems on the internet in which something has to be proved, and the solutions can usually be found as well. However, practicing with proofs is not very easy, currently: finding the right difficulty level is a challenge on its own, and if you don't know the solution, just looking at the solution does not make you a better problem solver.

I think Brilliant could fix this. Just like with the current topics, it knows at what difficulty level you are, and the grading could be done by the community: I would love to check some papers from other Brilliant users and give them feedback, and I'm sure many other Brilliant users think about it the same way.

Why this would be better than the current situation:

  • The difficulty level would be just right.
  • Feedback would be given rather than a full solution, hence you would still do a problem yourself, even if you didn't see it at first sight.
  • Marking proofs by others would improve your ability to explain what is really going on, which is a very useful skill.
  • For each problem, the best and most elegant proofs could be listed and perhaps voted on, providing the internet an awesome database of beautiful problems and proofs.

If this would all work, that would be amazing. I think it can be done by the Brilliant community, because we love maths and love helping others with it. Also, if I'm really proud of a proof I've done, I love sharing it with others. ;-)

I think many users visit this site in order to train for the olympiads, which will be a lot easier if proving things could be trained here as well, because that is essentially what olympiad mathematics is all about. Most math olympiads put their problems online, so those can be used already. And of course, users could come up with their own problems!

Hopefully this can become a thing, I think it would really be great.

Note by Tim Vermeulen
3 years, 4 months ago

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Thanks for the very thoughtful suggestion Tim. Peter Taylor Staff · 3 years, 4 months ago

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It sounds like much of what you wish could happen is already possible via the discussions board. If anyone has a beautiful proof that they created, they are more than welcome to post it to the discussion board to share with the brilliant community. Similarly, if someone thinks their proof needs a bit of work but they don't know how or where to fix it, they could post it and ask the brilliant community. I've seen these kinds of things happen several times before. The only thing that brilliant could be able to supplement would be actual proof style questions.

The only problem with the discussion board for this matter is that it is very informal and no points are awarded for any proofs of the like. Despite this, one could still generate a constructive and academic atmosphere in one's discussion of a proof or problem. I think that your idea is a great utilization of an already present discussion board. Bob Krueger · 3 years, 4 months ago

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@Bob Krueger

I think that your idea is a great utilization of an already present discussion board.

There are changes we could and probably will make to the discussion board to encourage using discussions to share your own open-ended work. Eventually, profiles will be re-done and we will probably implement a way for your profile to catalogue and link to all contributions you make to discussions. That way, a profile could effectively store your work in a browse-able way into posterity.

We might consider making a dedicated "proof swapping" section of discussions, if it seems like there is significant interest in it. We can layout and organize the discussions better to make more subcategories and have them all be more useful and more navigable.

Could Brilliant eventually become a system that catalogues proof-based problems by their difficulty, and integrates them with weekly challenges that are communally graded(Tim's idea)? Probably. It will not happen as quickly as changes to discussions to encourage similar kind of adhoc behavior as Bob suggests. Peter Taylor Staff · 3 years, 4 months ago

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@Bob Krueger Right, I have thought about this as well. However, the discussion board does not provide the users with problems, and definitely not necessarily for their particular difficulty level or interest. Tim Vermeulen · 3 years, 4 months ago

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Totally agree with you. Siddharth Kumar · 3 years, 4 months ago

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Proofs FTW! I can suggest that if you want more practice in proofs, visit AoPs. You can take an online class that has proofs every week, or if you dont want to pay, hang around on the community message boards. People post challenge problems 24/7. You have the opportunity to write your own solutions and receive feedback. Best of all, there are different levels like middle school, high school, calculus, and college to suit your needs. Here is the link for the website:

http://www.artofproblemsolving.com/ Fiyi Adebekun · 3 years, 4 months ago

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