Welcome! The intent of this group is to explore open (unsolved) problems in mathematics. The end goal for each open problem is to find a solution, and maybe publish it if it's a nice enough result! Even if we don't make it all the way there, we can have fun exploring unsolved problems and doing real research.
How to participate: find the most recent discussion thread related to OPEN PROBLEM #X. Post a comment about a solving idea you might have. Thoughts can be small or large.
If you want to post a long technical comment, or source code, make a new thread by clicking "Post discussion" on the group's home page. You can also use "Post problem" to challenge other users with a small, related problem that you've already worked out the answer to. (This is for problems you know the answer to - not unsolved ones!) Solving related problems or small cases can help build intuition for the big, unsolved case.
Some potential questions you may have are answered below.
ON THE OPEN PROBLEMS
I want to suggest open problems. Can I do that?
Yes please! We have a thread for that here; please post your suggestion there. For the moment, while we're getting started, I (Jason Dyer) will handle picking and posting the main ===OPEN PROBLEM=== problem threads. This will change in the future as this community is meant to be self-sustaining; eventually the community should be able to run everything, including picking their own problems.
What if I'm not sure if a problem is open?
Post it anyway! Even if it's a known result, there may be some subtle variation that is open, or it may be improving the proof is a worthy cause.
What kind of open problems would be good to pick?
I do not recommend one of the "famous" open problems like "prove the Goldbach conjecture" (one of the oldest unsolved problems in mathematics). I know from past precedent that things get stalled very easily; there's good reason some problems have resisted attack for 100+ years.
However, not every open problem is of that type; some of them are open simply due to the fact they haven't gotten much attention, and there's quite possibly an easy solution.
Does the open problem really have to have a definite yes/no style to it?
No, some open problems are simply "find a better algorithm to do X" or "prove this already proven result, but with different methods" or "we know the number in this pattern is at most Y; bring the number down some more".
Even a "catalog and categorize every math thing of this type" open problem would be interesting (some of the most important mathematics projects have been like this).
I only like the "big" open problems like the Collatz Conjecture. What should I do?
I would recommend picking a smaller side problem that's related to the big one you're interested in. Also, a "catalog and categorize" effort on some related mathematical object might make a worthy project (for instance, what about all the recurrence relations other than the Collatz one?)
How many open problems should be running at once?
From my personal experience it's best if the community is focused on one problem at a time, although it's possible we might be able to run two problems simultaneously. This is likely something we'll need to revisit once we find out how many regular contributors we have.
I want to help, but I don't know a lot of mathematics! What can I do?
First off, the problem have been chosen in such a way that you don't need much math background to get started, Also, there's all sorts of skills needed to solve one of these kind of problems:
Raw mathematical calculating and theorizing
Programming code that tests cases
Organizing already existing data in a format that makes it easier to see patterns
Error checking already existing formulas / data that might appear
Consolidating more technical arguments into simpler-to-understand ones
In other words, if you're interested, there's probably some area you can contribute.
I've got a general comment or idea, what should I do?
Post it to the current thread dedicated to solving the problem.
I've got some result that solves a small part of the problem, what should I do?
If it's short, you can just post it to the current thread as mentioned above.
Another approach would be to post your result as a full question using the Brilliant.org interface. For example, let's suppose you've proven Theorem X is false on a 5 by 5 grid. Post a true-or-false problem stating: Theorem X is true on a 5 by 5 grid. Set false be the correct answer, and then post a solution to your own question. Be sure to indicate in the subject title which open problem it's meant to be related to. Example subject line: 5 by 5 Case (Open Question #1).
Please don't post a question like this unless you think you know the answer! It's OK if there's a flaw, though - open question research is supposed to be hard.
I'm a teacher! Can my students join in?
Sure, that's highly recommended! One of the things we've found on Brilliant.org is that age matters very little; our Problems of the Week, for instance, often feature contributors as young as 13 and as old as 80.
Is it ok if I just post a sentence or two?
Again, this isn't about one person doing everything, but many individual contributions adding up to a group insight that can bust open a difficult problem.
The only thing to keep in mind is to not be ambiguous; you want enough information that someone else can fully understand your idea and work with it.
I've got an idea, but I think it's wrong. I don't want to look foolish.
Let me tell a story: in a similar project (run by professional mathematicians) I made a very naive observation based on a set of data that pointed out a simple arithmetic pattern. There were good (and obvious) reasons why my observation was wrong, but the observation made us realize that the software that generated our data had an error! So even wrong observations can be helpful.
ON PUBLISHING A PAPER
Will every successful open problem result in a published paper?
No. For instance, the results of problems like Open Problem #1 are tabulated and recorded by a mathematician (Erich Friedman, at Stetson University) but they aren't considered important enough for a real journal. This doesn't mean the problems aren't worthy of study, just that not every mathematical result ends up in print.
If a paper is published, who will be the author?
The Brilliant.org Open Problems Group.
If I've contributed, can I still refer to the paper on my resume/CV in the future?
Sure! You should include both the paper and the link to your contributions on the Open Problems Group.