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During my Ph.D. research I encountered a mathematical problem. I found solutions in some cases, but in other cases I couldn't find a solution. My advisors also could not help with this. I asked at Math.SE and MathOverflow, even offered a boundy, but got no hint.

Although I can proceed in my research without a solution to these cases (there are many other problems to solve anyway), they are very interesting and can contribute a lot to understanding the general problem.

The problem itself is very simple, such that even high-school children can understand it. So, I thought of the following idea: give the problem to talented high-school or under-graduate students that go to math olympiads, and offer a monetary bounty (in addition to co-authorship) to the first solver. Hopefully a fresh young mind can succeed where older minds have failed.

What do you think about this idea? Is this ethical? Useful? Done in the past?

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    Could you please add the links to these posts ? – Tom-Tom Jan 10 '14 at 13:34
  • Here is the most recent post: mathoverflow.net/questions/146607/… – Erel Segal-Halevi Jan 10 '14 at 13:45
  • Can you give a quick update on this? Did you do it? And dit it yield a solution to your problem? – Vincent Jul 7 '18 at 12:45
  • @Vincent several months after asking the question, I offered a bounty to students in a course on math problem solving, most of whom were talented highschool students. But, none of them came back to me with a solution. – Erel Segal-Halevi Jul 7 '18 at 21:29
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It sounds like a great idea to publicize the problem among young people who may have a lot of time on their hands, not a lot of other research ideas, and a desire to demonstrate their abilities through something other than timed contests.

One awkward possibility would be getting stuck in a situation in which someone can't convince you that their solution is correct, but you can't convince them it is incomplete or wrong. Of course this could happen with anyone, but it's particularly likely to occur with someone who is not an experienced researcher. If you offer a bounty, you should be clear about the conditions. (For example, that it all comes down to your judgment, and that you don't commit to spend unlimited amounts of time trying to understand a proposed solution.)

Coauthorship is certainly mandatory if you publish a paper that depends on this solution (unless of course the students publish their solution separately and you cite it). However, you may run into tricky situations in which it's not enough just to offer coauthorship for the first solution. For example, if several people solve the problem independently and around the same time, then they should all be treated evenhandedly. Or what if someone proposes a solution you don't believe, someone else proposes a clearer solution and you offer the second person coauthorship, and then the first person convinces you their solution was correct after all? Of course these problems are not hard to solve (in the last case, you can't retract the authorship of the second person, but you should make the first an author too if you still can). But the important thing to keep in mind is that you may need to make sensible decisions that go a little beyond the "first solver is a coauthor" rule.

A final observation is that if you succeed in attracting students to this problem, you will likely get lots of questions and comments from them along the way. If that appeals to you, then it's an advantage of this approach. On the other hand, if your goal is to get a solution with a minimum amount of time and attention on your part, then asking beginners might not be fruitful.

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For problem easy to formulate as this one, this may be a motivating problem for students who want to tackle with "real research", quite rewarding also for those who find a solution.

I don't know if this has been done in the past, but why not, as long as you give full credit of the proof to the students who find it and in case this is published, give them authorship. I would suggest, in case you have a winner, to endorse him or her on the arXiv, so the credit will be fully his or hers.

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You may want to explain on MathOverflow how you ran into the problem. If solving this is key in solving an important problem in some area you may get more attention and someone whose research interests are a good fit may want to put in some time and effort.

As for co-authorship my feeling is that the person who solves that particular problem should be the sole author (although you could be acknowledged for suggesting the problem). Then you can cite him/her in your work (as a paper of private communication).

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