Short version: a potentially upcoming 5-authors maths paper already have authorship dispute drama. I am not among the 5, but I believe I found the solution to the part they got stuck on. Should I tell them?
A is a new graduate student, never have any papers, decided to work with B a post-doc (who had a few paper before) on a topic. They did some significant amount of work, but still nowhere close to a result. A tells his friend C (another graduate student with 1 paper), not to ask for help but just to let him what kind of thing he is working on. But unexpectedly, C got interested and work on it for a while and get some new results, before telling A and B. The work is significant, but A claims that give it a few more weeks and they could have gotten that themselves. C's advisor come to his defense and insist that it is only fair that C got credits as well. Unfortunately, C asked D - an undergraduate in his junior year - to develop a computer simulation, which ended up allowing C to realize that a tweak to the hypothesis is necessary. C never told D what it is for, but D realized that after chitchat with B. Now D's advisor also want him on the paper, since he did some work, and it would really help him in applying to grad school if the paper can be done soon. Neither A nor B is happy about this of course. To make thing worse, E a post-doc from CS department, happen to see the problem and some work on the blackboard. He knew some papers in CS that happen to solve some aspect of the problem, and told B about it, which eventually cause their earlier work to be much more simplified. B thinks E deserve coauthorship too, much to A's resentment.
Now I was not there for the drama. A told me about this over beer a few months ago. I curiously asked what the problem is, and A told me everything including what they are stuck on, after making me promise to never work on this and never tell anyone else about the problem (A don't want to have a 6-author paper). I kept the promise, until recently, when reading some papers, and I found something and realized that this could be as well the key to the problem. After some more computation, I am 90% sure that this method would solve the problem, but I promised not to work on it, so I did not work out everything. Now I am torn. They are unlikely to ever know about this method (I'm in a very different specialty), but their effort had been walking in circle for a few months now. I had never had a paper before, so if I do put effort into this, at least I wanted to be recognized for it.
EDIT: as far as I know, there are only 10 people who know about the paper, us 6 and 4 professors who don't really care nor contribute; it is not public. I think A considered me his emotional crutch which is why he told me about it. The work is probably not good enough for publication, right now it consists of a bunch of different approaches and reformulation that all seemed very close to solve the problem, but did not.