I sometimes assign open problems as extra credit problems. Some thoughts:
1) Do you tell the students the problems are open?
YES- it would be unfair for a student to work on something they almost surely won't get.
NO- Some Open Problems are open because people are scared to work on them. Having said that, I think P vs NP is beyond the one smart person phase or even the if they don't know it's hard maybe they can solve it phase.
NO- See Page 301 of this interview with George Dantzig where he talks about his mistaking an open problem for a homework and ... solving it.
CAVEAT---There are OPEN PROBLEMS!!! and there are open problems??? If I make up a problem, think about it for 30 minutes, and can't solve it, it's open but might not be hard. See next point.
I tell the students:
This is a problem I made up but could not solve. It may be that I am missing just one idea or combination of ideas so it is quite possible you will solve it even though I could not. Of course, it could be that it really is hard.
A friend of mine who is not in academia thought that telling the students that I came up with a problem I could not solve, but maybe they can, is a terrible idea. He said that if a student solves it, they will think worse of me. I think he's clearly wrong. If I am enthused about their solution and give NO indication that I was close to solving it (even if I was) then there is no way they would think less of me.
Is there any reason why telling the students I could not solve it but they might be able to is a bad idea?
2) Should Extra Credit count towards the grade? (We ignore that there are far more serious problems with grades with whatever seems to make them obsolete: Calculators, Cliff notes, Cheating, Encyclopedias, Wikipedia, the Internet, ChatGPT, other AI, your plastic pal who's fun to be with.)
No- if they count towards the grade then they are not extra credit.
I tell the students they DO NOT count for the grade but they DO count for a letter I may write them.
What do you do?