One of my professors in grad school was an editor of a monthly publication dedicated to publishing undergrad-level math-related stuff. He has a really amusing drawer full of "proofs" of Fermat's last theorem, the Goldbach conjecture, and the parallel postulate.
In HS geometry, our teacher started showing us basic constructions with compass and ruler: bisecting a line segment, drawing a line perpendicular to another line, and bisecting an angle. Then, at the end of class, he mentioned that no one had ever succeeded in figuring out a method to trisect angles. He did not mention that it had been proven that you couldn't (in the general case).
I spent the entire evening trying to trisect the angle.
It seems like a useful way to spend an evening for someone learning geometry.
I think it was explained that way to me too. I also spent a lot of time trying different ways. In hind sight, that was probably one of the best ways I could have been taught. I was challenged to try and do something that others had not.
When Paul Wolfskehl offered a prize of 100k German marks for Fermat's "last theorem", the University of Göttingen had to print generic rejection forms saying:
"Thank you for submitting a proof of Fermat's theorem.