When I was a grad student, I lived across the street from an electronics store. The owner of the store had done some graduate work (in some sort of Engineering, I think). He ran a weekly ad in the local newspaper, and placed at the bottom of the ad a relatively hard math problem. And he gave anyone who could solve the problem the choice of a free radio or telephone (each of which would retail for less than ten dollars). After solving one of his problems, and speaking with him for a bit, I convinced him to place a problem from group theory (it was about equations over the group $Z_2$) in the next ad.
The next week the ad came out. My problem was there in print. My first publication. A few days later I went to talk to the owner of the store. He was furious. A whole bunch of people had come in with solutions and he gave away a lot of free radios and telephones.
The sad thing was that none of the solutions were correct.
EDIT- now for the mathematical urban legend...
A well-known topologist, let's call him X, told me that this had happened to him. He had been in a seminar for graduate students. The student speaker was proving one of X's theorems. X found this boring and fell asleep. And he started snoring.
They had to wake him up because no one could hear the speaker.