This question was posed by my high school geometry teacher, for extra credit:
Is it possible, using only a pencil, a straightedge (not a ruler) and a compass to trisect an angle of unknown value? If so, describe the technique, and mathematically prove its validity. If not, mathematically prove the impossibility.
I recall wondering at the time, why he was posing this question to a freshman high school geometry class, and if he genuinely expected any of us to come up with an answer. Nothing more was said about it and to my knowledge, nobody submitted an answer. But the question stuck in my mind, and I would often spend hours puzzling over it, trying over and over to come up with a valid solution.
I'll be 60 years old this July, and I'm no closer to an answer now than I was that day, so many years ago. If anyone knows the answer, I'd appreciate a simple explanation, as I am no mathematician.