Okay, that's another topic.
According to David Barash in the Chronicle of Higher Education, it's the mind-body problem. That is, what is the relationship between the experience of subjective consciousness and the physical brain? To put it in terms more friendly to current scientific consensus, "How does the brain create our experience of consciousness." (Most scientists will agree that "mind" is an emergent property of the brain.)
Well, I'll let you think about that problem for a while. I know the answer; I'm just not telling.
In any event, this problem may not be your cup of tea, but if you want to solve the first problem (How do we get more students to major in science?) then problems are actually the answer.
Difficult (and important) problems inspire curiosity, innovation, and motivation. And yes, also learning.
More class sessions, courses, majors, departments, and colleges should be structured on difficult problems.
In a sense, education is about learning the problem, and learning how to provide the best non-answers.