P versus NP problem
Consider the problem of scheduling exams (described above). But suppose, next, that there are 15000 students. There's a computer program that takes the schedules of all 15000 students. It runs in an hour and outputs an exam schedule so that all students can do their exams in one week. It satisfies lots of rules (no back-to-back exams, no more than 2 exams in any 28 hour period, ...) to limit the stress of exam week. The program runs for one hour at mid-term break and everyone knows his/her exam schedule with plenty of time to prepare.