There is a guessing strategy that the students can use to guarantee that the exam will not be a surprise. Each day, before class begins, the students declare "W

There is a guessing strategy that the students can use to guarantee that the exam will not be a surprise. Each day, before class begins, the students declare "We conclude that the exam will be today with probability 1. Therefore, we have advance knowledge of the date of the exam." They might be wrong a few times, but they will be right on the day of the exam, so the exam will not be a surprise. This means that the professor cannot guarantee a surprise exam, so his original claim is false. But note that the students' strategy requires them to always assume that the exam will be on the earliest possible date and to plan their study time accordingly, which is probably the behavior that the professor was trying to encourage anyway. It just so happens that he has to lie about the surprise exam in order to do so.