Several interrelated problems are posed to students. All involve the minute and hour hands of a clock and the path traced by the midpoint of the segment connecting the ends of the hands. Students begin by assuming that both hands are the same length and that the clock runs properly and shows the correct time. They extend the problem by dropping the constraints that the hands are the same length and the clock runs properly to see the variety of curves they can generate. Finally, they use algebra and trigonometric identities to prove the forms of various curves. Detailed figures, an activity sheet, and solutions are included. This mathematically rich problem was developed for the Ohio Resource Center to accompany the Mathematics Program Models for Ohio High Schools proposed by the Ohio Department of Education. (author/sw)