Tick Tock: Partitions of a Clock

Summary The analog clock is ubiquitous with its familiar circular face and hands announcing the time of day to passersby. Each 12-hour period, the hands subdivide the face into a variety of sector configurations. A curious observer may wonder if there exists a time when the hands divides the clock into three equal sectors. We show that there is no such time by establishing a general result which identifies which partitions of a clock are achieved. Additionally, we completely describe clock partitions which have equally sized sectors, hands coincide, and cyclically permutable ratios.