  We are motivated by a problem about running: If a race was completed in an
average pace of P minutes per mile, is there necessarily some mile of the race
that was run in exactly P minutes? The answer is no. We explain why, and
describe the history of this celebrated problem, known as the Universal Chord
Theorem. We also clarify and streamline the proof of a more powerful result by
Heinz Hopf from 1937.
