Consider a thin, flexible wire with one end at the origin and the other represented by the red dot. Thread a small bead onto the wire, so that it can slide frictionlessly along it. If the bead is released from rest at the origin, what shape should you bend the wire into for the bead to reach the other end in the shortest possible time? The answer is a cycloid, also known as a brachistochrone in this context, which comes from the Greek for “shortest time.”

Instructions: Click and drag the red dot to move the endpoint of the wire, and see the resulting shape that makes for the fastest trip. (The dashed line shows what the rest of the cycloid curve would look like if you continued it beyond the end of the wire.) Notice that when the two ends of the wire aren’t too far separated in height, it actually pays to drop the bead below the final point so that it can pick up speed before climbing back up to the finish line.

In 1696, Bernoulli posed this problem as a challenge to the great mathematicians and physicists of the day. The story goes that Newton solved it in a single night and then published his solution anonymously. But when Bernoulli read the anonymous paper he knew it had to be Newton’s work, remarking “I recognize the lion by his paw prints!”