If you want to roll a ball down a slope, what shape of slope will get the ball from point A to point B in the least time? This is the so-called brachistochrone challenge.
At first you might naively assume that a straight line would get the ball to its destination in the least time, because the shortest distance between two point is a straight line, but the ball is not moving at a constant speed.
In fact, because it needs to get rolling, the ball will travel down a concave ramp much faster than down a convex ramp. But which concave curve?
The winning curve is an inverted cycloid, the curve traced by a point on the rim of a circular wheel as the wheel rolls along a straight line.
(Hat tip to Charles.)