The moving missiles form the corners of a shrinking, rotating square. Ignore the rotation and focus on the shrinking. Each missile homes constantly on the missile to its left, which means it’s moving continuously along one side of the shrinking square. Each missile moves at 1 mile per second, so the square will shrink to nothing in 20 seconds.

From George Gamow and Marvin Stern, Puzzle-Math, 1958.

08/08/2014 A reader points out that Gamow’s proof doesn’t generalize — the radius of a regular n-gon with 20-mile sides increases rapidly with n and soon exceeds 20 miles. (Thanks, Stuart.)