The following problem can be solved by setting up some appropriate differential equation and solving it. There is an intuitive method of solving the problem without calculations.

Imagine n missiles at the vertices of a regular polygon. 3 at the points of an equilateral triangle; 4 on the corners of a square; 5 on a pentagon; et cetera. Say each polygon has a side 100 miles long, and assume that the missiles travel 500 miles per hour. The missiles pursue each other in a clock wise direction, each pursuing the one on the next vertex, correcting course instantaneously as the quarry moves.

Due to the symmetry of the situation, the missiles will always be on the vertices of a polygon which rotates and contracts as the missiles pursue each other. Very soon the missiles collide at the center of the original polygon.

Assume that the missiles can be treated as points, and courses are corrected continuously to keep each pursuer headed toward his quarry.

For any polygon, can you come up with an algorithm which provides the theoretical distance traveled by each missile before the collision? Is any polygon easier to deal with than the others?

Answer to last question is below. I will post intuitive solution in a few days. If you want to try the differential equations approach, have fun. I would not try that without being promised sex or money for the effort.























I was given the problem for the square in a differential equations course. Almost everybody in the class managed to solve the problem. Years afterwards the intuitive solution occurred to me, and it suggested an algorithm for solving the problem for other polygons. I am not sure the differential equations approach would work anything but the square.