# fly-between-trains problem

Two trains are approaching each another and a fly is buzzing back and forth
between the two trains. Given the (constant) speed of the trains and their
initial separation distance, and the (constant) speed of the fly, calculate
how far the fly will travel before the trains collide. This problem appears
to have been first posed by Charles Ange Laisant (1840–1921) in his *Initiation Mathématique*. There is a long-winded method of getting
the answer and a much shorter way. Suppose the trains start out 200 miles
apart and are each traveling at 50 miles per hour, and the fly – a
speedster of its kind – is moving at 75 miles per hour. The long method
involves considering the length of the back-and-forth path that the fly
takes and evaluating this as the sum of an infinite series. The quick solution
is to notice that the trains will collide in 2 hours and that in this time
the fly will travel 2 × 75 = 150 miles! When this problem was put to
John von Neumann, he immediately gave
the correct answer. The poser, assuming he had spotted the shortcut, said:
"It is very strange, but nearly everyone tries to sum the infinite series."
Von Neumann replied: "What do you mean, strange? That's how I did it!"