Sunday, October 7, 2007

Trains, Fly and John von Neumann

John von Neumann is one of the greatest mathematicians in the 20th century for his pioneering and popular work of Theory of Games and Economic Behavior with Oskar Morgenstern. However, you might not hear of some funny stories about him. Here is one:

Q: Two trains 200 miles apart are moving toward each other; each one is going at a speed of 50 miles per hour. A fly starting on the front of one of them flies back and forth between them at a rate of 75 miles per hour. It does this until the trains collide and crush the fly to death. What is the total distance the fly has flown?

A: The easy (maybe the easiest) way is as follows: Since the trains are 200 miles apart and each train is going 50 miles an hour, it takes 2 hours for the trains to collide. Therefore the fly wasflying for two hours. Since the fly was flying at a rate of 75 miles perhour, the fly must have flown 150 miles.

When this problem was posed to John von Neumann, he immediately replied,"150 miles.""It is very strange," said the poser, "but nearly everyone tries to sum the infinite series.""What do you mean, strange?" asked von Neumann. "That's how I did it!"

For more on von Neumann, here is Oskar Morgenstern's account of his collaboration with von Neumann. (needs authentication)

Anonymous said...

How did he figure it out?

Jeffrey Huang said...
This comment has been removed by the author.
Jeffrey Huang said...

Well, apparently, von Neumann calculated it very very fast: until the fly met the other train, it flew 200*(3/5), so, both trains moved 2/5 of the initial distance of 200; at this moment, the distance between two trains became 1/5 of the initial distance. Then the fly turned around, flew another 3/5 of this distance, left the distance between the trains 1/25 of the initial value; then it flew another 3/5 and so on...

so, the total distance the fly flew is:

200*[3/5+1/5*3/5+1/25*3/5+…]=150, eureka!