The following problem can be solved either the easy way or the hard way.
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?
The fly actually hits each train an infinite number of times before it
gets crushed, and one could solve the problem the hard way with pencil
and paper by summing an infinite series of distances. The easy 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 was flying for two hours. Since the fly was flying
at a rate of 75 miles per hour, the fly must have flown 150 miles.
That's all there is to it.
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!"
