The following problem can be solved either the easy way or the hard way.
Two cyclists 40 miles apart are riding toward each other on a straight track; each one is going at a speed of 20 miles per hour. A swallow starting above one of one of them flies back and forth between them at a rate of 50 miles per hour. It does this until the cyclists meet. What is the total distance the swallow has flown?
The swallow actually flies back and forth an infinite number of times before the cyclists meet, 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 cyclists are 40 miles apart and each cyclist is going 20 miles an hour, it takes one hour for the cyclists to meet. Therefore the swallow was flying for one hour. Flying at a rate of 50 miles per hour, it must have flown 50 miles. That's all there is to it.
When this problem was posed to John von Neumann by Max Born, Neumann immediately replied, "50 miles!"
"It is very strange," said Born, "but nearly everyone tries to sum the infinite series."
"What do you mean, strange?" asked Von Neumann. "That's how I did it!"
Source: John von Neumann Documentary starting at approximately 18 minutes into the film.
Based on the version of the anecdote from "Math Jokes".