The following problem can be solved 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 the trains 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?"
In a strict mathematical sense 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. This is they way that most trained mathematicians will solve the problem. Conversely a mathematical novice will most likely solve the problem the easy way - 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, at a rate of 75 miles per hour, and so the fly must have flown 150 miles. Easy.
When this problem was posed to John von Neumann, he immediately replied, "150 miles."
"Ah, I see you've heard this one before, Professor von Neumann. Nearly everyone tries to sum the infinite series."
"What do you mean?" asked von Neumann. "That's how I did it!"
An MIT student cornered the famous professor in a hallway:
Student: "Er, excuse me, Professor von Neumann, could you please help me with a calculus problem?"
John von Neumann: "Okay, sonny, if it's real quick -- I'm a busy man."
S: "I'm having trouble with this integral."
JvN: "Let's have a look." (a brief pause) "Alright, sonny, the answer's two-pi over 5."
S: "I know that, sir, the answer's in the back - I'm having trouble deriving it, though."
JvN: "Okay, let me see it again." (another pause) "Yep, the answer's two-pi over 5."
S (frustrated): "Uh, sir, I know the answer, I just don't see how to derive it."
JvN: "Whaddya want, sonny, I worked it out in two different ways!"
"Johnny was the only student I was ever afraid of," the mathematician George Pólya once recalled. "If in the course of a lecture I stated an unsolved problem, the chances were he'd come to me as soon as the lecture was over, with the complete solution in a few scribbles on a slip of paper."
Von Neumann was the subject of many dotty professor stories. He supposedly had the habit of simply writing answers to homework assignments on the board (the method of solution being, of course, obvious). One time one of his students tried to get more helpful information by asking if there was another way to solve the problem. Von Neumann looked blank for a moment, thought, and then answered, "Yes".
In the 1950s, von Neumann was employed as a consultant to IBM to review proposed and ongoing dvanced technology projects. One day a week, von Neumann "held court" at 590 Madison Avenue, New York. On one of these occasions in 1954 he was confronted with the FORTRAN concept (the first 'high level' computer programming language); its developer, John Backus, remembered von Neumann being unimpressed and that he asked "why would you want more than machine language?"
"The spectacular thing about Johnny was not his power as a mathematician, which was great, or his insight and his clarity, but his rapidity; he was very, very fast. And like the modern computer, which no longer bothers to retrieve the logarithm of 11 from its memory (but, instead, computes the logarithm of 11 each time it is needed), Johnny didn't bother to remember things. He computed them. You asked him a question, and if he didn't know the answer, he thought for three seconds and would produce an answer." - Paul Halmos, Mathematician.
Perhaps unsurprisingly, John von Neumann was strictly logical in his thinking. One afternoon his assistant, Paul Halmos, dropped von Neumann off at home. "Since there was to be a party there later, and since I didn't trust myself to remember exactly how I got there," Halmos recalled, "I asked how I'd be able to know his house when I came again. 'That's easy,' he said. 'It's the one with that pigeon sitting by the curb.'"
Henry Ford had ordered a dynamo for one of his plants. The dynamo didn't work, and not even the manufacturers could figure out why. A Ford employee told his boss that von Neumann was "the smartest man in America," so Ford called von Neumann and asked him to come out and take a look at the dynamo.
Von Neumann came, looked at the schematics, walked around the dynamo, then took out a pencil. He marked a line on the outside casing and said, "If you'll go in and cut the coil here, the dynamo will work fine."
They cut the coil, and the dynamo did work fine. Ford then told von Neumann to send him a bill for the work. Von Neumann sent Ford a bill for $5,000. Ford was astounded - $5,000 was a lot in the 1950s - and
asked von Neumann for an itemised account. Here's what he submitted:
Drawing a line with the pencil: $ 1
Knowing where to draw the line with the pencil: $4,999
Ford paid the bill.