# Trainspotting

I’m embarrassed to admit that it took me well over a minute to solve the following word problem.

A train leaves Philadelphia for Denver at noon, traveling at 65 miles per hour.   Two hours later, a train leaves Denver for Philadelphia, traveling 80 miles per hour on a parallel track.   At the moment the two trains pass each other, which one is closer to Philadelphia?

Hint: it doesn’t take a minute to solve.

Part of the problem, I think, is that years and years of mathematical training have lead me to automatically start considering the problem in its most abstract terms even before I’ve actually read what it is I need to solve.   I’m not even a half a sentence in before I’m mulling over the more general problem of

A train leaves point A heading for point B at time C, traveling at D miles per hour.   E hours later, a train leave from point B to point A (a distance of F miles), traveling at G miles per hour.   (a) When is the time H at which the two trains pass? (b) At time H, what is the distance between the trains and point A?

and I’m well into setting up the system of equations needed to solve this word problem before it dawns on me that the original question is not either of the ones I’ve posed… and is, in fact, infinitely more trivial.

This reminds me of a bit from The Hitchhiker’s Guide to the Galaxy:

“Bloody hell,” said Majikthise, “now that is what I call thinking. Here Vroomfondel, why do we never think of things like that?”

“Dunno,” said Vroomfondel in an awed whisper, “think our brains must be too highly trained.”

or the following Far Side comic:

In any case, in the spirit of further unnecessary generalizations, I also submit the following two abstractions of this archetypal word problem.

## Love story

Did you hear about the romance novel written by a famous mathematician?

It starts off with:

The two lovers ran towards each other like two trains, one leaving Boston at 3:36 PM traveling at 42 miles per hour, and the other leaving Chicago at 4:18 PM traveling at 53 miles per hour…

## John Von Neumann

Consider the following problem:

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. There is, however, a much easier solution: 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 by his colleague Paul Halmos, he immediately replied, “150 miles.”

“It is very strange,” said Halmos, “but nearly everyone tries to sum the infinite series.”

“What do you mean, strange?” asked Von Neumann. “That’s how I did it!”

* Told by Paul Halmos, in “The Legend of John von Neumann,” American Mathematical Monthly, 80 (April 1973) 386.

## PS

By the way, for homework, solve the general problem above. Might as well flex those mathematical muscles.

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