[mamon]

The article you linked to provides a trivial solution:

"the trains take one hour to collide (their relative speed is 100 km/h and they are 100 km apart initially). Since the fly is traveling at 75 km/h and flies continuously until it is squashed (which it is to be supposed occurs a split second before the two oncoming trains squash one another), it must therefore travel 75 km in the hour's time."

So if von Neumann was solving it by explicitly summing the series, as the anecdote claims, then he was doing it wrong :)

[xelxebar]

Well, if you're familiar with sums, then the "shortcut" isn't all that much faster; it's mostly a function of how quickly you can grok the salient points:

The fly travels 3/2 times the speed of a train, so every bounce the fly travels 3/5 of the remaining track and leaves 1/2 * 2/5 = 1/5 track to travel, so we just compute the geometric sum

    3/5 * \sum 1/5^r = 3/5 * 1/(1 - 1/5)
                     = 3/5 * 1/(4/5)
                     = 3/5 * 5/4
                     = 3/4.

One nice thing about this sum is that it encodes a bit more insight about the fly's flight path than the shortcut method.

[pvg]

I assume you're trying to beat the original responder in brazen literalism - now we have a citation request for a throwaway joke, an explanation (with citation) and a literal interpretation of the joke in the citation.