# Two trains and a fly

There’s a classic maths puzzle that goes something like this:

Two trains start 20 miles apart, and travel towards each other at 10 miles per hour. Just as they start, a fly takes off from the front of one train, flies at 15mph directly to the other, turns around, flies back to the first… and zigzags back and forth until the trains meet. How far does the fly fly?

The first time most mathematicians hear this puzzle, they attack it something like this:

- For the first journey, the fly and train are 20 miles apart and have an approach speed of 25mph, so the journey would take $\frac{20}{25}$ hours - 48 minutes - and the fly would have travelled 12 miles. Each train travelled 8 miles in the meantime, meaning they’re now four miles apart.
- The second journey goes much the same way: the distance is now 4 miles and the approach speed is still 25mph, so the journey takes $\frac{4}{25}$ hours - 9.6 minutes; the fly travels 2.4 miles, and each train travels 1.6 miles in the meantime, leaving them 0.8 miles apart.
- With a little work, you can show that each journey is a fifth of the distance of the previous one, meaning you have a geometric series to sum!

So that’s good, we have a formula for that: we know the initial journey $a$ is 12 miles, the common ratio $r$ is 0.2, and the infinite sum is $S_\infty = \frac{a}{1-r} = \frac{12}{0.8} = 15$ miles.

And then, whoever asked the question gives a smug smile and says “The trains take an hour to meet; the fly travels at 15mph, so yes, the answer is 15 miles.”

The story is, someone once posed this problem to Hungarian polymath John von Neumann, who immediately said “15 miles.”

“Ah, you’ve obviously heard it before,” said the person who asked it. “Most people try to add up the infinite series!”

Von Neumann looked puzzled. “That’s what I did!” he said.