After weeks of work, the S-Bahn is running properly once more! To celebrate, here’s one of my favourite little maths problems. It’s something I first noticed during the torturously long wait for a Merseyrail train before the timetable improvements. I haven’t a clue whether it has a real name, so for now, let’s call it the S-Bahn Paradox.

Suppose you want to go from Frankfurt West to Frankfurt Süd (this was the limit until the building work ended). Well, there’s a train precisely every five minutes during the day, and the journey takes almost exactly 15 minutes (more like 16, but let’s say 15). As you travel, you’ll pass trains travelling in the other direction. Supposing you leave just as the next train is coming in, how many will you meet along the way (including the ones at the start and end stations)?

15 minutes, a train every five minutes, that means that there are three five minute periods, and (not forgetting the train you meet at time zero), you meet four trains altogether, right?

Well, not quite.

Let’s do this the easy way, with pictures. Here’s our train line, straightened out and with the stations removed (as well as trains that don’t travel the full distance between West and Süd). Our train is on the left hand track, facing south. Each tick represents the distance that the train can travel in one minute. (It doesn’t matter that this distance may vary as the train speeds up and slows down – all that matters is average speed)

As you can see from the number on the right, we’ve just met one train, and there are three more waiting for us. The four theory’s looking pretty good right now. Let’s see what happens if we bump the clock along by one minute.

Hmm. We’ve moved, but so have the trains coming in the other direction. Let’s skip forward another minute and a half.

We’ve just met our second train after two and a half minutes? Why? Well, we’ve travelled south for two and a half minutes. The other train has travelled north for two and a half minutes. 2^{1}/_{2}+2^{1}/_{2} = 5 minutes, and so we’ve already covered the first five minute interval!

And after five minutes…

That makes three. I’m pretty sure you can tell where this is going.

Although trains only run every five minutes, and we only waited for fifteen minutes, we encountered seven trains over the course of this journey, meeting a new one every 2^{1}/_{2} minutes. The reason why might be clearer if we imagine that our train stays in place, but the trains in the opposite direction are travelling twice as fast. Then it becomes a bit more obvious how 5 minutes becomes 2^{1}/_{2} minutes.

This might seem like a pretty abstract piece of maths, but it has a very noticeable (and annoying) effect in the real world.

Everyone’s been there. You take the train or the bus to work each day, and the route feels like it’s really busy, with trains/buses racing past you every few minutes, and yet at the one time when you miss your train… there’s a 15 minute wait? It can’t be true!

Well, thanks to the S-Bahn Paradox, it is. Sorry!

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