
By Roje, Wikimedia Commons (CC-BY-SA-3.0)
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.
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