This is a little maths problem I was thinking about on the tram back today.
Until recently, there were two tramlines running through the centre of Frankfurt, the 11 and the 12. The two lines follow the same track through the old town, then split at a major crossroads outside the Judengasse Museum, at a plaza called Börneplatz. Since it’s in a convenient location for various museums and things, Börneplatz is an obvious place for a tram stop. This is easy enough to set up; just put a set of platforms on the shared track leading up to the junction.
Then Frankfurt Transport decided to open a new tram route, the 18. The 18 doesn’t share the same track as the 11 and the 12 through the old town. Instead, it comes in from the other side, using the same track as the 11 up to Börneplatz, and then switching and following the route of the 12.
Here’s where the problem starts. Börneplatz is an important stop, so the 18 had to stop there, but as things were, both of the platforms were on the wrong side of the crossroads. So, they built a second stop on the opposite side of the road, Börneplatz/Stoltzestraße, for the 18. However, this opens up a new problem – the 11 passes through both of these. To stop passengers getting confused and waiting in the wrong place, the 11 has to stop at both of them. This means it makes two stops, literally 20 metres apart, which just wastes everyone’s time and causes unnecessary congestion on a busy route.
Ideally, there would be some way to arrange the stops so that every tram could pass through the junction from any direction and arrive at platform once – and only once. One obvious solution would be a triangular station in the middle of the crossroads, but it’s a very tight junction and a very busy road, so there’s really no room for that, while the fourth spoke of the crossroads, the one that is (so far) without a tram, puts paid to the idea of having an extra long platform stretching right across the junction.
Nevertheless, there is an optimal solution to this problem – a way of setting out the station so that the trams all pass precisely one platform. And that is…