Let’s have a quick chat about the mathematics of transport fares.

Imagine you have a tram network with two stops on it. There’s really only one journey you can do – I mean, you can do it in two directions, but that’s the same distance, so we can assume it’s the same fare. That means that, ignoring all that complicated stuff like child fares and other discounts, then there’ll be a single journey.

Now let’s add a third stop to the network. Now there are three possible routes: A-B, A-C, B-C. So, that’s three fares. With a fourth stop, there are six, as you’re adding travelling between D and those three earlier destinations. With five stops it’s another four, so 10; with six, it’s another five, so 15.

You might dimly remember this sequence from your school days. These are the triangular numbers: the Nth triangular number is the number of dots you’d get in a triangle in which each side measured N dots, which is the sum of all the full numbers from 1 to N.

Three sides to every story. Image: Melchoir/Wikimedia Commons.

So, anyway, there are 93 stops on the Manchester Metrolink trams network. If every possible journey between each combination of two stops had its own individual fare, how many fares do you think you would need?

The answer is the 92nd triangular number, which is 4,278.

Which is a lot.

No transport network really wants to be administering a fare system involving 4,278 separate fares before you even get to discounts and so forth. So Metrolink, like most transport systems, has simplified things a little, by coming up with a zonal fare system.

But it has not, historically, simplified things as much as you’d think. Here’s the status quo:

Click to expand.

By my count that’s still 18 different fare zones, which is still 171 possible journeys, plus presumably fares for journeys which stay in each single zone. It’s bloody complicated, is my point here.

A lot of transport systems have congregated around a different model to simplify their fares. Since the 1980s, London has used a series of concentric rings, ranging from zone 1 (central London) to zone 6 (the outer suburbs; on some lines run by the city’s transport authorities, the inner ring of commuter towns are now in zones 7 to 9). You can find other concentric fare zone systems in other cities, including Paris and Barcelona.

Metrolink has now been instructed by the Greater Manchester authorities, including mayor Andy Burnham, to introduce its own concentric zonal fare system. It’s even produced a map, to show where the boundaries would be. Here it is:

Click to expand.

And here’s the geographic version:

Click to expand.

So, basically: zone 1 is central Manchester; zone 2 is within a couple of miles (including, pleasingly, the Salford Quays: the regional equivalent of Canary Wharf); zone 3 is a few miles beyond that and zone 4 the outermost bits of the network.

This isn’t certain to happen: there’ll be a “public engagement exercise” any day now, with the plan confirmed in July. But it seems probable that, from 2019, this is how Metrolink will work.

The fun thing will be to see if the new fare zones get tied up with matters of identity, as they have elsewhere. In Paris, zone 1 is restricted to the city proper. In London, the knowledge that your new flat is in zone 4, say, can have a direct impact on the how likely your friends are to visit it. Whether Greater Manchester will go a similar way remains to be seen.

But the point here, the most important thing of all, is that we had a chance to publish some maps.

Maps are great.

Sorry, where was I?

Jonn Elledge is the editor of CityMetric. He is on Twitter as @jonnelledge and on Facebook as JonnElledgeWrites. 

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