How many colours would you need to make London’s bus network more comprehensible?

Any colour you want as long as its red. Image: Getty.

Looking at Transport for London’s colour-coded bus trial in Barkingside, CityMetric editor Jonn Elledge found an interesting problem:

Even thinking about the maths does my head in – but it seems unlikely to me that every bus in London can be given a colour different from that of every bus it ever shares a stop with. At some bus stops, there’ll be two buses in violet.

So let’s have a go at this: given the current bus routes in London, how many colours would you need to so that no bus stop has two buses using the same colour?

TfL have their bus stop locations and routes available as open data so we can quickly get a feel for how hard a problem is actually is. The way I processed the file I ended up with 20,028 bus stops and 729 routes (we can probably have arguments about exact numbers, but go with me). A lot of those bus stops serve exactly the same routes as others – so there are only 4,129 unique nodes in this problem. Which is still lots, but feels more manageable.

While there probably an elegant mathematical approach, the boring brute force technique goes like this. Starting out with all bus routes sharing a single colour (let’s say blue), you get the computer to go through every bus stop and change routes to other colours to make sure that bus stop has only unique colours. You then repeat this until all bus stops have no duplicate colours.

You get different results for this depending what order you tackle the bus stops, so you run it with a few different orders to get a feel of about how many colours are needed. Running the program 1,000 times, the lowest number I get using this technique is 44. Lower numbers are probably possible through trying many more orders, but let’s say for the moment this is roughly right – you’d need 44 colours to apply this approach strictly to the entire London bus system. This is really too many colours to be able to usefully distinguish lines, so is probably a no-go.

We get more manageable numbers if we try a less strict version of the rule. If we let bus stops served by five or more routes have two routes of the same colour, the total number of colours required drops to 14. This is approaching a workable colour scheme in terms of actually being able to distinguish between all varieties.

When your number is up

But let’s move on from colours and think about what TfL is actually trying to do here: it wants to make it easier for people who don’t currently use buses to use buses.

It’s worth thinking about where bus route numbers currently come from:

When we introduce a new route – or make alterations to an existing route by splitting it – the last digit or digits of the historic ‘parent’ route are used wherever possible, so that passengers might associate the incoming route with its predecessor. This was the case in 2003, for instance, when route 414 was chosen as the number for the new route between Maida Hill and Putney Bridge, which was intended to augment route historic route 14 south of Hyde Park Corner.

In other words, bus routes numbers are path dependent on old naming decisions because of the desire to keep existing users happy. While this is probably a good idea, it can also end up in results that are very un-good for new users.

So if you ignored the past and the need to keep the millions of current users not confused, what could do if you just scrapped all the current route numbers and started from scratch? Specifically let’s look at two problems:

  • Ambiguous bus routes at the same stop that appear similar;

  • Hard to remember bus numbers.

If you don’t see why these things are problems, imagine yourself as a user for whom the concept of numbers is a bit fuzzier: for instance, dyslexic users for whom the number rearrange (where 365 and 635 might be similar), or those for whom the numbers are literally fuzzy because they’re less able to read the signs.


There are two key areas of ambiguity: digits that are visually similar to each other (66 and 68) and route numbers that are conceptually similar like 114 and 14.

For real world examples of conceptually confusing bus stops, there are 1,601 stops served by routes whose numbers wholly contain the number of another route at the same bus stop. While this is sometimes suggestive of similarity of route, in many instances it isn’t. If you’re at Church Lane the 71 and 671 share 88 per cent of stops in common – but if you’re at Southall Broadway, the 95 and 195 share just 0.1 per cent of their stops. Looking at all the stops with this problem, the average similarity is only 38 per cent.

As most journeys are short, differences at the far end of the route are probably not a problem for most users – but the point is, that vague, warm feeling that similar number routes at the same location should be similar is not backed up by the data.

There are also 205 stops that have routes which are anagrams of each other. The St Nicholas Center has the 407 and the 470, at Brooke Road you have the 76 and 67, and Lytton Grove has the 39 and the 93. This isn’t many in the grand scheme of things – but it’s not ideal.

While we’re thinking about which numbers are nicer than other, let’s look at research which numbers are easier to remember correctly than others. Milikowski & Elshout found that:

The order of memorability was

(1) Single digit numbers;

(2) Teen numbers (10-19);

(3) Doubled numbers (e.g. 44, 77, 22);

(4) Large tabled numbers (numbers which factor and therefore appear in the multiplication tables, such as 49, 36, 60, 84, 27); and

(5) Other numbers that do not fall into any of these categories.

While memorability for Single digit numbers was above 80 percent, that for Other numbers (no subcategory) was only around 40 percent.

This should inform our thinking about route numbers. The first thing our colour system lets you do is dump bus numbers above 100 and use colours as a replacement for the first digit. This immediately makes numbers easier to remember because we’re reducing the number of concepts you need to remember. Route 127 requires you to remember three things (one two seven) while Blue-27 requires you to remember two (Blue twenty-seven). This is more true with smaller numbers, but every little helps.

The next thing we need to do is jettison every number that is a reverse of another (we don’t want both 46 and 64). This gets rid of most numbers above fifty (while retaining doubles). The end result is each colour can now be followed by 62 numbers – which means 62 bus routes.

Ideally you’d also reduce ambiguous symbols such as (1 and 7) or (6 and 8) – but this really cuts down the number of usable numbers. Instead what we’ll do try and make sure ambiguous numbers like this do not appear at the same stop.

Seeing clearly

So here are our new constraints:

  • A colour can only have 62 routes;

  • There are 15 colours (up from 14, because the original solution required some colours to have more than 62 routes);

  • Bus stops with four or fewer buses can’t have multiple routes with the same colour, stops with more can have two;

  • One bus stop cannot have routes of different colours with the same number. You also can’t have both 21 and 27, or 46 and 48.

Is such an arrangement possible? It turns out it is.

To solve this one you randomise which routes get which numbers and score them according to how well they pass the above. Then you create random variations on the best performing plan, and so on, until it narrows in on a version that passes all the rules. This returned a viable arrangement of route colours and names after a few hours (and 161,663 attempts).

Can something like this be done in reality? Confusing all current users seems a bad idea – but maybe this kind of approach should affect how new bus routes are named. Rather than blindly following the history of a route, select rules you want to be true of your naming scheme (they might be different from mine) and get a computer to suggest the minimally confusing approach. It turns out it doesn’t take long to get answers to quite fiddly problems.

But the real point here is I don’t want to wear my glasses to wait for a bus, and changing the naming convention for every single bus route in London is a proportionate response to this problem.

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What it's been like living in one of the few places that never locked down

People enjoy sunny weather in Tantolunden park in Stockholm on May 30, 2020, amid the novel coronavirus pandemic. (Henrik Montgomery/TT News Agency/AFP via Getty Images)

While most of the Western world was confined to their homes for the better part of two months this spring, my friends and I in Stockholm continued hanging out. In stark contrast to most other places, we went to restaurants (occasionally, outside when possible), to one another’s houses (in our yards when possible), and even sent our kids to school. As the rest of the world opens up again, not much will change in Stockholm.

As an American expat living in the Swedish capital, I was initially angry at Sweden’s response to the Covid-19 pandemic. In my home country, early outbreaks in locations such as Seattle, New York City and the San Francisco Bay Area led to strict rules that were soon mirrored in many other states and cities. The Swedish strategy, meanwhile, boiled down mostly to recommendations: If possible, work from home; avoid unnecessary travel within the country; engage in social distancing; and if you’re above 70, stay home. I felt that, in the face of a global pandemic, a country known for its generous welfare policies – that took such good care of its citizens – wasn’t doing its part to protect us.

My friends and I are mostly expats with young families who, early on, pulled our children out of school against official policy. (Schools here only closed for those 16 and over.) We eagerly waited to hear what further action our current country would take. Surely a country known for its progressive social policies would take fast, decisive action to protect its citizens?

The regulations that were put into place in Sweden amounted to restricting public gatherings to no more than 50 people (reduced from 500, which concert halls skirted by restricting entry to 499), limiting restaurants to table service only, and no visiting retirement homes. People here did take the work-from-home guidelines to heart – no one I knew was going in to work. But bars and restaurants were full. My Instagram feed was a highlight reel of acquaintances clinking champagne flutes at the city’s major clubs and restaurants.

After the first few weeks, I slowly started meeting up with friends again. I sent my kids back to school, where they intentionally spent most of the day outdoors and drop-offs were restricted to outside only (parents weren’t allowed to enter the building). I was careful to take precautions like bringing hand sanitizer to playgrounds and wiping my hands after opening and closing the gate to school. Hardly anyone wore masks to the grocery shop or inside stores – the few times I’ve seen people wearing them I’ve done a double take. One busy Friday night in late April at the local supermarket there was a line out the door and someone regulating the number of customers allowed inside at the same time. I took a photo and sent it to my family in the US saying “Sweden finally catching up with the rest of the world!” (I haven’t seen entry to that store being regulated since.)

When I spoke to Swedish friends about the strategy many agreed with the relaxed approach, mentioning that other countries’ draconian measures would be unnecessary in Sweden. A recent poll showed that just 11% of people in Sweden felt they did not trust state epidemiologist Anders Tegnell, who is leading the strategy. In this country, the onus was placed on citizens themselves to follow recommendations. It's about personal judgement and individual responsibility within a framework that rested on mutual trust, rather than top-down control. Swedes’ high level of interpersonal trust and trust in authority was often cited in the press as the characteristic enabling the relaxed Swedish strategy in tackling the virus, as opposed to social distancing becoming a matter of surveillance and policing like in Spain or Italy, where any nonessential socializing was forbidden.

In early May, Sweden's ambassador to the US Karin Ulrika Olofsdotter said in an interview with the Washington Post that some media outlets made it look “like everyone in Sweden is out drinking and partying,” she said. “That is not the case.” But that was certainly how it felt to me. According to research by Esteban Ortiz-Ospina and Max Roser in 2016, in countries such as Norway, Sweden and Finland, more than 60% of respondents in the World Value Survey think that people can be trusted. And in the other extreme, in countries such as Colombia, Brazil, Ecuador and Peru, less than 10% think that this is the case.


Of course, many places in the US also took a similarly relaxed approach to tackling the pandemic, with conservative lawmakers and anti-lockdown activists citing Sweden as taking the right approach. Sweden, rarely finding cheerleaders among conservative US circles, suddenly stood as an example to follow. But since then, places such as Arizona, Texas and Florida have all seen significant spikes in cases following reopenings and are being deemed the new epicentres of the virus – while Sweden’s numbers have stabilised. According to some reports, the death toll in Sweden is one of the highest in the world per capita, but the total number of Swedish deaths remains at just above 5,000, compared to over 120,000 in the US, over 43,000 in the UK, over 28,000 in Spain and over 34,000 in Italy. The mortality rate in Sweden and the number of new intensive care cases in the country declined in the last week and contagion rates here are now “stable” according to the WHO.

Although it didn’t always feel like it at the time, Sweden issued clear guidance from the beginning, with the expectation that people would choose to follow it. It certainly was my experience that everyone I knew stopped going into the office and started working from home. William Hanage, an associate professor of epidemiology at Harvard’s School of Public Health, attributed Sweden’s slowing of the virus to implementing guidance early on. “Sweden’s policy is unusual in that it took a much less stringent approach to preventing transmission," he says, "but interestingly it implemented those measures at a very early stage in the pandemic, before large amounts of community spread had occurred.”

Now I go outside and all too often realise I’ve left my hand sanitiser at home. I even met a friend for lunch outdoors at a busy cafe one particularly sunny day, and another indoors one Friday night for dinner. In May I had a birthday bash in my garden with a dozen or so friends and we ended up at the local bar. I always felt guilty after, as if I’d done something wrong that I couldn’t tell my family in Baltimore about. When I watched international news or spoke to family back home I would feel a certain cognitive dissonance between my own seemingly low-risk reality and what I knew to be happening in the rest of the world. My family in the US calls me skeptically questioning why I’ve had people over in my garden, or been out to eat. I can’t explain the lack of logic that permits an entire city’s citizens to operate life as normal in the midst of a global pandemic. But Stockholm has become a bubble of exactly this.

Being relatively young and healthy, I’m not so worried about getting sick. Even though young and healthy people have gotten seriously ill, there haven’t been any reported cases at my kids’ or any of my friends’ kids’ schools. Nobody I know in Stockholm knows has gotten sick, allowing me to feel a certain distance from it. But my husband’s parents are in their mid-70s and weren’t able to see their grandchildren for two months save for a few visits to their hallway, where we wave and blow kisses to them standing at the door.

I’ve been grateful – but also felt a sense of guilt for – my freedom here. When there are no hard and fast rules about how to act, it’s easy to constantly question yourself: Is it really okay to be outside, sitting at this full cafe? Is it okay to invite a few friends over for a birthday? Is it okay to send my kids to school? These questions have surely gone through minds around the world in the past several weeks, and now it’s clear that that behaviour had dire consequences in some cities and not others.

While Swedish social media at times suggests an endless friend-filled party at summer homes and popular hangouts, the reality here is a balancing act between personal judgement and the freedom to continue life as normal. Self-regulation is what it comes down to in Sweden, anyway.

Elysha Krupp is a writer and editor currently living in Stockholm.