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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How the rise of anti-crime politics caused lasting harm to Black Americans

"I see an awareness that has developed in the Black community in the last 10 years or so about how deeply racist the criminal justice system has become," James Forman Jr. says. (David McNew/Getty Images)

The police killing of George Floyd, and the protest movement that emerged from it, has reinvigorated a national conversation around reinventing criminal justice policy in the United States.

At the same time, reports that violent crime is rising in many US cities have resurrected talk of the much-disputed “Ferguson effect,” a theory put forward by law enforcement professionals, and some researchers, who argued that police slowdowns in the wake of the first wave of Black Lives Matter protests resulted in elevated rates of violent crime. President Donald Trump is trying to weaponise this narrative, paired with images of federal officers clashing with protesters in the streets of Portland, to wage a 1968-style backlash election campaign.

“People who want to mobilise a lock-them-up style of either policing or prosecution have tried to weaponise those short-term increases,” says James Forman Jr., professor of law at Yale Law School. “Criminologists will say you have to be very, very cautious about short-term movement [in crime statistics]. We don't know whether or not what we're seeing right now [with violent crime increasing] is going to sustain itself. But the fact is, it's here and people are talking about it.”

In 2018, Forman won the Pulitzer Prize in nonfiction for his book Locking Up Our Own: Crime and Punishment in Black America. Drawing on his experience as a public defender in Washington, DC, he traced the emergence of anti-crime politics in late 20th century Black communities. Forman showed how newly empowered Black politicians fought for policies they believed would protect and uplift Black Americans, but inadvertently contributed to mass incarceration. 


CityMetric recently caught up with Forman to discuss crime trends, where he sees reason for hope in this moment and how the Black political class’s attitude toward crime and punishment has shifted since the latter part of the 20th century. 

This interview has been edited and condensed. 

There is talk right now about a resurgence of crime and violence in American cities. We saw similar, more localised concerns after the initial 2015 Black Lives Matter protests in Ferguson and Baltimore. Do you fear this could reinvigorate the kind of politics you describe in your book among segments of the Black community and political class?

I fear that it could be reinvigorated nationally and also in the Black political class. Look at the political conversations that are happening in Atlanta right now, for example, a city that also has seen a short-term uptick in crime as it is a site of a lot of protests about George Floyd and Breonna Taylor on the national level, as well as Rayshard Brooks and Ahmaud Arbery more locally in Georgia.

I think that you can already see in some of the language of the local elected officials this idea that we have to be very careful about pulling back. [They are saying] “while the protesters may make some valid points, we can't risk returning to the ‘80s and ‘90s.” Those decades really traumatised the United States, and particularly traumatised Black communities. There's a deep fear about returning to the levels of the violence that we saw in the crack years.

You write a lot about class divides among Black Americans, where middle income and elite Black people don't suffer as much from extremely punitive policies. They also have closer ties to the politicians who are creating these policies. There are very specific groups of people, even in marginalised communities, whose voices are heard.  As a result of these dynamics, you write about Black politicians fighting for things like mandatory minimum prison sentences or against decriminalising marijuana. Is there still that disconnect between those who suffer the most from criminal justice policies and those who are actually heard in political discourse?  

Let me just say a caveat, that when we talk about class divisions in the Black community it's important to hold two truths in our head at the same time. Bruce Western and others have shown the way in which class, educational status, income can dramatically reduce the likelihood of being hardest hit by the criminal system – namely incarcerated. Middle class and upper middle class Black people get some measure of protection. It's also true at the same time that Black people of all classes are worse off relative to their class counterparts in the white community. 

One area where class is least protective is policing and police stops. The police do not know how many degrees you have. They don't know how much money you have in your bank account. I want to be very clear that in making this point about class, I'm not making the argument that race or racism don't matter in this context. 

In terms of how it plays out now, I see an awareness that has developed in the Black community in the last 10 years or so about how deeply racist the criminal justice system has become. Twenty or 30 years ago they had a consciousness, but there's levels of understanding. Many of the people I write about in the book wanted to promote the interests of the Black community. They weren't motivated by indifference or callousness. When presented with mounting evidence of how awful this system has been in Black lives, they're reconsidering and recalibrating. 

Lots of former elected officials have said to me some version of “I didn't know at the time and I appreciate that you showed us in our full complexity. I appreciate that you showed the pressures we were under. If I had known then what I know now, maybe I would have been less quick to go along with some of these harsh measures.” 

The second thing that has affected the Black political class has been the emerging movements, led by Black people in particular and led by young people. They not only educated leaders, but pressured them and made them understand that there is a political cost. If you're not moved by the moral argument, then you'll be moved by the political argument. You'll be moved by the people protesting outside the office of District Attorney Jackie Lacey in Los Angeles, for example, where Black Lives Matter LA has held, I believe, a year of consecutive protests against a Black district attorney who has had really some of the worst practices.

From what I can tell, she's been pressured by the movement to change some of her positions on important issues like prosecution of low-level drug offenders, for example, and the aggressiveness with which she prosecutes police officers for acts of violence.

What do you make of the calls to defend or even abolish the police?

What I find so compelling about abolition, initially in the prison context and extended to the police as well, is that it shifts the conversation and forces us to go through experiments in which we imagine what it would take to build that world. I think that exercise is very important, because it pushes us further than we are naturally inclined to go. Cultivating a broader imagination is an incredibly important part of this work, because as you know from my book, often it was lack of imagination that caused people to fall back on [punitive policies]. 

That's what caused D.C. Councilmember David Clarke to call the police rather than public health experts when he was overwhelmed with letters about heroin addicts in public space. He was anti-drug war, but he couldn't imagine responding to a call for help with heroin addicts with anything other than police. That's a very common move from even really good and progressive people. 

People who are for defunding, for abolition, are absolutely right about reinvesting that money into alternative structures that support communities. But the reinvestment part doesn't follow naturally from the terms. We might want to come up with a term that captures the new stuff we want to do. I think that's particularly important because one of the reasons Black communities have ended up supporting more police is that Black communities have always wanted their fair share of the resources.

Then, the evidence suggests the United States has too many police officers doing prophylactic, preventative, or stop-and-frisk style policing. The style of policing that leads to district level harassment, pulling people over for no reason. But we have too little investment in the parts of police departments that investigate unsolved crimes. I'm talking about the investigator or the detective who comes to your house after there's been a robbery, an assault, a rape, or homicide. 

As compared to European countries, in the United States we actually underinvest in those parts of our police departments. Jill Leovy’s book Ghettoside shows this in dramatic detail. She describes an LAPD that's stopping and frisking Black drivers wantonly and yet the homicide detectives are still relying on a fax machine and the fax machine is broken. They have to go with their own money to Staples to buy a printer. Meanwhile, other aspects of the department are kitted out in this ridiculous riot gear that makes them look like they're in Fallujah. 

That under investment is particularly damaging to Black communities because we're disproportionately victimised by crime. Because of racism and this allocation of resources, the police are less likely to respond in Black communities. The kids I used to work with in the charter schools in DC, we talk about no snitching, but one of the reasons they would never call the police after they'd been victimised by crime is they would say, “They're not even going to come. You're wasting time.” 

I did a Q&A with Jill Leovy too and her argument is one I've struggled to articulate in our present moment. She argues the state doesn't have a monopoly on violence in low-income Black neighbourhoods, because investigations of violence are deemphasised and crime victims or their loved ones often take retribution into their own hands.  But right now, establishing or preserving the state's monopoly on violence isn't an appealing talking point. 

Yes, this is another thing nobody's talking about. Whatever we're going to do instead of the police has to be accountable to the public. The best, most direct way to have accountability is to have the individuals be public employees. As long as we have 300 million guns in this country at least some of those state employees are going to themselves be armed. It's unreasonable to ask them to do the job without it. Not as many need to be armed as are armed now, but some of them need to be. But they can't be hiding behind union contracts or civil service protections which make it impossible to remove even the worst performing, most abusive officers. 

We can not call them police if we want to. That's semantic, but maybe symbolism matters. But those people have to be state employees. They can work with community-based nonprofits, but there are also communities that don't have as robust of a nonprofit network, and they deserve protection too. These [community] groups have to be accountable to the state and, when they don't exist, the state has to be there. 

Progressives get all the points I just made when it's applied to education. The notion that things be public and accountable to the state is understood when it comes to schools. It's exactly why so many people on the left are opposed to charter schools, because they say they don't have public accountability. They want these things to be a state function. But this point about the difficulty in removing this entirely from the hands of the state is, I think, one that liberals and progressives understand from other contexts.

Jake Blumgart is a staff writer at CityMetric.