# Q: Why do buses come in threes? A: Mathematical inevitability

That's the problem with the internet in summer, isn't it? Nothing but repeats.

Anyway, here's something we originally published in June 2015. But it's dead good, honest.

The old joke that you wait ages for one bus, then three come along at once, is bordering on cliché. But it's also, as it turns out, true – not just because of bad planning, but also because of maths.

The phenomenon is so common, in fact, that it has a choice of names. Bus bunching, clumping, convoying, platooning – all relate to the depressing reality that, over any length of time, buses serving a single route are likely to end up tootling along directly behind each other.

The reasons why this should be can be difficult to get your head around – so Lewis Lehe, a postgrad working on a PhD in transport engineering at the University of California, Berkeley, has built a game of sorts to demonstrate. You can play it here. But if, like us, you are both lazy and impatient, here's how it works, with a few helpful screenshots to illustrate.

The game features two buses, serving a circular route with four stops. At the start of the game, the two buses are evenly spaced, at opposing ends of the loop. Passenger flows at each stop are identical to those opposite: when one bus has to pause for a set period at one stop, the other is pausing for the same length of time across the map.

The result is that the two buses are always carrying the same number of passengers. Left to their own devices, they will remain equally spaced forever, in perfect equilibrium.

And we know this, because at this point we got distracted, had a conversation on Twitter, went to make a coffee, and then suddenly remembered we were in the middle of doing something about buses, flicked back to the game in a panic, and discovered that it looked like this:

Our game was still ticking along nicely, the two buses now holding seven passengers apiece.

Eternal equilibrium is really boring, though, so let's mess things up a bit. What happens if we delay Bus One, just for a single second?

Err, well, this as it turns out. This is a couple of moments after the unexpected delay.

Two things to notice here. One is that the gap between the two buses has closed, slightly: that momentary delay allowed Bus Two to catch up, slightly. The other thing is that Bus One now has more passengers.

Those two things are actually features of a single phenomenon. In those few moments in which Bus One was delayed, Bus Two started to catch up – not by much, but for enough for it to have an effect. The consequence is that it's now reaching stops which last saw a bus relatively recently – and where there are fewer passengers for it to pick up.

But here's the insidious thing: that process is self-perpetuating.

Think this through for a moment. If Bus Two is serving stops that have relatively fewer passengers waiting, it can get moving again faster. Bus One, by contrast, is serving stops that haven't seen a bus in a while (it was delayed, after all), and so it's picking up more people. That means it needs to stop for longer, both to let its larger contingent of people off, and to collect the next lot.

And so, Bus Two gets closer...

...and then catches up altogether.

Look at the distribution of passengers now. Bus One is now arriving at stops with populations bigger than some small Chinese cities. Bus Two, just behind, arrives moment later to find them deserted. Without intervention, this situation will persist, essentially, forever.

In real life, of course, there would be an intervention (Bus Two would almost certainly overtake, as soon as there’s room). And real life is messier than this model in other ways, too. Bus routes are served by more than two buses; passenger flows aren't so evenly spread.

But it’s this inevitable messiness that provides that slight delay that kicks the whole process off. A slight disparity in passenger numbers, a traffic jam, bad luck with the lights; whatever the cause, one bus will be delayed, and those behind it will start to catch up.

Until suddenly you get three buses showing up at once and everyone makes tutting noises.

There are ways of mitigating this feedback loop. One is setting maximum or minimum stopping times in advance, to regulate the service. Another is building in waiting time at one end of a route, so that a bus doesn't immediately turn around and set off again (in effect, doing the second half of a circular route), thus bringing a measure of predictability to the time it takes to do an entire circuit.

These things can help – but the ubiquity of bus bunching suggests that they can’t elimate the problem altogether.

So the next time someone in your hearing asks why it is that buses all come along at once, you can reply, "Mathematical inevitability". Then you can look smug about it, as they, presumably, decide that they’d rather walk.

# To see how a city embraces remote work, look to Helsinki

A deeply rooted culture of trust is crucial to the success of remote work. (Sean Gallup/Getty Images)

When I speak to Anssi Salminen, an account manager who lives an hour outside Helsinki, he’s working from a wooden platform on the edge of a Finnish lake. With a blanket laid out and his laptop set up, the sun low in the sky, Anssi’s remote work arrangement seems blissful.

“I spend around half of my time working somewhere else other than the office,” he says. “I can work from home, or on the go, and I also travel to the Netherlands once a month and work from there.

“The emphasis in my work has always been that it doesn’t matter when or where I work, as long as I get things done.”

For many people around the world, the shift to remote work was sudden, sparked by the coronavirus pandemic. Finland, however, is finding the transition much less significant. Before Covid-19, the Nordic nation already displayed impressive levels of remote working, with 14.1% of its workforce reporting usually working from home. Only the Netherlands has a comparable percentage of remote workers, while the UK lagged behind at 4.7%, and the US’s remote workforce lingered at around 3.6%

Anssi works for one of many Helsinki-based companies that offers its employees flexible policies around when and where they work. That arrangement is in part due to the Finnish capital’s thriving start-up scene. In spite of being a relatively small city by global standards it is home to over 500 technology start-ups. These companies are leading the way when it comes to keeping employees connected wherever they choose to work.

“Our company has a completely location-free working policy,” says Kasper Pöyry, the CEO of Helsinki-headquartered software company Gapps. “All meetings are made available for online participants and facilitated accordingly. Some employees have worked extensively from abroad on a working holiday, whilst others prefer the comfort and social aspects of the well-stocked office. Whatever works for our employees is what works for the company.”

Like Gapps, many Helsinki-based firms are deeply preoccupied with providing the necessary technology to attract talent in a vast and sparsely populated country. Finland has only 15 inhabitants per square kilometre, and companies understand that in order to compose teams of specialised expertise, they may have to seek talent outside of the city. Local governments take a similarly proactive stance toward technological access, and Helsinki offers free, unrestricted, high-speed Wi-Fi from city-wide hotspots, while the country as a whole boasts some of the best coverage in Europe.

But encouraging remote work isn’t just about optimising the potential of Finland’s workforce – companies in Helsinki also recognise that flexibility has clear benefits for both staff and employees.

“The idea of a good work-life balance is ingrained in Finnish culture,” says Johannes Anttila, a consultant at organisational think tank Demos Helsinki. “It goes back to our rich history of social dialogue between labour unions and employers, but also to an interest in delineating the rules of working life and pushing towards people being able to enjoy their private life. Helsinki has been named the best city in the world for work-life balance, and I think that this underlies a lot of the mentality around remote work.”

For Peter Seenan, the extent to which Helsinki residents value their free time and prioritise a work-life balance prompted his move to the city ten years ago. He now works for Finnair, and points to Finland’s summer cottages as an example of how important taking time to switch off is for people in the country. These rural residences, where city residents regularly uproot to enjoy the Nordic countryside, are so embedded in Finnish life that the country boasts around 1.8 million of them for its 5.5 million residents

“Flexible and remote work are very important to me because it means that I don’t feel like I’m getting stuck in a routine that I can’t control easily,” he says. “When I’m working outside of the office I’ll go down to my local sauna and go ice swimming during the working day, typically at lunchtime or mid-morning, and I’ll feel rejuvenated afterwards… In winter time especially, flexibility is important because it makes it easier to go outside during daylight hours. It’s certainly beneficial for my physical and mental health, and as a result my productivity improves.”

The relaxed attitude to working location seems to pay off – Finland is regularly named the happiest country in the world, scoring highly on measures such as how often its residents exercise and how much leisure time they enjoy. With large swathes of unspoiled countryside and a national obsession with the outdoors, sustainability is at the forefront of its inhabitants’ minds, leading to high levels of support for measures to limit commuting. In January, Finland passed a new Working Hours Act, the goal of which was to help better coordinate employee’s work and leisure time. Central to this is cementing in law that employees can independently decide how, when, and where they work.

Yet enacting the new ruling is not as simple as just sending employees home with their laptops. For Kirsimarja Blomqvist, a professor of knowledge management at LUT University, perhaps the most fundamental feature that remote work relies upon is a deeply rooted culture of trust, which Helsinki’s residents speak of with pride. The anecdotal evidence is backed up by data which suggests that Finland boasts one of the highest levels of trust and social cohesion in Europe, and equality and transparency have always been key cornerstones of political thought in the country.

“Trust is part of a national culture in Finland – it’s important and people value it highly,” she explains. “There’s good job independence, and people are valued in terms of what they do, not how many hours they work for. Organisations tend to be non-hierarchical, and there is a rich history of cooperation between trade unions, employers, and employees to set up innovative working practices and make workers feel trusted and valued.

“It’s now important that we ensure that this trust can continue to be built over technology, when workers might have been more used to building it face-to-face.”

As companies begin to look hopefully toward a post-Covid future, the complexities of remote work are apparent. Yet amid issues of privacy, presenteeism, and social isolation, the Helsinki model demonstrates the potential benefits of a distanced working world. The adjustment to remote work, if continued after the crisis, offers a chance to improve companies’ geographical diversity and for employers to demonstrate trust in their workforce. On these issues, Blomqvist believes other cities and employers can learn a lot from Helsinki.

“People are now beginning to return to their workplaces, but even as they do they are starting to consider the crisis as a jumping point to an even more remote future,” she says. “The coronavirus pandemic has been an eye-opener, and people are now interested in learning from Finland’s good practices… We are able to see the opportunity, and the rapid transition to remote work will allow other countries to do the same.”

Katie Bishop is a freelance writer based in Oxford.