Where the Pregolya river meets the Baltic sea, the Russian enclave of Kaliningrad looks out across the Vistula lagoon towards Gdansk in Poland. In past times these two cities formed part of a vast network of trading cities stretching from Estonia to England, flourishing through the later Medieval period thanks to a commercial and defensive confederation of merchant guilds. Reaching the height of its prosperity in the late sixteenth century, thanks to its export of wheat, timber, hemp and furs, Kaliningrad was originally called Koenigsberg and was visited by more than one hundred ships annually. The resulting wealth allowed the people of the city to construct seven bridges connecting the banks of the river and two large central islands together.

As a result of this construction, the townsfolk were able to travel between these formerly distinct regions of the city more easily. According to local legend, this pastime of city centre strolling became so popular that curiosity began to emerge among the inhabitants about the different routes they could take. At some point, this evolved into a type of Sunday afternoon game, with the goal of devising a route by which all seven of the bridges would be crossed once and once only. However, despite all their efforts, it did not seem possible to complete the route following these rules. The puzzle remained unsolved for so long that the riddle was elevated up through the academic classes and ultimately to one of the greatest mathematical minds of the time, Leonhard Euler.

Although Swiss, this prolific mathematician, physicist, astronomer, logician and engineer was at that time a resident of St Petersburg, further north along the Baltic coast in Russia. Euler was a tremendously industrious academic, publishing around 500 books and papers over the course of his life, so to many it seemed odd that his interest would be piqued by such a trivial logical problem. In fact, when the mayor of Gdansk originally contacted him to probe his mind for a solution to the problem in a letter of 1736, Euler’s response indicated his initial reluctance due to the problem “bearing little relationship to mathematics”, suggesting that the solution would be “based on reason alone” and that its discovery “does not depend on any mathematical principle.” However, it seems that Euler’s inclination towards certainty and understanding was too strong to resist, as a letter written soon after to a friend, Giovanni Marinoni, explains that in Euler’s view the Koenigsberg problem “...is so banal, but seemed to me worthy of attention in that neither geometry, nor algebra, nor even the art of counting was sufficient to solve it.”

On August 26, 1736, Euler shared a paper presenting a systematic explanation of his thoughts and interpretation of the problem. He ended by concluding that the citizens of Koenigsberg were not mistaken: there was indeed no route that could achieve the desired goal. Euler began his analysis by stating that the nature of this problem certainly concerns geometry, but not of the familiar kind involving measurements and calculations, before making the connection to a topic recently discussed by the German polymath, Gottfried Leibniz - that of an innovative field of mathematical study called ‘geometry of position’, but of which little record exists. Euler then explained the need for this kind of abstraction in order to solve the problem, taking us to the crucial and groundbreaking part of his analysis.

It was clear to him that the barrier preventing the people of Koenigsberg from understanding the puzzle came about as a result of their unrefined perception of it. He thus started his analysis by removing all extraneous features of the map, leaving us with the ‘land masses’ (islands and river banks) and the bridges connecting them. This may seem extraordinarily simple to us - as we interact with abstracted diagrams and schematics everyday, in the form of transit maps and appliance manuals - but, at the time, equivalent documents were largely decorative and not created with efficient practicality in mind. Euler then described the land masses as dots and the bridges as lines and gave each of the former a letter A, B, C and D and the latter a number one through seven.

Although a piece of paper, a pen and a keen mathematical mind would be needed to understand the following fifteen paragraphs of analysis, they are not necessary to understand the significance of his method. The method Euler employed could be adapted as a framework to solve any problem that involved networks and routes through the analysis of all permutations. This was crucial to the later development of logistical analysis, in which the goal is to find the most efficient path or paths in complex systems, and its application can be seen in activities as diverse as the arrangement of postal routes, electrical circuits, DNA sequencing and chemical compounds.

Euler’s endurance thus paid off and the brain-teaser that had perplexed the citizens of Koenigsberg was no more. Sadly, though, the inhabitants of Kaliningrad can no longer enjoy the pursuit that first put Koenigsberg on the intellectual map. Although in 1875 a new bridge was constructed - meaning that the route was at last achievable - two of the original constructions fell to bombing in World War II, while a further two were demolished and replaced by a modern highway.

A The ethics of autonomous vehicles – those that effectively drive themselves without human intervention – has become a growing concern recently as such cars start to appear on public roads. This has provoked an urgent need to address the contentious issues surrounding this developing technology before it becomes widespread. However, the issues raised are not easy to resolve, highlighting the frequent inadequacy of our own moral judgements and perhaps asking us to take responsibility for the pre-programmed morals of a machine.

B The branch of knowledge which deals with moral principles is known as ethics. Ethical concerns about autonomous cars are explored within an interdisciplinary field called “robot ethics” which brings together experts in engineering, computing, law, philosophy, sociology and psychology as robot ethicists seek to understand the implications of autonomous robots in human society. The reliability of machines and the level of decision-making authority they may have are key concerns and autonomous cars are a clear case of the predicaments posed by this new technology.

C The potential autonomous cars have to significantly increase road safety is not disputed. According to Bryant Walker Smith from the University of South Carolina, 90% of all traffic accidents are caused by human error at least in part, and removing the error component would dramatically reduce the number of those accidents. Another investigation, by leading management consultancy McKinsey & Co, found autonomous cars could save up to 30,000 lives a year in the USA alone.

D However, in order to eliminate the effect of human error, all vehicles would need to be fully autonomous. Yet it seems unlikely the public would accept a situation in which we relinquish all manual control of vehicles. Not only do very few people trust machines more than themselves (despite all the evidence proving machines are safer), there is also the problem that putting the vehicle in complete and unsupervised control raises questions about the level of the driver’s responsibility. This leads us to the ethical question: who is ultimately responsible for the moral decisions programmed into and made by the vehicle we are, to a degree, controlling?

E In September 2016, Germany was the first country to propose introducing a set of ethical principles for autonomous vehicles into traffic laws. The transport minister Alexander Dobrindt suggested three key rules for autonomous vehicles: first and foremost, when faced with imminent danger, autonomous vehicles must always choose to damage property rather than injure people. Secondly, the vehicles must never distinguish between humans in any way, and must consider all to merit the same level of protection. Lastly, at all times there must be a human overseeing the vehicle's driving. This must be guaranteed by rendering the car manufacturer liable for any accidents which take place if nobody is holding the steering-wheel.

F Robot ethicists were quick to point to some flaws in this draft proposal. While it is easy to argue that a car should crash into things not people, what should it do if it has to choose between crashing into one group of people over another? This thought experiment is usually called ‘The Trolley Problem’, and was first proposed by Philippa Foot in 1967. It is traditionally presented thus: A runaway trolley is hurtling down a railway track towards five people who are tied to it. You are standing next to a lever which will divert the trolley if pulled, but doing so would kill one person tied to the parallel track. Do you pull the lever? The utilitarian view, which seeks to achieve fewest deaths, says it is obligatory to pull the lever, killing one person instead of five. However, drivers are unlikely to welcome self-driving cars with fully utilitarian morals, potentially leading to situations in which the car chooses to swerve into a barrier and kill the driver to save the lives of five pedestrians.

G Bryan Casey of Stanford University considers the Trolley Problem irrelevant for autonomous vehicles, and holds that the problem is not even ethical, but simply legal. He believes all points of contention will ultimately be tested and resolved by the courts, and argues that the rules programmed into cars will be designed not to adhere to an impossible and slippery sense of innate morality, but to keep the car within the law. Instead of attempting to minimise injury, cars will inevitably be programmed to minimise legal liability for manufacturers. However this is done, it must be agreed across the car industry and nations as far as possible to make the extensive rolling out of the vehicles viable.

H If the law is written in such a way that the manufacturer is held responsible for all injury and any damage caused by the autonomous car regardless of other considerations, car-makers will program the vehicles to eliminate situations of risk. Manufacturers such as Volvo and Google have already stated they will accept responsibility for accidents involving their vehicles. Their reasons for this stem from an attempt to earn the public’s trust in the technology, shifting some responsibility away from the customer in the case of an accident. However, they would only accept responsibility due to a flaw in the vehicle, not when the technology was used inappropriately, or if the collision was caused by a third party. If others follow their lead, some headway may be made in this legal minefield of responsibility, allowing for the undoubted spread of autonomous cars to continue its journey.