The bridges of Paris (problem)
Can we cross them all once? And only once?
“I have mapped out the path to follow. I shall now start down it, and nothing will stop me from pursuing it to the end.” – Immanuel Kant, Thoughts on the True Estimation of Living Forces, 1749.
The stakes are, as always, life and death, even for the small stuff. To see this clearly, especially given the day you’re having and with what’s on your mind right now, the questions you cannot decline, the conflicting pressures, the dark entanglements, perhaps what you need to do – what we all need to do – is start, like K., with a vigorous walk at eight o’clock in the morning – or, some say, three o’clock in the afternoon, or just after his habitual dinner of three dishes, wine, &c., with a small second course at five in the afternoon – but sharp and attuned, like clockwork itself, never changing, so dependable the townspeople set their watches by it, etcetera, from Printzessin Strasse, where he lived and worked, to the northern island in the Pregel, where he is buried.
Today, the island officially bears his name. In K.’s day, however, when it was Königsberg’s bustling centre, home to the university and intellectual life and the major shopping and financial districts – and not yet biopsied off the face of the earth by Soviet and British bombers – it was known as Kneiphof, which is an old Prussian word for “swampy”. This is the name the townspeople still use when they cross one of its bridges to stroll through its gardens or visit its rebuilt cathedral and new interactive museums.
Walk alone, as he did, but do not imitate his other idiosyncrasies. Breathe, like him, in through the nostrils, but not exclusively. The mouth, as Dylan tells us, will do, it works just fine, even if someone else is speaking with it or it’s ten thousand miles deep in a graveyard or made of mercury in the missionary times. Nor should you try to carry the foot to the ground perpendicularly with a kind of “stamp”, as K. did, to secure a larger basis, by setting down its whole sole at once. Most importantly, try not to fall on your face as he once did, onto the street with such resounding force that, unable to regain his feet, he required the assistance of two young ladies, one of whom he presented with a rose that just so happened to be clutched in his left hand at the time.
Stop where he stopped. Probably on the waterfront somewhere. A bench in a shady spot under an ornamental pine. There are probably some boats and ships nearby and some chain-link fences. Sit with him on the bench and rest. Think about things. Not necessarily the same things that obsessively occupied this five-foot-tall elderly gent beside you, clad in his old overcoat – thoughts of space, inertia, the integration of metaphysics with natural science, the existence of God and an afterlife. The hiatus in the planetary system between Mars and Jupiter. The absurdity of miracles and prayer and the resurrection of Christ. His dead electric cat.
You will want to think instead about your own everyday stuff. Personal matters. Things in the news. Your body. Your anxieties. Keep in mind, however, that these are temporary states and conditions. Tomorrow might be different. It might just be a happy day, and the day after that even happier, and the next one happier and happier.
It must be said, too, and in fact, a man named Reichart did say it, that K. was drier than dust both in body and mind:
“A more meagre, arid, parched anatomy of a man has not appeared upon this earth. The upper part of his face was grand; forehead lofty and serene, nose elegantly turned, eyes brilliant and penetrating; but below it expressed powerfully the coarsest sensuality, which in him displayed itself by immoderate addiction to eating and drinking.”
We wholeheartedly sympathise. Eating and drinking, yes, but especially drinking, which, like opium, eventually renders us not just parched but quiet, reserved, and untalkative, and then introduces you to the “dream-like land” where the “illusory feeling of increase in our vital force” leads us to “no longer feel life’s obstacles”, followed by a pounding headache, vertigo, nausea and the “unnatural state of one’s inability to classify your senses according to the laws of experience.”
Coffee, then, and none too soon. “Coffee! Coffee!” K. would often shout, according to Thomas de Quincey’s febrile account, extrapolated from the bedside-vigil notes of K.’s private secretary and estate manager J. G. Wasianski:
“If another cried out – ‘The coffee is coming immediately.’ – ‘Yes!’ he would retort, ‘and so is the next hour: and, by the way, it’s about that length of time that I have waited for it.’ Then he would collect himself with a stoical air, and say – ‘Well, one can die after all: it is but dying; and in the next world, thank God! there is no drinking of coffee, and consequently no – waiting for it.’ Sometimes, he would rise from his chair, open the door, and cry out with a feeble querulousness – ‘Coffee! coffee!’ And when at length he heard the servant’s step upon the stairs, he would turn round to us, and, as joyfully as ever sailor from the masthead, he would call out – ‘Land, land! My dear friends, I see land.’”
We shall see it, too. But first, before moving forward, or retracing our steps with K. to the university where he taught, now gone, blown to bits along with everything else on this island by the RAF and the Red Army, join him in thinking deeply about the bridge he and you crossed to get here – Blacksmith’s Bridge, one of this island’s five bridges, among which is Connecting Bridge, the one that links this island to the larger island just to its south, which you can see from the bench.
Back then, in K.’s day, that other island was called Lomse – another old Prussian word for “swampy” – but the Soviets renamed it Oktyabrsky after the October Revolution. It has, or had, three bridges.
Now, turn your thoughts with K. to that island’s two other bridges and this island’s four other bridges, all of which, at different points on the map, connect to the north and south banks of the river. We will cross these seven bridges when we get to them, but first let me introduce you to the great Swiss polymath Leonhard Euler, who in 1735, when K. was a precocious 11-year-old pupil at the local Pietist Latin school, was living and teaching three weeks away by horse and carriage in St Petersberg. At the end of August of that year, in a paper presented to the St. Petersburg Academy, Euler (pronounced “oiler”) provided a negative solution to the problem that the townspeople of Königsberg had set for him a few months before: devising a walk through Königsberg that would cross each of the seven bridges once and only once. He did this through abstraction – stripping away everything extraneous, much as we had done at the start of this so-far failing and unconvertible essai, and reformulating the problem, which, in its most basic terms, is this: on the small island on which we find ourselves seated, every arrival over a bridge requires a departure over another bridge, unless we’re starting or ending our walk. The next time we arrive, it must be by a different bridge again, and we’ll leave by yet another one. Which means every time we set foot on the island, there must also be a corresponding departure. In other words, the number of arrivals equals the number of departures – unless the island is either our walk's starting or ending point. For this to work smoothly, the island must have an even number of bridges connecting it to other land masses: half for arriving and half for leaving. Clear so far?
This same logic applies to the other land masses: you can start or end your walk on a land mass with an odd number of bridges, but every other land mass you visit must have an even number of bridges.
But, here, in Königsberg, which the Soviets renamed Kaliningrad in 1946, and which, because of WWII and the collapse of the Soviet empire, is now a Russian exclave and an enclave within the European Union, surrounded by Poland, Lithuania, and the Baltic Sea, geographically separated from mainland Russia, and home to its Baltic fleet: all the land masses have an odd number of bridges. This makes it impossible to plan a walk that crosses each bridge exactly once.
Euler’s reformulation – he replaced each land mass with an abstract vertex or node and called these A, B, C, and D and rewrote the question in the simplest terms as “Which land mass connects to which other land masses and how often?” – proved that there was no solution to the problem, that it was impossible to cross every bridge once and once only, that at best six could be crossed but never the seventh, no matter which route you took.
He did this by turning the map into a mathematical structure we now call a graph.
Not in the sense of x-y plane plots or bar graphs, more like a “network”, as in a social network, DNA sequencing, malware, misinformation and ransomware design, and the route optimisation on your phone’s GPS.
Thus were two new branches of math – graph theory and topology – born.
Idiot wind, blowing every time you move your mouth Blowing down the backroads headin’ south Idiot wind, blowing every time you move your teeth You’re an idiot, babe It’s a wonder that you still know how to breathe – Bod Dylan, "Idiot Wind", Blood on the Tracks, 1975.
Now, this Sunday, the day of the week that K. stretched himself out as if taking a position for his final act and settled into the precise posture that he preserved to the moment of death, is as good a time as any to revisit the topological problem of bridges and networks on two islands closer to my heart and home: Île de la Cité and Île Saint-Louis, which are linked to each other and to the Left and Right Banks of the Seine by 15 bridges.
Can we cross all these bridges once and once only? Where should we start? Where should we end? Does it matter? Does anything matter?
Of course, everything matters, and commenters who answer correctly why this is so – or supply interesting answers to any two of the questions above – will automatically advance to a new competitive round, which will take place once my tendonitis has healed and see us traverse, together, holding hands and breathing in and out from our noses or mouths as we see fit, each of Paris’s 336 bridges once and once only – the 37 crossing the Seine and connecting its banks to the city’s three islands, the 33 crossing the canals, the 149 crossing the périphérique, the 58 crossing Parisian streets, and the 33 SNCF railway bridges, 10 RATP bridges and 49 pedestrian footbridges.
And then we’ll sit down together to three dishes, wine, &c., with a small second course. And maybe a dram of Hungarian wine or a cordial. Sound good? Good.