# That’s Maths: When a walking route in central Paris is bridge too far

## Swiss mathematician’s bid to resolve problem known as The Seven Bridges of Königsberg is first real puzzle in topology

Leonhard Euler, the outstanding Swiss mathematician, considered a problem known as The Seven Bridges of Königsberg. It involves a walk around the city now known as Kaliningrad, in the Russian exclave between Poland and Lithuania. Since Kaliningrad is out of the way for most of us, let's have a look closer to home, at the bridges of Paris.

In the centre of Paris, two small islands, Île de la Cité and Île Saint-Louis, are linked to the left and right banks of the Seine and to each other. We consider only bridges that link the two islands to each other or to the river banks, and ignore all other bridges spanning the Seine. The figure shows there are seven bridges from the left bank, seven from the right bank, 10 from Île de la Cité and six from Île Saint-Louis. This makes 30 in all but, as each bridge is double-counted, there are in total 15 bridges.

Inspired by Euler, let us seek walking routes through the centre of Paris that cross each bridge exactly once. We distinguish between open routes that link from one point to another, and closed routes that lead back to the starting point. We pose three problems. The first is this: starting from Saint-Michel on the left bank, walk continuously so as to cross each bridge exactly once. The second problem is not so simple: start at Notre-Dame Cathedral, on Île de la Cité, and cross each bridge exactly once. The third problem is this: find a closed route crossing each bridge once and returning to its starting point.

Euler reduced the problem to its basics, replacing each piece of land by a point or node and each bridge by a link joining two nodes. He saw immediately that any node not at the start or end of the walk must have an even number of links: every link into the node must have a corresponding exit link. Thus, there can be at most two nodes with an odd number of links. Moreover, if any node has an odd number of links, there are no closed routes.