Bristol’s 42 crossings – not a bridge too far for maths ace

David Clensy meets the Bristol University mathematician who challenged himself to cross all of Bristol’s bridges - going over each one just once

CROSSING seven bridges doesn’t sound like the most taxing challenge. But crossing seven bridges when you’re only allowed to cross the bridges once, in one direction, was a mathematical problem that challenged the greatest minds in the Prussian town of Königsberg in the 18th century.

When mathematician Dr Thilo Gross arrived in Bristol a year-and-a-half ago to take up his new post teaching engineering mathematics at Bristol University, he soon noticed that his new home had some notable geographical likenesses to Königsberg.

But while Königsberg had seven bridges and two islands of land caught between two rivers, Bristol had three such “islands” – Spike Island, St Philip’s and Redcliffe – and it had a more taxing 42 formal bridges that could be crossed by foot.

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“I came here to Bristol because it was one of the best places in the world in terms of my own particular specialism – network mathematics,” he says.

“That’s essentially what the Königsberg bridge problem is – it’s all about understanding the mathematics of networks.

“I teach the problem to all my students, so when I realised it could be done here in Bristol, I couldn’t resist trying to work it out.”

It took Dr Gross a fortnight with a pen, a map and countless screwed up and discarded pieces of paper – but eventually he worked out his solution.

But rather than leaving it as an academic puzzle, he couldn’t resist putting on his walking boots and taking to the streets of the city for himself – even though completing the route would mean a 33 mile walk.

“It was a very long day,” he tells me, as he recovers from his self-inflicted trial by rambling, a few days after his very long walk.

“It took me more than 12 hours to complete the walk – but it was absolutely worth it when I stepped over the finishing line at Clifton Suspension Bridge. To be honest I never thought I’d be able to complete the walk until the very moment when I walked over the Avon Gorge.”

Eventually mathematician Leonhard Euler proved that the problem for Königsberg has no solution.

In the history of mathematics, Euler’s negative solution of the Königsberg bridge problem in 1735 is considered to be the first theorem of graph theory – a discipline without which much modern computer science would be impossible.

His treatment led to the birth of the branch of maths known as discrete mathematics, and by extension modern day network science that is relevant to a wide range of application areas from the running of the internet to controlling epidemics.

So for Dr Gross to be able to demonstrate to his students how here in Bristol the problem can be solved, was something of a “eureka!” moment for him.

“The key is to understand first and foremost how you are going to cross from each island to the next without trapping yourself,” Dr Gross explains.

“My initial plan had me starting my walk in Bedminster, but I quickly realised I cornered myself – especially when I discovered that Bristol had a bridge I didn’t know about.

“It is a small railway bridge near Temple Meads railway station, which to my amazement had a small footpath running alongside it.”

The discovery of that one bridge sent Dr Gross back to the drawing board.

“There is nothing intuitive about this problem,” he says. “It’s not that you start at one end and work your way neatly to the other – it’s far more complex than that, and this is why it takes 33 miles of walking.”

He eventually started his walk at Spike Island, criss-crossing back and forth along the bridges of the New Cut in an easterly direction, before looping back around the Temple Quarter, across Bristol Bridge, Pero’s Bridge and Avon Bridge before tackling Plimsoll Bridge and walking to Sea Mills and then Avonmouth Bridge, before returning back on the North Somerset side of the river and crossing the final bridge – Brunel’s Clifton Suspension Bridge.

“There will be other ways to do it,” he says. “But I particularly wanted to finish the day with the best known bridge in the city.”

In fact, the mathematical treatment showed that ending the walk with the suspension bridge was not possible until recently, when a new footbridge was built behind Temple Meads Station.

The solution of the bridge problem, may not be a great mathematical discovery, but Gross says it was a great personal achievement.

“In a city with a name – Brycstow – that means ‘place of the bridge’ and which has a bridge as its major landmark, I just had to do it,” he says.