In 18th Century Prussia, Loenhard Euler decided to take a walk along the Seven Bridges of Königsberg, crossing every bridge exactly once. He had trouble devising a route that would end with him back where he started, though, so he invented graph theory just to prove that it wasn’t possible.

Euler’s idea to represent real-world phenomena as graphs — mathematical structures composed of nodes and the edges connecting them — is fundamental to a great deal of today’s science and technology. Graphs (or networks) are central to modern biology, linguistics, sociology, etc. The Internet is one big graph.

An Eulerian path traverses every edge (“bridge”) in a graph exactly once, while an Eulerian circuit also ends up back on the same node (“landmass”) as its starting point. Euler showed that if every node connects to an even number of edges (its degree), it is an Eulerian graph and there exists an Eulerian circuit (thus earning him all the naming rights?). Too bad for Königsberg, which had all odd-degree landmasses and thus no circuit….

Pittsburgh’s Eulerian Circuit

A recent brush with the so-called “Königsberg bridge problem” got me wondering about my home, Pittsburgh, the “City of Bridges.” With 400+ bridges, 3 rivers, and 3 main landmasses, is there a walkable Eulerian circuit somewhere in there? First, I winnowed it down just to the bridges which:

Cross a river (more than 90% are actually land bridges)

Have pedestrian access (a few are limited to railways or motor vehicles)

Connect to the city of Pittsburgh on both sides

It turns out there are 15 such bridges, and they connect in such a way that each of the three landmasses have even degree! 14 bridges connect to “The Triangle,” and 8 bridges each touch the North Side and South Side:

This circuit (map) is just over 20 miles (very doable in a day), largely follows the Three Rivers Heritage Trail system, and passes through several key parks and neighborhoods. So I decided I wanted to hike it….

The Eulerian Circuit Hike

I lead the research group at Duolingo, and when I mentioned my Eulerian hike idea to our VP of Engineering (who did some pioneering research on online social networks), she was enthusiastic about hiking it with me. She also nudged me to spam our all-hands list to see if others wanted to join. Once we settled on a date (one NYC-based colleague even spent the week at our PGH HQ specifically to do this with us!), a total of 13 “Duos” and 4 guests met up in Lawrenceville at 8:00 a.m. for what promised to be a unique day….

(Note: “Eule” is the German word for “owl,” which is Duolingo’s mascot, so of course we brought a Duo plushie along for the adventure. And to clarify: this wasn’t on official company outing, just something a few of us did for fun!)

Built in 1923, this arch bridge commemorates George Washington’s near-death experience crossing to meet French forces along the Allegheny River (a precursor to the 7 Years’ War). It was also part of my running route for my first 7 years in Pittsburgh, when I lived in hipster-turned-yupster Lawrenceville. We got honked at while crossing, and nearly lost our Head of HR to a ravine that she thought might be a shortcut down to the riverside trail….

Our second bridge was another arch style, built from 1927-1928 to replace a series of older bridges (previously aligned with 30th Street) that were destroyed by flood and fire, respectively. So far so good with this one. It was actually the longest bridge we crossed all day (2,681 feet; just over half a mile), with the highest water clearance as well (72.6 feet), and a nice distant skyline view.

From there we wandered a mile southwest to the heart of the Strip District, where we paused and split up for coffee, biscotti, breakfast pizza, grass jelly drinks, or whatever other assorted morning fare before continuing to….

This is a fun one. Built in 1923 (naturally, to replace a covered wood bridge that burned down), it has a pair of bronze sculptures on each pylon, based on the Fontaine de l’Observatoire by Jean-Baptiste Carpeaux in Paris. During a 1978 restoration, contractors “cleaned” the sculptures on one side by stripping off the protective verdigris patina and coating them with bronze-colored lacquer instead. Art historians noticed what was happening and prevented the same treatment on the other side, so now the two pylons don’t really match. That was 40 years ago, and the lacquer has yet to wear off to let the weathering resume on the city side. Nice one, Pittsburgh.

The next three bridges are known as the “Three Sisters” since they’re essentially identical 1920s yellow suspension bridges; the only triplet of bridges in the United States. In fact, they shut down all three of them in 2009 to film different parts of the same car chase scene for the film The Next Three Days in tandem. This mucked with bus routes and stranded the singer of my band on the North Side, so we had to cancel band practice. True Story. (The final battle scene from Inspector Gadget was also filmed on one or more of these bridges.)

Anyway, the first bridge is named for biologist and environmentalist Rachel Carson, who grew up a few miles upstream along the Allegheny River. (Some of my fellow bridge hikers just recently completed the “Rachel Carson Challenge,” a 35-mile wilderness hike on the summer solstice.)

The next bridge, which honors Andy Warhol, took us back across toward the museum that also honors him. This is in fact the only bridge in the United States named for a visual artist, and said museum is the largest dedicated to a single artist. Out of the “three sisters,” it also has the most recent paint job. Allegedly, the city considered painting it pink, but decided to stick with tradition instead. It was also the site of a spectacular yarn-bombing in 2013.

The last of the sister bridges honors Roberto Clemente, an 18-season Pirates right fielder, and the first Latin American inducted into the Baseball Hall of Fame. It brought us back across the river from PNC Park (where the Pirates now play), which many Pittsburghers wanted to be named after Clemente himself. The (re)naming of the adjacent bridge was apparently a compromise. On game days, this bridge is closed for pedestrian-only traffic.

We wandered along the concrete riverside up to Point State Park for a rest stop, water refill, and giant fountain photo op! The park is built around the remains of Fort Duquesne, a French star fort that was first established in the 1750s. That fort was the namesake of our next bridge, obviously, which took us across the Allegheny river for the 7th and final time for the day.

Construction on the bridge was mainly 1958-1963, but it wasn’t opened until 1969 due to delays in constructing the approach ramps on the North Side. So for six years it was known as the “Bridge to Nowhere” because that’s quite literally what it was. It was also our shortest bridge of the day: a mere 826 feet across a narrow point in the Allegheny. There’s a nice pedestrian/bike ramp directly from “The Point” across to the northern trail, which goes past Heinz Field (home of the Steelers), the Carnegie Science Center, and a sketchy casino on the way to…

This “bowstring arch” bridge was our only bridge across the Ohio River, connecting us from the North to the South Sides directly. It was also the first bridge of the day that I had never personally walked across before (nor had anyone else), so it took us a minute to find the staircase up to the pedestrian sidewalk. The bridge itself isn’t that much to look at, but it’s a great place to look from, offering fantastic views back to The Point from the water (without a boat).

When we got off the bridge a little after noon, a small faction started to grumble about lunch. When we reached the Duquesne Incline, 3 folks parted ways to go eat at the top of Mt Washington. (They promised to rejoin us, but never did.)

The rest of us continued along to the other bridge that goes directly into Point State Park. Fort Pitt was the English military base that replaced Fort Duquesne during the 7 Years’ War (and later grew into Pitt’s Boro — and finally Pittsburgh). It was also our first of 7 bridges across the Monongahela River, which we followed for the rest of the day. (This bridge and the tunnel connected to it were featured prominently in The Perks of Being a Wallflower.)

At that “point” (pun intended), we decided to break for lunch with a short detour to Market Square, which was filled with yummy but quickie restaurants, and was more or less on the way to Smithfield Street, connecting us to….

Not only is this one the more distinctive bridges we crossed all day, it’s also the oldest. It opened in March 1883, two months before the Brooklyn Bridge in New York. However, construction on that bridge began 14 years earlier (compared to 2 years), so NYC actually wins the trophy for America’s oldest roadway bridge. That’s okay, we still have more bridges and an Eulerian circuit, so you can the the oldest bridge if you want to, Big Apple. The “double-helix” look of this lenticular truss bridge was quoted for the “oo” in Google’s logo. (It also makes movie appearances in Flashdance and Striking Distance, among others.)

Here is where things got decidedly less pedestrian-friendly. At the end of the Smithfield Bridge, it wasn’t obvious how to climb the 150-foot elevation gap to get onto the Liberty Bridge (our highest point of the day). We considered taking the Monongahela Incline up and finding a connection point somewhere on the top of Mt Washington, but the station was swarming with tourists and getting all 14 of us up there might have taken ages.

With some help from GPS and Google Maps, we managed to zig-zag up some winding, sidewalk-impoverished streets up to a well-hidden staircase that gave way to the pedestrian path on (only) one side of the bridge. Now, this is the bridge that caught fire in 2016, and even though it re-opened a month later, there was (unbeknownst to us) still some pedestrian path repair work to do, because we dodged several construction workers along the way. (We were also honked at by a passing Uber full of our lunchtime defectors from earlier in the day.)

Finally, the staircase back down to street level dropped us straight into the enchanting Firstside Park, an obscure little sculpture garden tucked away underneath the busy highway overpasses. This was also just over the halfway point for our journey (in mileage, not bridges), so we decided to break for a little relaxation and scenery. A few of us did this lying down. We also lost two more of our party at this point: both out-of-towners who had evening plans and knew from the start that they wouldn’t be able to go all the way….

With a dozen hikers left, we followed Second Ave toward the next bridge. Then our hopes were dashed by a road sign: “10th Street Bridge Closed.” From a distance, the yellow suspension bridge was definitely half-encased in blue padding. Our Eulerian circuit was in jeopardy. Thankfully, though, it was only closed to southbound road traffic. Half the bridge was still open (including the sidewalk), so our circuit was still a go! About that bridge though: I’ve heard that it was a “dry run” for San Francisco’s Golden Gate Bridge (which broke ground after the 10th Street opened in 1933), since they’re both Art Deco suspension bridges with cables provided by the same firm. But I suspect that story is apocryphal. (Oh, and a Jack Reacher car chase was filmed on it).

From 10th Street we had a lovely stroll through Pittsburgh’s historic South Side along Carson Street, where there are so many bars that you could go to a different one every weekend and not repeat yourself for years. This gave us a glimpse of the Tudor-cum-Gothic Oliver Bath House on our way to a café for much-needed iced coffees, water refills, and bathroom breaks before heading up the street to our youngest bridge of the day….

Once you’ve walked a dozen bridges, you notice some trends. For example, newer bridges — like this one from 1977 — were built for wheels, not feet. Now, there were clearly protected pedestrian sidewalks on this bridge, but no clear way to get to them. So we scratched our heads in front of a high-end furniture store for half an hour just trying to figure out how to cross it safely. (The furniture sales folks called us “a pretty sophisticated crowd for a bar crawl,” whatever that means.) We finally figured it out: if you take 22nd Street north behind the library, past Ornsby Park, then take a right and ignore any illicit happenings in the parking lot beyond the baseball field, you will find an overgrown and lavishly secluded staircase that ascends to the bridge. Duh.

Getting back off the bridge was another matter. On the other side we wandered, sidewalk-free, spiraling beneath a mess of concrete spaghetti until we found ourselves at a bus stop somewhere on a deserted part of Second Ave. We could see the Great Allegheny Passage trail (converted from an old railway line) above and behind us, but couldn’t find away to get onto it. So we headed east through an industrial mystery zone until we spotted a fenceless dirt slope (which we inelegantly climbed) up to the trail, and could follow it in peace to….

Completed in 1887 (making it our second-oldest bridge of the day), the Hot Metal bridge was built to carry exactly that. Rail cars transported crucibles of molten iron from the blast furnaces on one side of the river to be fashioned into steel on the other side. It’s estimated that 15% of all American steel made during the World War II effort went across this bridge.

And after the three previous bridges of harrowing and near-inhospitable walking conditions, this bridge was a frickin’ dream. Not only did it have a dedicated bike/pedestrian path of exceptional berth, but it was clean, well-maintained, and connected directly to the hiking trail on both sides! We stayed on the Great Allegheny Passage for another a 1.5 miles or so before taking a hole-in-the-fence detour to Page Dairy Mart. Pro-tip: Ice cream after hiking 18+ miles across 14 bridges is really, really yummy. If I’m being honest, this stop was possibly the whole reason I even did this thing.

Somewhere around 5:00 p.m. we made our way back to the trail for the last 2+ miles to our final bridge, connecting the 31st and 15th wards of the city. Despite being tired — we were already 9 hours in by this point — we were able to pleasantly zone out, chat, and keep our eyes peeled for bald eagles for the better part of an hour. Once we exited the trail underneath the 1966 cantilever monstrosity that is the Glenwood Bridge, it was again not obvious how to gain access by foot. But by then we’d developed bridge-walker superpowers because we found the brush-covered, clandestine staircase without any help from GPS or Google Maps this time. And we got honked at while crossing. Full circle.

So we did it! We hiked Pittsburgh’s Eulerian circuit!

At first, wrapping up such a long and varied day in Hazelwood felt anti-climactic. It was once a bustling steelworker neighborhood, but the last operating steel mill within city limits closed there in the 1990s. Today, it’s pretty sparse and not well-served by public transit. We stopped at a parklet next to a senior center (in a defunct railway company building) to wait for a fleet of Uber/Lyft drivers to take us back into more familiar and “developed” parts of the city.

Yet like so much of Pittsburgh, Hazelwood has a deep history. Pulitzer-winning playwright August Wilson moved there as a teenager when his family was displaced by Civc Area construction in the 1960s. Rapper Wiz Khalifa grew up there decades later under very different conditions (it’s the backdrop for parts of his “Black and Yellow” music video). Even though Hazelwood doesn’t even have a proper grocery store at the moment, the hip French bakery La Gourmandine just moved their baking operations there from the original Lawrenceville location, which satisfyingly connects us back to the start of our journey. Hazelwood — like Lawrenceville, the South Side, East Liberty (where Duolingo and Google are), and much of the city, to be honest — might be on the verge of its own renaissance. Walking a century’s worth of bridges may well offer a little perspective on where a place has been, where it is, and where it’s going….

Conclusion

If you like bridges, Pittsburgh, or urban hiking, then I really recommend doing this circuit sometime. It’s not just a nerdy thing to do for the sake of saying you’ve done it. (I mean, it is… but it’s not just that!) This particular route is pretty well-paced and hits some cool highlights of the city, in my opinion. If I do it again, I think I’ll take a detour up Mt Washington (between the West End and Fort Pitt bridges) along the Emerald View Trail and back down the Duquesne Incline for some more elevated views of the city.

Some summary stats:

Hiker retention : 71% (12 out of 17)

(12 out of 17) Bridge quotient: 22% (4.5 miles out of 20.4)

(4.5 miles out of 20.4) Decade with the most bridges crossed: 1920s (7)

(7) Style breakdown: arch (7), suspension (4), cantilever (2), truss (2)

Back at Duolingo HQ, there are already talks of NYC or Seattle Eulerian hikes, since those cities have the largest contingents of Duos outside Pittsburgh. Although neither have Eulerian circuits like Pittsburgh (only paths).

(Note: If you allow only one side to connect to Pittsburgh proper, there’s also an 18-bridge Eulerian circuit that’s 34 miles long. But it’s probably more of a bike than a hike, and doesn’t look to be as fun since it basically inverts the path, adds one bridge to each river, and requires some backtracking.)

Photo credits: Variously taken by Cindy Berger, Cindy Blanco, Molly Lindsay, Jeesoo Sohn, Amanda Strader, and myself.