Our first exposure to a bustling Seoul was abrupt. The Korean airport taxi driver suddenly stopped on a busy shopping street and started unloading our luggage. We looked around baffled for our accommodation. He pointed up a steep alleyway, demanded payment and screeched off into the traffic.

We stood on the pavement with two large suitcases, two small children, two rucksacks and a car seat while waves of shopping Koreans parted around us then re-converged behind like a river. Our www.airbnb.com host had sent us a series of pictures to guide us to his apartment. Flipping frantically through these, I eventually found the alley in the dark. We lumbered off.

Within 50 metres, I tripped. Stumbling helplessly towards a plate glass window, I fell quite literally into a passing tourist. The baby started howling with alarm and the extremely startled visitor took pity on us, dragging our cases through the pictorially represented maze of alleyways.

If our host, Youngjin, was surprised by a sweating, cursing, crying Western family pulling a load that would have defeated a good-sized carthorse through the heart of Seoul’s answer to Soho, he hid it well.

It was not what you would call an auspicious arrival in the Korean capital. The thought of navigating this foreign city alone with two kids the next day filled me with trepidation.

But then I discovered the subway, which somehow came to symbolise Seoul. It was a marvel. Move over London with your dirty sheets of free newspaper blowing everywhere; the Koreans have done underground transport in style.

Maps were displayed in English on gigantic Samsung touchscreens. Every platform had dozens of HD TV screens running public information videos suggesting that drinking coffee in the train or taking a photograph under a woman’s skirt were perhaps not the done thing (the kids were transfixed). The huge carriages were sparklingly clean, each journey cost just €1 and the trains were on time.

That’s what you get when you achieve an economic miracle. Since the Korean War in the 1950s, Korea has gone from being one of the poorer countries in the world, dependent on foreign aid, to the 15th largest economy in the world.

The result is that almost everything is jaw-droppingly modern and well planned. So instead of an underground creaking and groaning all the way from the 1860s, they’ve got the most extensive system in the world, entirely built since the 1970s and still having shiny new lines added today.

In a sobering reminder of the tension with neighbouring North Korea though, gas masks sit in glass cabinets on every platform.

So we sat on the train while everyone tried very hard not to stare at us. With our blonde hair, we stood out like beacons. Everyone else had black hair and was, well, Korean looking.

On the train, people tended to just nod, smile and offer us their seats. Outside though, it was a different matter. After a respectful period of double takes, eventually someone would break ranks and touch the kids’ hair.

The next thing you knew, we were surrounded, the children were clutching several treats and we were immortalised in dozens of smartphone snaps looking slightly overwhelmed. “Mummy”, said my son, “there’s another swarm coming”. The name for these incidents stuck. We got used to it though as we roamed around, sampling what Seoul had to offer.

If our host, Youngjin was surprised by a sweating, cursing, crying Western family pulling a load that would have defeated a good-sized carthorse through the heart of Seoul’s answer to Soho, he hid it well

For kids, the city has got it taped. The website www.seoulchildrensmuseum.org (€3) found us making fake sushi, balancing balls on jets of air and watching our own faces superimposed onto a cartoon.

At the World Cup Stadium Park, we swooped down a zip line and climbed up ropes to slide down clear Perspex tunnels. We were ostracised in the ball pit (apparently, Korean families don’t take their kids to mingle if they have a tickly cough and are covered in plague-like mosquito bites) and in the Science Museum (www.ssm.go.kr/v2/eng/sub22.asp, €0.70), both kids got happily saturated turning water wheels with well-directed jets of water.

Even the public toilets were a revelation with most having a parents’ room where a normal loo sat next to a miniature one for the kids. They also regularly sported potted plants.

Meanwhile, we were constantly assailed by a titanic clash between the old and the new. Spectacular traditional palaces in vibrant colours were dwarfed by skyscrapers topped with enormous flat-screen TVs. An ancient city gate was marooned in a roundabout of traffic. Serene Buddhist temples were wedged in between modern apartment blocks.

The sleek Namsan cable car (€4 one way) carried tourists to the Seoul Tower (€6), which is built over one of the city’s oldest fortresses, a place where warning fires would have been lit in case of invasion in the 1300s. Now it houses a modern communications aerial and young lovers hang padlocks on the wire fences here to symbolise their love; so many have sealed their union this way that in places the spectacular view has been obscured.

It’s possible to find other slices of Old Seoul still standing. Buckchon Hanok Village is filled with traditional Korean houses. These gorgeous properties are built with an adobe of soil, timber and rock. Visiting them pulls you briefly into a whisper of Korea’s past.

Nearby, in Gyeongbokgung Palace (€2), stony-faced guards in traditional costume tolerate photos with tourists and delight in initiating a noisy inspection procedure without warning, to the shock of gawping children (mine).

Our favourite window onto Old Seoul was Deoksugung (70 cents), however. This palace of “Virtuous Longevity” is the only one open in the evening. When the crowds had gone, lovers strolled hand in hand and soft lighting illuminated the gorgeous paintwork and detailed carving.

And then there was the food. Ah, the food. In the airport, during an early swarming, I’d asked a tween what his favourite food was. “Bibimbap,” he answered without hesitation. I ordered it expecting some kind of greasy teenage fast food. In fact, it was a tasty combination of fresh and pickled vegetables, mushrooms on long, delicate stalks, rice and seaweed, topped by a fried egg.

Korean barbecue was of course a must. The waiter took pity on us and actually cooked the meat at our table (it later became apparent that we should have handled this part ourselves) and the entire restaurant staff then took up residence at an adjoining table to watch the show.

We inexpertly manhandled the meat onto two layers of unidentifiable, but delicious leaves with chopsticks. We then heaped on the condiments and attempted to transfer it to our mouths. Eventually, they could handle it no longer and someone was dispatched to the table to inform us good-naturedly that we were slathering on far too much “heavy” sauce and not enough veg, while the rest all giggled helplessly behind her back.

We were soon expert wrappers, however, from potato pancakes hot off a market griddle, to bindaettok, a crispy beansprout omelette. We tried everything, from abalone porridge for breakfast to beef bulgogi for dinner and buckwheat noodles in an icy tomato soup for lunch, drawing the line only at raw sea urchin.

We also drank vats of excellent coffee with what was described as “sweet potato glutinous stick donuts”; despite their reputation for being a tea-loving nation, the Koreans are really passionate aboutcoffee shops.

And then there was the shopping. At Dongdaemun Market, you can buy pretty much anything but the best bargains are to be found in the fashion aisles. I had to dig around, but wedged between the nasty sequined T-shirts made in China were some cutting-edge Korean design pieces at extraordinary prices.

When I ran out of Won, the stallholders obligingly relieved me of my dollars too. Outside the multistory market, I was revived with freshly squeezed lime juice loaded with strange spices and sipped from a plastic bag.

Despite the fact that massively rapid development has swallowed huge tracts of countryside, the Koreans adore hiking and the subway is regularly packed with people wearing identikit walking outfits in a rainbow of hi-tech fabric colours, on their way to locations like Bukhansan National Park.

This is no lonely trail; with 10 million visitors a year, the park needs an infrastructure of metal stairs and boardwalks, but it’s a great place to exchange urban sprawl for mountaintop views and fresh, misty air.

Back in the city, we hired a tandem and a bike with a baby seat in Yeouido Park for €4 an hour, and suddenly we were free. The park segued into the riverside cycling path, an incredible stretch of open space which runs for a long section of the Han River.

After six days, we left Seoul pretty much as we entered; laden down with more luggage than we could realistically carry. An unbelievably efficient bus service made short work of the trip to Incheon airport for just €10 though.

Airports are rarely much to write home about, but there’s a reason why Seoul is regularly voted one of the world’s favourites; even the fast food has a twist on it here.

My burger was wedged between two rice patties slathered with bulgogi sauce and while we ate, the kids threw themselves round an inspired playroom and thus fell into an exhausted sleep the minute we got onto the plane.

Seoul city planners, I salute you for thinking of everything.

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