Ten days in Denmark: a country you can cross in three hours, and shouldn't

Both of the afternoon's stops are two minutes from the central station. The Glyptotek is a brewer's private collection made public: Carl Jacobsen of Carlsberg spent his money on Roman portrait busts, Egyptian sarcophagi and, when it was still unfashionable to do so, on Gauguin, Degas and van Gogh. The building was finished in 1897 and the domed winter garden at the middle of it, palms and a fountain under glass, is where the café is.

Then Tivoli, across the road, which opened in 1843 on the ground just outside the old city ramparts and has been the middle of Copenhagen ever since. Go in the evening. It is a garden with rides in it rather than a theme park with flowerbeds, the lake is the flooded remains of the city moat, and the reason to be there after dark is the lighting, thousands of coloured lamps strung through the trees. There is a wooden rollercoaster from 1914 with a brakeman riding in it.

Eat before you go in, or eat cheaply inside. Tivoli has more than thirty restaurants and the good ones are priced for an occasion.

Christiansborg sits on its own island in the middle of the city. It is the fifth building on the site, finished in 1928, and it is the only place in the world where a parliament, a prime minister's office and a supreme court share an address. Buy the combined ticket: it gets you the royal reception rooms with their tapestries, the kitchens with their copper, the stables, and the excavated remains of the medieval fortress in the basement, which is the reason Copenhagen exists at all. The tower is free and has the best cheap view in the city.

Walk north through Strøget, the pedestrian spine, to the Round Tower. It went up in 1642 as an observatory, and instead of stairs it has a cobbled spiral ramp coiling up the inside, wide and shallow enough for a horse and carriage. A Russian tsar is supposed to have ridden up it. There is still a working telescope at the top and a low platform over the copper roofs.

Finish at Rosenborg, ten minutes further north, a small moated castle that Christian IV built as a summer house and then never really left. Three silver lions guard the coronation throne in the long gallery upstairs, and the Danish crown regalia are in the strongroom underneath, still in use. The park outside is where the city spends its afternoons.

Do the Little Mermaid early or not at all. She is a bronze a metre and a quarter high on a rock at the water's edge, put there in 1913, and by ten in the morning there are coaches. At half past eight there is nobody, the light comes off the sound, and the walk out along the harbour past the star shaped citadel is the good part anyway.

Walk back south to Amalienborg for noon. The royal residence is four matching mansions from the 1750s round an octagonal courtyard, and there is no wall, no moat and no gate: the square is a public street with the head of state living on it. The guard changes at twelve, having marched down from the barracks at Rosenborg, and the flag tells you whether the monarch is in.

Save Nyhavn for the end of the day. The canal was dug in the 1670s for cargo, silted into disrepute, and was full of sailors and tattoo parlours within living memory; Hans Christian Andersen lived at three separate addresses along it, which will mean more to you in four days' time. Between eleven and four it is close to unwalkable. Come at six, when the light runs down the water and everybody has gone to dinner.

Christianshavn was laid out in the 1610s as a fortified merchant town on its own island with canals on the Dutch model, and it is a short walk or one metro stop from the centre.

The Church of Our Saviour has a black and gold spire with an external staircase spiralling round the outside of it, four hundred steps, the last flight in open air with a handrail and a great deal of sky. The story locals tell is that the architect discovered the spiral turned the wrong way and threw himself off, which is not true and improves every time it is told. The view takes in the whole city and, on a clear day, Sweden.

Then walk the canal to the Danish Architecture Center on the waterfront. Copenhagen has spent thirty years building housing and bridges that other countries send delegations to look at, and this is where it explains itself. There is a slide down four floors, which is more fun than it has any right to be.

Pack tonight. Tomorrow's train is short but the good part of Odense is the afternoon.

The train takes an hour and a quarter and passes over the Great Belt on the way, eighteen kilometres of tunnel, island and suspension bridge that took a decade to build.

Odense is where Hans Christian Andersen was born in 1805, to a shoemaker and a washerwoman, in one room of a small yellow corner house in the poorest part of town. He left at fourteen for Copenhagen with almost no money and a determination to be famous that everybody who met him found exhausting, and he became the most translated author in the world after Shakespeare. He also disliked Odense and rarely came back.

The house is still there, still yellow, and is the most photographed building in the city. Around and under it, a new museum opened in 2021, designed by the Japanese architect Kengo Kuma as a set of circular rooms sunk into the ground with hedged gardens above, so that the old house stays the tallest thing on the site. Inside it is deliberately not a biography: the rooms are staged as his stories, dark, disorientating and occasionally frightening, which is closer to what he actually wrote than the Disney version.

The garden along the river below is the gentle version, with a paper boat sculpture in the water.

St Canute's Cathedral is a brick Gothic church named for a king who was killed in Odense in 1086, cornered in a church by rebels who objected to his taxes and speared at the altar. He was made a saint fifteen years later and his skeleton is in the crypt in a glass topped casket, with his brother Benedict in another beside him, both of them plainly visible and unexpectedly small. The altarpiece above is the other reason to come: a gilded oak triptych carved by Claus Berg around 1520, three hundred figures deep, and one of the finest late Gothic altarpieces anywhere in northern Europe.

Brandts, five minutes away, is a nineteenth century textile mill converted into the city's art and media complex: a museum of photographic art, a museum of media, changing exhibitions in four large halls, an art cinema and a good deal of coffee. The conversion is one of the better ones in Denmark and the buildings themselves are worth the walk.

In the afternoon, and this is not a joke, the Danish Railway Museum, which is directly behind the station in the old roundhouse. It holds dozens of locomotives and carriages over ten thousand square metres, including royal saloons, and it is one of the largest railway collections in northern Europe. In summer it runs vintage steam trips out into Funen. Even people who have no interest in trains come out impressed by the roundhouse.

Ninety minutes on the train, across to Jutland and north. Drop the bags and start at the cathedral, a hundred metres from the main square: the longest and tallest church in Denmark, begun in the twelfth century and finished over the following hundred years. Inside it is white, high and unusually plain, apart from a five panel altarpiece by Bernt Notke of Lübeck from the 1470s and a set of medieval frescoes whitewashed at the Reformation and uncovered again in the twentieth century.

In the afternoon, ARoS, ten minutes away. The permanent collection runs Danish golden age painting through to contemporary installation, and the nine black walled rooms in the basement would justify the ticket by themselves. But the reason everybody comes is on the roof: Olafur Eliasson's circular glass walkway, a hundred and fifty metres round, glazed in every colour of the spectrum in sequence, so that walking it turns the whole city red, then orange, then yellow, and on round.

Go up an hour before closing. The low light is what makes it work and the crowds have gone by then.

Den Gamle By is not a reconstruction. It is seventy five real Danish buildings, dating from 1597 to 1909, taken down in towns all over the country when they were about to be demolished and rebuilt here into a coherent market town with cobbled streets, a millstream and working shops. There are staff in period clothes but not oppressively many, and they will talk to you about the work rather than perform at you.

What people are not prepared for is the twentieth century part. Two further quarters have been added, one set in 1927 and one in 1974, with a jazz club, a record shop, a hairdresser and a fully furnished flat in each. The 1974 apartment, with its brown kitchen and its cassette recorder, does something to visitors of a certain age that the seventeenth century houses do not. There is also a small flat set up as the home of a woman with dementia, used for reminiscence therapy, which is quietly the most affecting thing in the place.

Three hours minimum, and eat inside; the bakery and the 1920s coffee house are both real. Afterwards the botanical garden is immediately next door, with hilly lawns, ponds and a glasshouse complex whose tropical dome has a spiral ramp climbing through the canopy. It closes later than the museum and costs nothing.

Moesgaard is seven kilometres south and is the single best reason to come to Aarhus. The building, finished in 2014, is a huge wedge driven into a hillside with the roof planted in grass and sloping back down to meet the ground, so that people picnic and in winter sledge on the roof of the museum. Inside, a wide staircase descends through human evolution with life sized reconstructions of hominins standing at the landings.

At the bottom are the two things you came for. Grauballe Man was found in a peat bog in 1952, died around the third century BC with his throat cut from ear to ear, and is the only bog body anywhere preserved completely, down to the hair, the stubble and the fingerprints. Beside him is the Illerup Ådal deposit, thousands of weapons from a defeated army thrown into a lake as a sacrifice, laid out across an entire wall. Between the two is a reconstructed Bronze Age burial mound you can walk inside.

The parkland below runs down to the beach, is free and open all year, and has reconstructed prehistoric buildings along the way.

Coming back, get off in the Marselisborg forests. The deer park is a fenced section of woodland where fallow and sika deer wander loose and come close, free and unfenced from first light until dark. The royal family's summer residence is a couple of kilometres further, a modest white house whose gardens open to the public whenever the monarch is not in.

Two last things, both near the station.

In the basement of a bank across from the cathedral is a single room of Viking Aarhus, assembled from what came out of the foundations when the building went up: the town rampart, part of a house, pottery, a skeleton. It takes twenty minutes and it explains the name of the city. Aros, the river mouth, was a fortified settlement here by about 900.

Then walk down to the water for Dokk1, the harbour front library, which is the largest public library in northern Europe. Calling it a library undersells it: it is the city's living room, with a children's section that is essentially a playground, a robotic underground car park, and a bell that rings across the harbour every time a baby is born in an Aarhus hospital, set off by the parents from the maternity ward.

The station is five minutes from the door, and the capital is a little over three hours away.