Four days in Copenhagen: the harbour is clean enough to swim in, and people do

Start with the two things nearest the station, which is convenient and also happens to be the right order. 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 the French were still unfashionable, on Gauguin, Degas and van Gogh. The building was finished in 1897 and the domed winter garden at its centre, 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 moat, and the point of it is the lighting after dark, thousands of coloured lamps in the trees. There is a wooden rollercoaster from 1914 with a brakeman riding in it.

Eat before you go in or inside on the cheap. Tivoli has more than thirty restaurants and the good ones are priced for a special occasion.

Christiansborg first, 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 the parliament, the prime minister's office and the 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 ruins of Absalon's medieval fortress in the basement, which is the reason Copenhagen exists. 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 was built in 1642 as an observatory and instead of stairs it has a cobbled spiral ramp coiling up the inside, wide and shallow enough that a horse and carriage could take it. A Russian tsar is supposed to have ridden up. At the top there is still a working telescope and a low platform looking out 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. The long gallery upstairs has three silver lions guarding the coronation throne, and in the strongroom under the building are the Danish crown regalia, still in use. The park it sits in is where the city spends its afternoons.

Do the Little Mermaid early or not at all. She is a bronze figure 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 at Kastellet is the good part anyway.

Walk back south to Amalienborg for noon. The royal residence is four matching mansions from the 1750s facing 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 ceremony is longer and more serious when the monarch is actually in residence, which the flag will tell you.

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 different addresses along it. The painted houses are the photograph everybody takes, and between about eleven and four it is close to unwalkable. Come at six, when the light goes down the water and the crowds have 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 the last morning because the metro from here reaches the airport in about twelve minutes. Leave the bags at the hotel or the station and come over the bridge.

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 of them in the open air with a handrail and a lot of sky. The story locals tell is that the architect discovered the spiral turned the wrong way and threw himself off it, which is not true and gets better 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, which is the polite way of saying you should look at the new city as well as the old one. Copenhagen has spent thirty years building housing and bridges that other countries send delegations to see, 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.