Twelve days in Scandinavia: Three capitals on one railway, and each one contradicts the last

Both of the afternoon's stops are two minutes from the central station, which is convenient and 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 middle, 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 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 cheaply inside. Tivoli has more than thirty restaurants and the good ones are priced for a special 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 the only place in the world where a parliament, a prime minister's office and a supreme court share an address. Take the combined ticket: the royal reception rooms with their tapestries, the kitchens with their copper, the stables, and the excavated remains of the twelfth century 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 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 cart. A Russian tsar is supposed to have ridden up it. There is 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 never really left. Three silver lions guard the coronation throne in the long gallery upstairs, and the Danish crown regalia are in the strongroom beneath, 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 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 regardless.

Walk back south to Amalienborg for noon. The royal residence is four matching mansions from the 1750s facing an octagonal courtyard, with 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. 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 the 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 it, which is not true and improves with every telling. The view takes in the whole city and, on a clear day, Sweden, which is where you are going tomorrow.

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, because the Stockholm train is a morning departure and a long one.

A little over five hours: out across the sound to Sweden, then north through Skåne and up past the lakes. Have lunch on the train.

Gamla Stan is a five minute walk from Stockholm Central and it makes sense of the rest of the city. The old town is a small island of tall ochre and blood red houses on lanes barely wide enough for two people, and the first written mention of Stockholm dates from 1252.

Stortorget is the middle of it, and the gabled houses on the north side are the backdrop to the worst two days in Swedish history. In November 1520 the Danish king Christian II, freshly crowned here, had about a hundred Swedish nobles and clergy executed on this square. The outrage put Gustav Vasa on the throne three years later and ended the union with Denmark, which is a long way of saying that Sweden as a country begins with a massacre carried out by the city you left this morning.

The Nobel Prize Museum is on the same square in the old stock exchange: one floor, small, and better than it sounds, with laureates' donated objects running past on an overhead conveyor and the banquet ice cream in the café.

The palace went up between 1697 and 1754 after fire destroyed the medieval castle, and it is one of the largest still in use by a head of state anywhere. The royal family live out at Drottningholm, which means the state apartments, the treasury with the regalia and the museum of the burnt castle in the cellars are all open on one ticket. Two hours is honest, and the guard changes in the outer courtyard around midday.

Walk west to Riddarholmen, the small island of noble palaces almost nobody crosses to. The church there is the oldest building in Stockholm, a late fourteenth century abbey with an iron openwork spire, and it stopped being a parish church long ago. It is a mausoleum now: most Swedish monarchs from Gustavus Adolphus onwards lie in it in black and green marble, with the arms of the Order of the Seraphim hung round the walls.

In the afternoon cross the water to the City Hall, the dark red brick building with the square tower that appears in every photograph of the city taken from the south. The Nobel banquet is held in the Blue Hall each December and the dancing afterwards upstairs in the Golden Hall, which is lined with eighteen million gilded mosaic tiles. Inside is by guided tour only; the tower is a separate ticket and worth it in summer.

Djurgården is a royal hunting park that became the museum island. Take the ferry from Slussen rather than the bus: it goes every quarter of an hour, costs the same as any other journey and puts the waterfront in front of you on the way over.

Go to the Vasa first and go at opening. The ship was the flagship of the Swedish navy, ordered by a king in a hurry, and it sailed thirteen hundred metres on its maiden voyage in August 1628 before a gust laid it over and the water came in through the open gunports. It sat in the harbour mud for three hundred and thirty three years, which is what preserved it, because the Baltic is too brackish for the shipworm that eats wooden wrecks everywhere else. It came up in 1961 and almost all the timber is original. Nothing prepares you for the scale in the room.

Then Skansen, twenty minutes uphill, which opened in 1891 and invented the open air museum. A hundred and fifty buildings were taken apart in the Swedish countryside and rebuilt here, a Dalarna farmstead, a whole town block of workshops, a wooden church people still marry in, with staff doing the work that went with them. There is a zoo of Nordic animals attached, so the wolves, lynx and wolverines you will never see in the wild are here. Choose half of it.

The ABBA museum is by the funfair at the bottom of the hill and is entirely unembarrassed about what it is.

Twenty minutes south on the green line is the strangest thing on this route and one of the best. The Woodland Cemetery was won in competition in 1915 by two architects who were both thirty at the time, and it is on the World Heritage list not for a building but for what they did with a pine forest and a slope. The path from the gate runs along an open meadow towards a low hill with a stand of trees and a plain granite cross beside the way, and the whole composition makes the walk itself the point. The graves are in the woods, unfenced, under the pines. Greta Garbo is here.

Go early, allow ninety minutes, and treat it as a working cemetery, because it is one.

Come back up the line to Södermalm for the afternoon. The southern island is a granite ridge with the city built up its sides, which is why it has the views. Monteliusvägen is a narrow path along the western cliff top with benches looking straight across at the City Hall and the old town. Round it are second hand shops, a great deal of coffee and the ordinary business of a neighbourhood people live in.

The Oslo train leaves in the morning and there are only a few a day, so check the departure tonight.

A long westward run through Värmland and over the border, then down to the head of the Oslofjord. It is forest, lakes and red painted farmhouses almost the whole way, and it is the quieter of the two long legs.

The Munch museum opened in 2021 in a tower that leans out over the harbour and was thoroughly disliked when the drawings were published. It holds the collection Munch left to the city in his will, over a thousand paintings plus many times that in prints, spread up thirteen floors that get quieter as you climb. He made several versions of The Scream and they are rotated rather than shown together, so which one you meet is luck. The point of a whole museum rather than one room is that Munch stops being a single picture: the sickroom paintings, the frieze of life sequence and the late self portraits are all here.

Then walk five minutes to the opera house and go up the outside of it. It is faced in white Carrara marble on slopes that start at street level and run to the ridge of the roof, with no ticket, no barrier and nobody telling you not to. Go in the last hour of light.

Bygdøy is a wooded peninsula ten minutes across the water and it holds most of Norway's national story. Take the boat from behind the City Hall rather than the bus.

The Fram first. Nansen had it built in 1892 with a rounded hull designed to be squeezed upward rather than crushed when the pack ice closed, then deliberately froze it in and drifted for three years to find out where the Arctic current went. Amundsen later took the same ship south and left from it for the Pole in 1911. It is indoors, complete, and you go aboard and down into cabins considerably smaller and darker than any account suggests.

Next door, the Kon-Tiki museum has Thor Heyerdahl's raft of nine balsa logs, which he and five others sailed four thousand miles from Peru into Polynesia in 1947 to argue that the Pacific could have been settled from the east. Most archaeologists think he was wrong about that. The raft in the room is very small.

Finish uphill at the open air museum, some hundred and sixty relocated buildings including the Gol stave church of around 1200 with dragon heads on its gables, and a street of apartments each furnished to a different decade between the 1850s and the 1980s. After a week of Danish and Swedish equivalents you will find yourself comparing all three, which is the pleasure of doing them in one trip.

Gustav Vigeland spent the last twenty years of his life on one commission and the city gave him a park for it. Frogner park is free, never closes and holds around two hundred figures in bronze, granite and wrought iron, all by him, arranged along an axis climbing to a seventeen metre monolith cut from a single block of stone with a hundred and twenty one bodies twisting up it. Subtle it is not, and there is nothing else like it in Europe. Children climb the statues and nobody objects.

In the afternoon take the metro up line 1. It climbs steadily out of the city until the last stations are in pine forest four hundred metres up. The ski jump at Holmenkollen first opened in 1892, has been rebuilt many times over, and the current steel structure went up in 2010; a lift takes you up the tower, and the oldest ski museum in the world is dug into the hill beneath it.

Stay on to the end of the line for waffles with sour cream and cloudberry jam at Frognerseteren, a dragon style timber lodge from the 1890s with the whole city and the fjord below it. In winter the ski trails start at the platform and people come out in the evening to use them, which is the single most Norwegian fact on this route.

The National Museum opened in 2022, absorbed four older institutions and is now the largest art museum in the Nordic countries. Stick to the second floor: Norwegian and European painting from the fifteenth century on, a room of Munch including one of the painted Screams, and before it the nineteenth century landscape painters whose pictures of the fjords are the reason the rest of the world started coming to Norway at all.

Then walk east along the water to Akershus, the fortress on the headland, begun around 1300 and never taken by a besieging army. The grounds are free and open late, the ramparts look straight down the fjord, and the resistance museum in the old arsenal is worth twenty minutes: this is where the occupation authorities executed Norwegian resistance members, and where Quisling was shot in 1945.

Oslo Central is ten minutes from the fortress gate and the airport trains leave every ten minutes.