Rosenborg is the Renaissance castle Christian IV built as a summer retreat in 1606, and never properly left. Red brick, copper spires, the Dutch Renaissance style the Danish kings imported when they wanted something newer than what the previous king had. It sits in the middle of Kongens Have, Copenhagen's oldest park, free and busy on any warm afternoon.
Two and a half kilometres, seventy-five minutes if you do the basement crown jewels. Start at Nørreport — the city's busiest transit knot, where M1, M2, M3 and the S-tog all meet. The park's main gate is two minutes east on Gothersgade. Walk east through the lime avenues; the lines are Christian IV's even if the trees themselves have been replanted.
The castle is at the centre. Pay to go in; the audience halls upstairs are the architectural payoff, the Skatkammeret basement is where the crown jewels and the Coronation Throne are. The Herkulespavillonen is a small 1773 folly in the southern grounds. North across Øster Voldgade, the university's Botanisk Have opens onto the old rampart land — the Palmehus (1874) is the small Danish version of Kew's. Back to Nørreport via the boating lake.