For three weeks each summer the sleepy town of Gråsten (population 4200) is abuzz as Queen Margrethe and Prince Henrik (and usually the extended family) head for some R&R at their summer residence. When they’re not visiting, the lovely palace gardens are open to the public; it makes a good half-day trip from Sønderborg, 16km away. Seasonal closing times vary, from 4.30pm in winter to 8pm in summer.