I agree with their recommendation on the Nyhavn neighborhood. You can also catch boat rides there that take you around the harbor.
I really enjoyed the Danish National Museum. It is always interesting to see how countries present their culture and history to outsiders. Like most modern museums in Europe, the information is mostly in both the local language and in English making it quite accessible.
Do stop at Tivoli Gardens. One is tempted to call this an amusement park but it is more than that. Think Seattle Center but with gambling. Indeed, a stop at Tivoli on my high school student group tour of Europe was literally the first time I ever saw slot machines in person. My recollection from grade 11 is that they were crowded with little old ladies gambling away their pension checks. It is great fun just to walk and watch here, and reasonably good Danish food can be had.
By all means wander through Christiania as well. This 60s throwback is what Berkeley always wished it could be. Christiania is not, itself, all that it used to be. In the national museum they have an example of a pot-selling stand taken from Christiania's past, but even without the open air drug market it is still great counterculture fun that has not yet (though one can imagine it may at some pointed) jumped the shark into a 60s-theme tourist trap. Best not, though, to demand to be taken here straight from the airport as one economist, who shall remain nameless, did when visiting Denmark. Your gentle hosts will repeat the story forever after to the amusement of future guests.
If you are flush with money, the downtown Marriott, where I stayed at the expense of the much-abused Danish taxpayer while teaching a course, is quite nice.
Who was my favorite student this term?
11 months ago