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Baden means "bath" or "bathe" in German. The Romans, those intrepid explorer/conquerers, spent a lot of time in the springs in this area. According to Wikipedia, Caracalla himself came up for a soak.
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It actually wasn't Baden-Baden until 1931. Before that it was just Baden. Baden-Baden is the short form of Baden in Baden, as Baden is in Baden which has actually been Baden-Wuerttemberg since 1952. (Which, yes, if you're keeping score here, makes the name whole Baden-Baden in Baden-Wuerttemberg. For further etymological enjoyment, Baden-Wuerttemberg is also known as Laendle which is Swabian for "small land." Land, that is, not name.) Calling this city Baden-Baden differentiates it from Baden bei Wein (Baden by Vienna) and Baden Switzerland. And yes, the baden in Baden-Baden still are in use to this day.
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There is a funicular railroad that runs up Mount Merkur from which you can enjoy a spectacular view of Baden-Baden below.
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On an interesting note (which will be proven a very bad pun), Wikipedia notes that "The anthem of Baden is called 'Badnerlied' (English: Song of the people of Baden) and consists of usually four or five traditional verses. However, over the years, many more verses have been added--there are collections with up to 591 verses of the anthem." (Suddenly "99 Bottles of Beer" doesn't seem quite so long, does it?)
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Of course, when you're not taking train rides up the sides of mountains or singing the song of the people of Baden, you could take a walk through the Lichtentaler Allee, a historic park and arboretum and admire its plantings and footpaths.
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A walk in the park after a spa visit? I guess my arm could be twisted into such a thing. (Just don't make me memorize all the verses to that song!)