You can tour the whitehouse if you make prior reservations. I'm almost certain it's 100% free, but I don't remember. When I went, I got to go through all these back corridors to get to the elevator, and I saw this room with all of these crazy sci-fi type computers and stuff, then they quickly shut the door. I'm not kidding you, it looked like 80's sci-fi.
Anyway, the usual monuments are rather boring. Do a once around the mall, and you're fine. The Masonic Museum was boring, in my opinion. As was the air and space museum. Both were small (Big exhibits, but if you count up the exhibits, rather small) in comparison to the other smithsonian museums. Natural History museum was possibly the coolest museum I've ever been to. Amazing stuff, there. I'm starting to get foggy with my museums, because I've been to a LOT of places.
A majority of the city is absolutely dangerous, so don't travel alone. I'm telling you, DC is probably the most dangerous place I've ever been. Worse than some of the bad places in New York, which I've also been to. In general, don't go anywhere on foot alone. Don't bring a car, either, because the streets are narrow and packed. Rely on public transportation, or someone you know. Trying to drive DC's streets on your own is HARD.
If you're interested, there's LOTS of nice music in DC. I personally just wandered around to different open doors with music. This is mostly across the river, though.
That's all I've got. I'll ask my friend what he liked, and report back.
Edit: I just remembered, GO TO THE HOLOCAUST MUSEUM. Possibly the most moving thing I've ever seen. Literally after a quick three hour walk through (You'd need 9 to obsorb all of the information in this thing.) I was spontaniously crying. Like, I was subconsciously so sad, that I was crying without really noticing it. It's REALLY insightful. There's a particular room with all of the shoes they've collected from a camp, and it's catistophically moving. You have to see this museum.
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