Re uniqueness, this seems to be a bit like numberlink for me, so perhaps bear with me on some future puzzles. I'm pretty confident these are all good though!
The first puzzle is a somewhat narcissitic-yet-trivial example - although hopefully it clarifies exactly what you are supposed to do with loops.
#268 Curve Data – rated (stupidly) easyMaybe this next one isn't quite a medium, but seeing as it's the first one for the blog I'm going to relax my strict regime just this once!
EDIT: A previous version had rather more solutions than I thought. This one I think is unique!
2nd EDIT: It isn't! Back to the drawing board... :)
3rd EDIT: I got someone to test solve this first :p
#269 Curve Data – rated mediumAll puzzles © Tom Collyer 2009-13.