Surprisingly difficult puzzle

Back in Sapporo, I was given a puzzle to solve in a bar I often visit to have dinner. This puzzle turned out to be quite difficult, and it took me several visits to this bar before I solved it. The goal is to get all the pieces into the frame, of course.

Since the pieces do not fit by just reordering them using the "obvious" way to put them into the frame, I thought the round corners and non-round corners and the fact that if you put to cat heads towards each other you get some slack etc. had something to do with the solution. but neither of these things had anything to do with the puzzle solution. These red herrings cost me two nights of puzzle failure, haha.