Jigsaw puzzle solving, the problem of constructing a coherent whole from a set of non-overlapping unordered fragments, is fundamental to numerous applications, and yet most of the literature has focused thus far on less realistic puzzles whose pieces are identical squares. Here we formalize a new type of jigsaw puzzle where the pieces are general convex polygons generated by cutting through a global polygonal shape with an arbitrary number of straight cuts. We analyze the theoretical properties of such puzzles, including the inherent challenges in solving them once pieces are contaminated with geometrical noise. To cope with such difficulties and obtain tractable solutions, we abstract the problem as a multi-body spring-mass dynamical system endowed with hierarchical loop constraints and a layered reconstruction process that is guided by the pictorial content of the pieces. We define evaluation metrics and present experimental results on both apictorial and pictorial puzzles to indicate that they are solvable completely automatically.