Symmetric Assembly Puzzles are Hard, Beyond a Few Pieces

The goal of a 2D assembly puzzle is to arrange a given set of polygonal pieces in a way that they do not overlap and form a target polygonal silhouette. For example, there are over 5,000 Tangram assembly puzzles [4], and many more similar 2D assembly puzzles; see, e.g., [2]. A recent trend in the puzzle world is a relatively new type of 2D assembly puzzle which we call symmetric assembly puzzles. In these puzzles the target shape is not specified. Instead, the objective is to rearrange the polygonal pieces so that they form a symmetric silhouette (as before, overlap of the pieces is not allowed). The first symmetric assembly puzzle, “Symmetrix”, was designed in 2003 by Japanese puzzle designer Tadao Kitazawa, and was distributed as his exchange puzzle at the 2004 International Puzzle Party (IPP) in Tokyo [3]. In this paper, we aim for an arrangement that creates mirror symmetry (reflection through a line), but other symmetries such as central symmetry or 180◦ rotation (or all of the above) could be considered. The lack of a target shape specification makes these puzzles quite difficult to solve in practice, even for relatively few and simple pieces. In this paper, we study the computational complexity of symmetric assembly puzzles in their general form. Given n simple polygons P1, P2, . . . , Pn, with m1,m2, . . . ,mn vertices respectively, the goal is to find a mirror-symmetric polygon that can be exactly covered by P1, P2, . . . , Pn. We may either allow or forbid the pieces to flip over (reflect). Given the difficulty humans have with few low-complexity shapes, we consider two different generalizations: bounded piece complexity (mi = O(1)) and bounded piece number (n = O(1)). In the former case, we prove strong NP-hardness, while in the latter case, we solve the problem in polynomial time (but the exponent depends on the number of pieces).