Mathematical characterizations and computational complexity of anti-slide puzzles

For a given set of pieces and a frame, an anti-slide puzzle asks us to arrange the pieces so that none of the pieces can slide in the frame. Since the first anti-slide puzzle that consists of dozens of cuboid pieces in 3D was invented, tons of anti-slide puzzles using pentominoes have been proposed. Some of them are not in a frame, which we call that interlock puzzles. In this paper, we investigate computational complexity of anti-slide puzzles and interlock puzzles in 2D. In previous work in theoretical computer science, a few models have been proposed for dealing with the notion of anti-slide, however, there exist gaps between these models and real puzzles. We first give mathematical characterizations of anti-slide puzzles and show the relationship between the previous work. Using a mathematical characterization, we give a polynomial time algorithm for determining if a given arrangement of polyominoes is anti-slide or not in a model. Next, we prove that the decision problem whether a given set of polyominoes can be arranged to be anti-slide or not is strongly NP-complete even if every piece is x-monotone. On the other hand, a set of pieces cannot be arranged to be interlocked if all pieces are convex polygons.(c) 2022 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).