In this study, we investigate the computational complexity of some variants of generalized puzzles. We are provided with two sets S_1 and S_2 of polyominoes. The first puzzle asks us to form the same shape using polyominoes in S_1 and S_2. We demonstrate that this is polynomial-time solvable if S_1 and S_2 have constant numbers of polyominoes, and it is strongly NP-complete in general. The second puzzle allows us to make copies of the pieces in S_1 and S_2. That is, a polyomino in S_1 can be used multiple times to form a shape. This is a generalized version of the classical puzzle known as the common multiple shape puzzle. For two polyominoes P and Q, the common multiple shape is a shape that can be formed by many copies of P and many copies of Q. We show that the second puzzle is undecidable in general. The undecidability is demonstrated by a reduction from a new type of undecidable puzzle based on tiling. Nevertheless, certain concrete instances of the common multiple shape can be solved in a practical time. We present a method for determining the common multiple shape for provided tuples of polyominoes and outline concrete results, which improve on the previously known results in puzzle society.

翻译：在本研究中，我们探讨了一些广义拼图变体的计算复杂性。给定两个多联骨牌集合S₁和S₂。第一个拼图要求使用S₁和S₂中的多联骨牌拼出相同形状。我们证明：若S₁和S₂中多联骨牌数量为常数，该问题可在多项式时间内求解；而在一般情况下，该问题为强NP完全问题。第二个拼图允许复制S₁和S₂中的拼块，即S₁中的一块多联骨牌可多次使用以拼出某个形状。这是经典拼图“公共倍数形状拼图”的广义版本。对于两个多联骨牌P和Q，公共倍数形状是指可由多个P副本和多个Q副本拼出的形状。我们证明第二个拼图在一般情况下不可判定，该不可判定性通过归约一种基于平铺的新型不可判定拼图得以证明。尽管如此，公共倍数形状的某些具体实例可在实际时间内求解。我们提出了一种为给定多联骨牌元组确定公共倍数形状的方法，并概述了具体结果，这些结果改进了拼图领域此前已知的结论。