The study demonstrates undecidability of two polycubes tiling in three dimensions, suggesting broader implications for higher dimensional cases.
In this paper, we prove that it is undecidable whether a set of two polycubes can tile Z³ by translation. The proof involves a new technique that allows us to simulate two disconnected polycubes with two connected polycubes. By expanding this technique to higher dimensions, we also prove that a set of disconnected tiles in Zⁿ can be simulated by the same number of connected tiles in Zⁿ for n ≥ 3.
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Y. Kim (2025) studied this question.
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