Tiling the Plane with a Fixed Number of Polyominoes

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Abstract

Deciding whether a finite set of polyominoes tiles the plane is undecidable by reduction from the Domino problem. In this paper, we prove that the problem remains undecidable if the set of instances is restricted to sets of 5 polyominoes. In the case of tiling by translations only, we prove that the problem is undecidable for sets of 11 polyominoes.