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Zbl 1066.05049
Reid, Michael
Klarner systems and tiling boxes with polyominoes. (English)
[J] J. Comb. Theory, Ser. A 111, No. 1, 89-105 (2005). ISSN 0097-3165

Summary: Let $\cal T$ be a protoset of $d$-dimensional polyominoes. Which boxes (rectangular parallelepipeds) can be tiled by $\cal T$? A nice result of {\it D. A. Klarner} and {\it F. Göbel} [Nederl. Akad. Wet., Proc., Ser. A 72, 465--472 (1969; Zbl 0185.03105)] asserts that the answer to this question can always be given in a particularly simple form, namely, by giving a finite list of ``prime" boxes. All other boxes that can be tiled can be deduced from these prime boxes. We give a new, simpler proof of this fundamental result. We also show that there is no upper bound to the number of prime boxes, even when restricting attention to singleton protosets. In the last section, we determine the set of prime rectangles for several small polyominoes.

MSC 2000:: *05B45 Tessellation and tiling problems
52C20 Tilings in 2 dimensions (discrete geometry)
05B50 Polyominoes

Keywords: Rectangle; Prime rectangle

Citations: Zbl 0185.03105

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