Tiling Rectangles and Half Strips with Congruent Polyominoes

Tiling Rectangles and Half Strips with Congruent Polyominoes, by Michael Reid
Journal of Combinatorial Theory, Series A 80 (1997), no. 1, pp. 106-123.
[DOI] [Math Reviews] [Zentralblatt]
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Abstract
The problem of tiling a rectangle with congruent copies of a single polyomino prototile has attracted a fair amount of attention. It is easy to give some infinite families of polyominoes, each of which tiles a rectangle, but there are also sporadic examples, which seem to be much rarer. In this paper, we begin with the observation that two of the nine previously known sporadic examples are related by a 2 × 1 affine transformation. Two other sporadic examples also yield "rectifiable" polyominoes, when transformed in the same way (and another case has since been found). Several other examples of rectifiable polyominoes are also given, including a new infinite family. Inspired by these tilings, we also give three infinite families of polyominoes, each of which tiles an infinite strip. It is unknown if every such polyomino also tiles a rectangle, although this seems unlikely, in view of these families. Several open questions are also posed (some of these have since been solved partially or completely).
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Updated January 8, 2008.