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Tiling a Square with Eight Congruent Polyominoes

✍ Scribed by Michael Reid

Publisher
Elsevier Science
Year
1998
Tongue
English
Weight
76 KB
Volume
83
Category
Article
ISSN
0097-3165

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✦ Synopsis

The problem of finding polyominoes that tile rectangles has attracted a lot of attention; see [1] for an overview, and [2, 3] for more recent results. Several general families of such polyominoes are known, but sporadic examples seem to be scarce. Marshall [2, Fig. 9] gives a polyomino of rectangular order 8 and asks if it can be generalized to a family of rectifiable polyominoes.

Here we show one way to generalize Marshall's construction, which yields an infinite family of polyominoes of rectangular order 8. Marshall's construction is the first square in Fig. 1. FIG. 1. Infinite family of polyominoes of rectangular order 8.

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In the second edition of Golomb's classic ``Polyominoes'' [9], several infinite families of rectifiable polyominoes are given, but only nine sporadic examples are known. Curiously, two of these sporadic examples are related by a 2\_1 affine transformation (Fig. 1). This led us to consider the image

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We show that a square-tiling of a p\_q rectangle, where p and q are relatively prime integers, has at least log 2 p squares. If q>p we construct a square-tiling with less than qΓ‚p+C log p squares of integer size, for some universal constant C.