Journal of Combinatorial Theory, Series A
(1997), no. 1, pp. 106-123.

Tiling Rectangles and Half Strips with Congruent Polyominoes,
by Michael Reid

Journal of Combinatorial Theory, Series A
(1997), no. 1, pp. 106-123.

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).

Updated January 8, 2008.