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**Polyominoes: puzzles, patterns, problems, and packings.
2nd, rev. and exp. ed.**
*(English)*
Zbl 0831.05020

Princeton, NJ: Princeton University Press (ISBN 0-691-08573-0/hbk). xii, 184 p. (1994).

Fans of recreational mathematics will be very happy with this 2nd (revised and extended) edition of this book on polyominoes. As a generalization of the domino, an \(n\)-omino is a configuration of \(n\) equal-sized squares, each joined together with at least one other square along an edge. As the title of this book explains, it discusses all kinds of puzzles and patterns that can be made with polyominoes. The list of chapters will give a good overview of the contents of this book. The following chapters did also appear in the first edition (1965) (see also the review of the Russian translation (1975; Zbl 0326.05025). Polyominoes and checkerboards; Patterns and polyominoes; Where pentominoes will not fit; Backtracking and impossible constructions; Some theorems about counting (in an appendix answers to exercises from this chapter are given); Bigger polyominoes and higher dimensions; Generalizations of polyominoes. Two new chapters in this edition have been added. In the chapter on tiling rectangles with polyominoes the author discusses the order of a polyomino, defined by D. A. Klarner as being the minimum number of congruent copies that can be assembled (allowing translation, rotation, and reflection) to form a rectangle. Nice examples of polyominoes of high order are given. Moreover open questions, such as what odd numbers can occur as odd orders of polyominoes, are discussed. In the 9th chapter, Some truly remarkable results, some recent theorems are discussed. Some examples: If any two squares of opposite color are removed from an \(8 \times 8\) checkerboard, what is left can always be covered with dominoes (Gomory). An \(a \times b\) rectangle cannot be tiled using \(1 \times n\) tiles unless \(n\) divides one of the sides of the rectangle (De Bruijn).

Some nice appendices have been included in this 2nd version. On the one hand there is the treatise by A. Liu, on the recent status of 12 unsolved problems from a problem compendium (dealing with the fitting together of pentominoes and related polyominoes) posed by the author (these problems appeared also in the first edition). On the other hand we mention the appendix on “Klarner’s constant and the enumeration of \(n\)-ominoes” which gives an update on the status of the enumeration of both plane and solid \(n\)-ominoes.

The bibliography of the first edition has been greatly expanded with new titles.

Some nice appendices have been included in this 2nd version. On the one hand there is the treatise by A. Liu, on the recent status of 12 unsolved problems from a problem compendium (dealing with the fitting together of pentominoes and related polyominoes) posed by the author (these problems appeared also in the first edition). On the other hand we mention the appendix on “Klarner’s constant and the enumeration of \(n\)-ominoes” which gives an update on the status of the enumeration of both plane and solid \(n\)-ominoes.

The bibliography of the first edition has been greatly expanded with new titles.

Reviewer: Frank De Clerck (Gent)

### MSC:

05B50 | Polyominoes |

05-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to combinatorics |

05B45 | Combinatorial aspects of tessellation and tiling problems |

00A08 | Recreational mathematics |