## Saturday, August 25, 2007

### Polyominoes

I just finished Polyominoes by Solomon Golumn.

Polyominoes are, of course, the general extension of what we know as dominoes. Where dominoes are simply two attached squares, polyominoes are any number of attached squares. The study of polyominoes is the study of how many different unique ways these squares can be attached given a certain number of squares, and how they can be used to cover certain areas or space, such as a rectangle.

If you're familiar with the game Blokus, the pieces given to each player in a game of Blokus consist of all unique polyonimoes from size one (monomino) through size five (pentomino). Arranging the twelve unique pentominoes into a rectangle is an age old puzzle with thousands of unique solutions.

The book is presented in its description and by the publisher as a math book. Yet, the first few chapters seem more concerned with basic explanations followed by long series's of puzzles and solutions.

Only midway through the book does it begin to start to feel like a math book, as it presents combinatoric theory, symmetry issues, and other kinds of mathematical studies. In fact, the book goes through a number of proofs that are heavy on the mathematical equations and much less interesting to the casual reader.

Instead, just the ideas of what type of puzzles you can play with are what makes the book interesting to me. These include, aside from the standard "how can you make a rectangle" such things as how you can make cubes out of three dimensional versions of the pieces, polyominoes beyond three dimensions, attached repeated shapes from other polygons, and so on.

It's not actually a long book, but it has some pretty cool puzzles and ideas. Surely Blokus isn't the only game that could benefit from using these types of shapes.

More on polyominoes at Wikipedia.

Yehuda