Polyominoes: Puzzles, Patterns, Problems, and Packings, by Solomon Golomb is the definitive book on the subject by its creator who brought these wonderful objects to the mathematical world in 1953. Polyominoes are formed by groups of squares joined along their edges. Single squares are called monominoes. Two squares form the familiar dominoes. Then we have, in order, Trominoes (3), Tetrominoes (4), Pentominoes (5), Hexominoes (6), and so on. The graphic above shows the 12 distinct pentominoes.
Some of the simpler problem investigations teachers can ask involve finding out how many distinct tetrominoes and pentominoes there are, not counting rotations/reflections as different. Other interesting questions involve classifying polyominoes according to the type of symmetry they have.
The amount of significant mathematics developed concerning polyomininoes is simply amazing, and a large number of websites feature these rectangular wonders. There are some types of Sudoku that use polyomino-shaped regions on the grid. The game Tetris is based on seven tetrominoes (some are reflections). The graphic below shows some of the ways pentominoes can be formed into rectangles.
Links to further information:
Wikipedia article on Solomon Golomb
Google Books article containing preview pages, website links, and places to purchase
Give polyominoes a try in your classes and see why they have been so popular over the years!
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{ 4 comments… read them below or add one }
what were Solomon Golomb hobbies
Hi Joseph,
The following is from wikipedia; I don’t know if this qualifies as hobbies or not:
He is a regular columnist, writing Golomb’s Puzzle Column in IEEE Information Society Newsletter. He was a frequent contributor to Scientific American’s Mathematical Games column. Among his contributions to recreational mathematics are Rep-tiles. He also contributes a puzzle to each issue of Johns Hopkins Magazine, a monthly publication of his alma mater, for a column called “Golomb’s Gambits.”
-Mr. L
Amazing! Do you happen to know if the twelve pentaminoes can also tile a SQUARE? The book we are using stated that forming rectangles would be possible, but left the formation of squares in question! We are curious.
Hi Gloria,
Twelve pentominoes would equal 60 square units, which is not a perfect square; so you can only make non-square rectangles with all twelve pieces.
Cheers,
- Mr. L