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Everyone knows what a “domino” is: a game piece made of two adjacent squares. A domino has anywhere from zero to six dots in each square, and most of the games played with dominoes involve matching two dominoes with the same number of dots.
Mathematicians like to generalize. In 1953 a mathematician named Solomon Golomb invented the idea of polyominoes: geometric objects made by connecting a certain number of squares with their sides matching. You can think of polyominoes as solid objects that can be picked up and moved around. There are many puzzles that can be made using polyomino pieces.
Here are a few sample polyominoes:
Monomino |
Domino |
Tromino |
Tetromino |
Pentomino |
Hexomino |
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You can form polyominoes with any number of squares. As you can see, there is only one way to form a monomino or a domino; however, there are increasingly more ways to form every other type of polyomino. For instance, there are two ways to form trominoes:
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A Little Warm-up Problem
There are exactly five different tetrominoes (shapes made from four squares). Can you find them? Can you prove that there are no more than five? Draw your results on graph paper.
You may want to make or get some square tiles or cubes and actually build your tetrominoes before you draw them on graph paper.
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An interesting question
How can you be sure that two tetrominoes are different? For example, here are two tetrominoes that look different, but mathematically they are the same. Can you see why?
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A |
B |
Tetrominoes A and B are congruent. If you picked up tetromino A and rotated it 90 degrees to the right, you could show that it would fit exactly on top of tetromino B. This is one common test that can be used to determine whether any two 2-dimensional shapes are congruent. Think of tetrominoes as objects that can be picked up and moved around. If you can pick up one of the shapes and rotate it, flip it over, and/or slide it so that it fits exactly on top of the other shapes, then the two shapes are congruent.
The Main Problem
Find all the possible pentominoes: shapes made by linking exactly five squares so that they match along at least one edge. How many different pentominoes can you find? How will you know you’ve found all of them? Draw your results on graph paper.
A big hint
If you take a checkerboard and remove the four corner squares, you’ll have a pattern on which all the different pentominoes will fit exactly. If you can fit all your different pieces onto this shape, you will have found all the possible pentominoes. (This is not an easy task however. It takes trial and error, spatial visualization, and lots of patience.)
An Extra Challenge
After you are sure that you’ve found all the pentominoes, see if you can find all the possible hexominoes, shapes made from six squares. Draw your results on graph paper.
Background
There is a wealth of mathematical problems that can be posed and solved using polyominoes. You’ll find hundreds of them in a book by Solomon W. Golomb titled Polyominoes: Puzzles, Patterns, Problems, and Packings (Princeton University Press, 2d ed., 1994). In it you’ll find literally hundreds of problems, many of which can be tried out for fun as geometric puzzles. If you’re a serious mathematician, you’ll find plenty of proofs, as well as many intriguing conjectures that have not yet been conclusively proven (at least as of the publication of this book).
There are three powerful mathematical ideas that are involved in this month’s puzzles. The first of these is the concept of congruence, introduced above.
Mathematically, two shapes are congruent if they have the same size and shape, regardless of their orientation. If you can pick up one of the shapes and rotate it, flip it over, and/or slide it so that it fits exactly on top of the other shapes, then the two shapes are congruent. This is one simple test for congruence. There are many others.
The second is the idea that you sometimes have to test to be sure whether two mathematical objects are the same or different. Just because two polyominoes look different, you can’t be sure they are different until you’ve picked one up and made sure it’s not congruent to the other. For polyominoes, congruence is the test—but mathematicians have many other tests they can use for different types of problems.
The third is the idea of finding all possible ways to solve a particular problem. This is a common type of problem in combinatorics. For example, find all the possible ways to arrange the numbers from 1 to N. Or for a more everyday example, a restaurant serves pizza with four different toppings: onions, mushrooms, sausages, and extra cheese. How many different types of pizzas can they make using 1, 2, 3, or 4 of the toppings? (Of course, a real restaurant would probably provide many more than four toppings.)
Combinatorics problems involving geometry can be much more complex. For example, if N is the number of squares in a polyomino, there is no known rule or formula that can tell you how many different polyominoes that can be constructed for any value of N. (Mathematicians have worked out the exact number of polyominoes for many values of N, but no general rule has ever been found.)
In addition to Solomon Golomb’s original book, Polyominoes, there have been several other books written about them for serious and recreational mathematicians. If you search the Web for polyominoes, you’ll find thousands of references and even some downloadable polyomino games. If you are familiar with the computer game Tetris, you’ll recognize that the playing pieces are all tetrominoes.
After you’ve tried this for yourself, .
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