Polyominoes are geometric objects made by connecting a certain number of squares with their sides matching. Last month’s puzzle involved making all 5 tetrominoes (4-square polyominoes) and all 12 pentominoes (5-square polyominoes). If you have not done last month’s puzzle, you might like to try it first. This month we will be using polyominoes as puzzle pieces to solve geometric problems.

Here are the 5 tetrominoes.
Here are all 12 pentominoes:

Polyominoes were invented by mathematician Solomon W. Golomb back in 1953. His book Polyominoes: Puzzles, Patterns, Problems and Packings, (Princeton University Press, Second Edition, 1994), contains hundreds of problems. 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). You can also find many more polyomino problems by surfing the Internet. This month we’ll try some of the simpler puzzles using just tetrominoes and pentominoes.

You can think of the polyominoes as jigsaw puzzle pieces—except unlike most jigsaw puzzle pieces, polyominoes can be turned over and used on both sides.

To begin, you’ll need to cut out a set of tetromino and pentomino pieces from Attachment 1. It’s best if you print them on card stock. If your printer can’t handle card stock, print the attachment on paper, cut out the pieces and use them as patterns or templates to cut stiffer pieces using index cards or manila file folders.

You’ll also need to print out the game boards on Attachment 2 and Attachment 3 and the graph paper, Attachment 4.

This month we have several different puzzles, and a game for two people.

Puzzle 1
Fit all 12 pentominoes on the two large rectangular grids on Attachment 2. Remember that you may have to turn some of the pentominoes over in order to fit them on the grid. Record your solution by drawing the pentominoes on a copy of Attachment 2.

Puzzle 2

1. Are there any other rectangular grids on which you can fit all five pentominoes? Why do you think so? Outline such a grid using graph paper (Attachment 4) and try to fit all 12 pentominoes on it.
2. Is there any rectangle with an area of 60 squares that you cannot fit 12 pentominoes on? Can you prove that it is impossible?

Puzzle 3
This problem uses tetrominoes. Is it possible to place all 5 tetrominoes on a 4 x 5 grid (Attachment 3)? If you don’t think it’s possible, can you explain why?

Puzzle 4
This problem uses both tetrominoes and pentominoes.

1. Place all 4 tetrominoes and just one pentomino on a 5 x 5 square grid (Attachment 3).
2.  Are there any pentominoes that you can’t use in this way? How do you know?

Puzzle 5
Here’s a very challenging problem. Divide your pentominoes into two sets of six. Then try to fit all the pentominoes onto two 5 x 6 grids (Attachment 2).

Puzzle 6
Invent your own puzzle with pentominoes and/or tetrominoes. Make a shape using some or all of your pieces. Trace around the outside of the shape and then give the pieces and the outline to a friend and ask him or her to solve the puzzle. (Or mix up the pieces yourself and see if you can solve your own puzzle.)

Puzzle 7
Here’s an unusual game for two players and one set of polyominoes. It’s played on an 8 x 8 grid
(Attachment 3).

Players take turns placing pentomino pieces anywhere on the game board. Your goal is to make it so the next player can’t place another piece without out overlapping another piece or going off the board. The player who places the last legal piece wins.

Hint: If you think you would enjoy solving puzzles or playing games like these online, there are a number of websites that provide interactive games. One good example is Pentominoes. This web site also has links to many solutions. Another set of Polyomino games can be downloaded from the Polyominoes Home Page. Another page on this site contains links to several other Web sites and books about Polyominoes.

Math Background

The critical mathematical skill developed here is the ability to visualize geometric shapes, as they relate to each other and how the appear in different positions.  As you work with these puzzles you will begin to improve these visualization abilities. 

The ability to visualize shapes in different positions is heavily dependent on the symmetry of the shapes. For instance, six of the pentominoes are asymmetric.
Each of these has eight possible positions in which they can be placed. For example:
Four of the pentominoes have exactly one line of symmetry.
These have four possible positions and are therefore easier to visualize in different positions.
There is one pentomino with two lines of symmetry. It can be placed in only two different positions, which are very easy to visualize.
There is also one pentomino with four lines of symmetry. It can be placed in only one position.

After you’ve tried this for yourself check our solutions.

…try some Science Lab projects about…
Air & Space
Earth Science
Electricity & Magnetism
Properties of Liquids

…try some interactive experiments in our…
Virtual Lab

…or test your skills at our…
Engineering Challenge