What are Pentominoes?

# What are Pentominoes?

For years, children have been introduced to the concepts of numbers through the game of dominoes. In dominoes, a rectangular shape is divided into two parts which each have a specific number of dots. The dots are arranged in patterns which help the children learn to develop conservation of groups of objects. The arrangement of the dominoes is based on the number of dots and the rules of the games. Although ending in the same suffix as dominoes, pentominoes are significantly different in their purpose and goal.

Pentominoes are a unique way to think about shapes. Creating the pentominoes requires thinking geometrically as well as critically. Pentominoes are typically created on a grid where the squares are connected following specific rules in order to create the unique and individual pentomino pieces. To create a pentomino one must connect five of the blocks on the grid together where they only touch on one side. Each of the five squares must be joined with the other with that one common side. There are two rules for making pentomino shapes. These rules are:
1. If one shape can be rotated or turned to look like one of the other shapes, it is not a new shape and therefore is not different.
2. If it can be flipped to look like one of the other shapes, it is not a new shape and therefore is not different.

In the end there are twelve distinct pentomino shapes. As each shape covers five squares, this means that only a total of sixty squares can be utilized to create all of the pentomino shapes. These twelve pentomino shapes are typically given a standard letter name. The letters used to define each pentomino are: F,I,L,N,P,T,U,V,W,X,Y,Z. These letters are used to represent the pentominoes as the shapes created look visually similar to the letters by which they are labeled.

Rather than relying on the number of dots, as dominoes do, pentominoes describe the number of blocks which are now joined. The prefix, pent-, refers to the number of blocks joined, five. As the shapes are created, they can be manipulated to form puzzles and squares. Logic can be utilized and developed by arranging the shapes in such a manner as to cover the least area possible. Manipulating and moving the shapes allows for concrete learning about the principles of connecting a variety of shapes, which is beneficial for many professions and life events. For example, the arrangement of furniture within a room is a similar experience to arranging the created pentominoes.

Pentominoes were developed in the early 1900s as a way to begin to think mathematically through solving puzzles. The name was coined later by Solomon Golomb. Pentomino activities help to develop many different geometric concepts including:
Tessellations
Transformations
Congruence
Similarity
Perimeter
Area
Volume

Since it is a more relaxed and fun puzzle like activity pentominoes provide the ability to increase spatial-ability skills and mathematical reasoning skills in a nonthreatening environment. The traditional format of a game using pentominoes is to take a standard six by ten grid and arrange the twelve pentomino pieces in a variety of ways to completely cover all sixty squares. There are reportedly two thousand three hundred and thirty-nine different arrangements which could be used. There are many other games which can be played using pentomino shapes. One game is known as the thirteen holes problem. In this problem, the ultimate goal is to fit the twelve shapes together in such a manner as to leave as many holes as possible (in this case thirteen holes). The rules follow:
1. No two holes are allowed to touch each other along any of their perimeters
2. Every hole must be surrounded by eight squares.

There are two possible solutions for creating thirteen holes. Students may need to work up to successfully creating thirteen holes. Mathematicians have proven over the years that one cannot arrange the tiles in a way which creates more than thirteen holes. In other words, there is no way to arrange the tiles to create fourteen or more holes.

Some commercial games have been created using the concept of pentominoes, one game is called Blokus©. This game has the players use the already identified tiles on a game board using strategy to see who can lay the most tiles before there are no more plays left for each player. The tiles created are similar to pentominoes but are created by combining various numbers of blocks. For example, the petominio cross is found in the game, but so are three and four block pieces to use as part of the game. The concept and problem solving approach used in this game is similar to that as used in the games designed for pentominoes.

Another game which can be played using pentominoes is called pentomino squeeze. Remember that each pentomino has five sides but may be arranged in a variety of shapes. In this game, a game board is used which has squares the same size as the squares used to create the pentominoes. Typically, the game board has sixty-four squares and is in an eight by eight grid. The twelve pentomino shapes are dealt to the two players in a random order. The player who is given the cross must play it as the first tile anywhere on the board. Each player must take turns playing their pentominoes one at a time on the board. No blocks or shapes can overlap. The players continue taking turns laying down their shapes one at a time. When it comes to a time that no player can lay down anymore of their pieces or a player has used all of their pieces, a winner is decided. The winner is the player who last played a piece successfully on the game board.

In addition to pentominoes, which are created with five blocks, other –ominoes can be created by joining different amounts of blocks. Biominoes would consist of two blocks joined with one side. Triominioes would consist of three blocks joined with one side. You can see the pattern. The types of –ominoes which can be developed is infinite. While many people have tried to determine a formula based on the number of blocks being connected with the possible outcomes, there is currently no formula which correctly predicts how many possible outcomes there will be given the number of blocks which will be joined.

Pentominoes provide not only the ability to begin to think about shapes and objects in a geometric manner. They also provide for logical thinking, deductive reasoning and many others of the more critical thinking operations. In this way, geometric concepts are being built, developed and reinforced; while at the same time, the ability to think, construct ideas, problem solve and rationally determine the solutions to problems is also being developed. Often throughout later geometric thinking and processing, it is more the ability to think and construct ideas upon each other which are required, rather than the simple understanding of basic shapes. Therefore, an activity such as pentominoes is an excellent stepping stone between the beginning levels of geometry and the later more complex proofs and theorems.