A u s t r a l i a n M at h e m at i c s T r u s t. Pentomino Game. Teacher s Notes

Size: px

Start display at page:

Download "A u s t r a l i a n M at h e m at i c s T r u s t. Pentomino Game. Teacher s Notes"

Jordan Phillips
6 years ago
Views:

1 A u s t r a l i a n M at h e m at i c s T r u s t Pentomino Game Teacher s Notes Background Polyominoes are the shapes which can be formed from a number of equal size squares placed edge to edge. Generally, we count them as different if they cannot be rotated or reflected to be the same. Using this rule, there is 1 domino, there are 2 triominoes and 5 tetrominoes. Students are usually familiar with tetrominoes because they are the building blocks of the popular game, Tetris, though this has seven different shapes (including two reflective pairs). In this game, we use 5-square shapes, called pentominoes. There are 12 different pentominoes and, as the initial sheet shows, they can be tessellated to form a 6 10 grid. There is a chapter in the Newton book of the MCYA Enrichment series on polyominoes which provides many activities with these shapes. Introductory Activities Before introducing students to the games, you may like to get them to find the 12 pentomino shapes. This will no doubt involve some discussion of the rules for classifying shapes as the same or different. They may also be interested to explore which of the shapes tessellate by themselves. Playing the Game Game 1 is a simple strategy game in which students try to place their pentominoes on the grid in such a way as to prevent others from doing so. It can be played with two, three or four players, though if played with four, it is better for them to be in two teams playing alternately. If you can print out the sheet with the pieces on card, they will obviously be a little more durable. Game 2 is similar, but uses the tetris concept that pieces played must fall to the bottom of the board. This will make the game go for a few more moves. It is an interesting question for both games as to what is the largest or smallest number of moves a game can take. Students could monitor this and it could generate some interesting discussion. Do you have an advantage if you go first? Game 3 is an individual challenge. Obviously if they can see the initial sheet they have an answer, so this needs to be introduced after they have done their cutting out! We have provided an alternative 5 12 grid. The solutions to this, and the 4 15 and 3 20 tessellations are in the Newton book or are available from the Trust if needed. university of canberra locked bag 1 canberra gpo act 2601 australia tel: fax:

2 Additional Activities 1. Animal, Vegetable or Mineral An interesting problem with polyominoes is their classification as Animal, Vegetable or Mineral. In this activity, we explore whether we can move one piece from a polyomino and replace it so that we have the same polyomino but in a different position. If we can continue to move pieces (one at a time) always having the same polyomino but making it walk or slide across the table, then it is an animal. If it can move, but only in a limited way, it is a vegetable and if there are no possible moves, it is a vegetable. Below, there is an example of each for the tetrominoes. Animal At each stage the coloured piece moves to the dotted square. Vegetable The red square can never be moved. Each of the outside squares can be moved to fill the vacant space (the dotted square). Mineral No square in this shape can be moved. The challenge for students is to classify each of the 12 pentominoes as Animal, Vegetable or Mineral. 2. Mutations A related problem is to see what other polyominoes can be generated from a particular polyomino by moving only one square. With the pentominoes, all 12 can be generated from just two parent pentominoes. Which of the pentominoes has the most children? 2

3. Perimeters Look at the perimeter measurement of the different pentominoes. What possible answers are there? Why is the perimeter always an even number?

The maxima are easy to find but the minima are quite tricky). 4. Other tessellation problems In addition to the tessellation problems which formed part of the game, there are many others.

3 3. Perimeters Look at the perimeter measurement of the different pentominoes. What possible answers are there? Why is the perimeter always an even number? What would be the maximum and minimum perimeter of larger polyominoes of a given size? (Beware this is a little more difficult than it might seem. The maxima are easy to find but the minima are quite tricky). 4. Other tessellation problems In addition to the tessellation problems which formed part of the game, there are many others. Which of the pentominoes tessellate? What size rectangles can be generated from the tessellations? Feedback If you and your class enjoy these activities, you are very welcome to give us some feedback your class enjoy these you are very welcome to give us some feedback at feedback@amt.edu.au. at mail@amt.edu.au We may will publish any interesting interesting discoveries discoveries your your students students have have made madein inour ourteachers Teachers newsletter Maths Matters or on our website newsletter Maths Matters or on our website. 3

4 A u s t r a l i a n M at h e m at i c s T r u s t Pentomino Game Instructions - Game 1 - for 2, 3 or 4 players 1. Carefully cut the pentominoes from the game sheet (for durability you may wish to laminate this sheet). 2. Either 2, 3 or 4 players may play. Take turns to select a pentomino from the pile (or these can be dealt out randomly) 3. Players alternately place one of their pentominoes on the game board, ensuring they do not overlap or go outside the board. They can be played either way up. 4. If a player cannot make a move, play passes on to the next player. 5. When no moves can be made by any player, the last player who placed a pentomino on the board is the winner. Instructions - Game 2 - Tetris variant 1. Deal out or distribute the pentominoes as in Game Players alternately place one of their pentominoes on the game board. However, pieces must fall to the bottom of the board as far as possible without overlap. They can be played either way up in any orientation. 3. The board will fill from the bottom up. The last player who can place a pentomino on the board is the winner. If playing games 1 or 2 with 4 players, an alternative is that they form 2 teams of two, with players 1 and 3 playing against 2 and 4. Instructions - Game 3 - for 1 player (or a group) Try to arrange the pentominoes to exactly fit on the gameboard. Of course, it is better if students have not seen the original pentomino cutout when they try this. However, an alternative 5 12 grid is provided which should provide an additional challenge. It is also possible to arrange the pentominoes to fit a 4 15 board or even a 3 20 board. 4

5 Pentomino Game Y Z V U I N X T F W P L 5

6 Game Board A 6

7 Game Board B 7

Notes ~ 1. Frank Tapson 2004 [trolxp:2]

Notes ~ 1. Frank Tapson 2004 [trolxp:2] Pentominoes Notes ~ 1 Background This unit is concerned with providing plenty of spatial work within a particular context. It could justifiably be titled Puzzling with Pentominoes. Pentominoes are just