Placing Four Tiles 2x1 on a 4x4 Square: An InvestigationAuthor(s): Beryl BoydSource: Mathematics in School, Vol. 11, No. 1 (Jan., 1982), pp. 22-23Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30213679 .

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Placing four tiles

2x1 on a 4x4

square: an investigation

by Beryl Boyd, Primary School Consultant, ILEA

Starting Point How many different ways can you place four tiles 2 x 1 on a 4 x 4 square? Assumed rules: Tiles must not overlap, and each tile must occupy two whole squares vertically or horizontally.

e.g. not

I started marking out 4 x 4 squares on squared paper and

colouring possible positions of tiles. When I had made about 40 patterns, I stopped to review the results.

Some were rotations or reflections of others; should I count these as different?

Because I had coloured the positions of the tiles, the direc- tion in which they had been placed was ambiguous in some cases.

e.g.

A B

Had the square been made by A or by B?

A B C

Had the rectangle been made by A or by B or by C? In other words, each time I made a square from two tiles or

a rectangle from three tiles, I needed to make it clear in which direction I had placed them and also to include any possible variations. From this stage onwards I decided to shade rather than colour.

Groups of patterns were emerging;

e.g. those including a 2 x 2 square those including a 3 x 2 rectangle those including a 4 x 1 rectangle, etc.

How many different patterns were there going to be?

22 Mathematics in School, January 1982

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Would I be able to tell when I had found them all? I decided to make a chart, numbering the possible positions of the rectangular tiles

1 23-4

9 10i 11112

5 6 78

13 21 145 22

15 23

16 24

17 - -1

19

20

I started to list possible number combinations:

(1 2 3 4), (1 2 3 5) - not possible because 5 would overlap 1. (1 2 3 6) - not possible because 6 would overlap 2. (1 2 3 7) - not possible because 7 would overlap 3. (1 2 3 8)

I stopped, and made a list of impossible combinations

1 would overlap 5, 13, 14 2 would overlap 6, 13, 14, 17, 18 3 would overlap 7, 17, 18, 21, 22 4 would overlap 8, 21, 22 5 would overlap 9, 14, 15 6 would overlap 10, 14, 15, 18, 19 7 would overlap 11, 18, 19, 22, 23 8 would overlap 12, 23, 24 9 would overlap 15, 16 and so on.

I resumed work on my list of possible number combinations, omitting the impossible ones or crossing them out if I had in- cluded them in error.

(123 4),(123 8),(123 9),(12310), (12311), (1 2312)... Several days and two exercise books later, I reached (21 22 23 24).

I started to draw the patterns beginning with the most obvious outline, a rectangle 4 x 2.

and reflection

(13, 14, 15, 16) by rotating them

1/4 turn

(21, 22, 23, 24)

(1, 2, 3, 4) (9, 10, 11, 12)

I crossed out these sequences from my list, noting that by rotating the patterns a 1/2 turn the sequences would re-occur.

I investigated different ways of making the same rectangle and crossed off the combinations representing the identity, rotation and reflection of these.

Then I made the position of the rectangle central along one side; this led to ways of representing; 2 squares 2 x 2 2 rectangles 4 x 1 1 rectangle 3 x 2 and 1 rectangle 2 x 1 1 square 2 x 2 and 2 rectangles 2 x 1 1 rectangle 4 x 1 and 2 rectangles 2 x 1

All these I drew with their reflections, and crossed off the number combinations which they and their rotations repre- sented.

At this stage I was left with many more unrepresented number combinations than I had anticipated, and I had a file full of patterns.

Part II of the Investigation

Ways of Placing Four Rectangles 2 x 1 on a Square 4 x 4, Without Making a Square or Larger Rectangle As I came to patterns which were the same if rotated, I collected them together and worked through the rest of my number combinations. (The supply of squared paper was a recurring frustration.)

At the end I had drawn more than 600 patterns (many were reflections) and with their rotations they represented more than 2 500 number combinations. I kept a record of the numbers representing each pattern, in a book with pages numbered the same as the file of patterns.

I found the "perfect" pattern in;

(5, 8, 17, 20)

Rotation = reflection = identity. A pattern with vertical, horizontal and diagonal symmetry.

Activities for Children (I have tried these in primary classes) Draw a square 4 x 4 (e.g. on 2 cm squared paper).

Cut out four rectangles 2 x 1 from a separate piece of paper. Colour your rectangles one colour. 1. Place the rectangles within the square so that each rectangle

covers two whole squares, and the rectangles do not overlap. Record the position of your rectangles by shading part of the square. Repeat several times, changing the position of the rectangles each time. What fraction of the 4x4 square have you shaded each time? What area have you shaded each time?

2. Work with a partner. One person places three of the rectangles according to previous rules. In how many different positions is it possible for the second person to place the fourth rectangle? How can the first person make it easier/more difficult for the second person to find several ways?

3. Work with a partner - two rectangles each. First person places two rectangles within square. Second person must place his/her rectangles so that the total shape is symmetrical in some way.

Where to go next? What would happen using a different size square

and/or different size rectangles and/or different number of rectangles.

What would happen using rhombi within an equilateral triangle?

What would happen if ?

Mathematics in School, January 1982 23

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