Mathematics in a Matchbox: Part 2Author(s): William GibbsSource: Mathematics in School, Vol. 21, No. 4 (Sep., 1992), pp. 2-4Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30214900 .

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mathe in a

mathe Part 2

by William Gibbs

Overseas Education Unit, University of Leeds

Have you ever noticed what happens to a matchbox when it gets squashed? I was walking along the streets of Nairobi one day and noticed a flattened matchbox on the pavement and was surprised to see that it was an almost perfect square. What happens if you then cut this flattened match- box cover in two along a diagonal and then open up the two triangles you have cut out? (figure 1). They open up to give two larger isosceles right angled triangles which of course fit together to make an even larger square. This is the beginning of the Matchbox Tangram.

Fig. 1. Matchbox Dissection

2

The 6 Piece Matchbox Tangram Cut each of the two large right angled triangles along the edge creases to give two small right angled triangles, two large right angled triangles and two trapezium, as shown in figure 2.

Fig. 2. Matchbox Tangram

Mathematics in School, September 1992

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These six pieces can now be put together in many different ways to make a whole variety of shapes some of which are shown in figure 3.

Fig. 3. Tangram Shapes

Mathematics in School, September 1992

The variety of shapes that can be created is increased if you have chosen your matchbox wisely and the hypoteneuse of the small triangle is equal in length to the side of the large triangle. This is the case if the sides of the matchbox are in the ratio - 1 : 1:

"2. A quick way to check if this

is true is shown in figure 4. The first arrangement checks if the width plus the height of the box equals the length and the second arrangement tests if the length of the box fits along the hypoteneuse of the triangle shown. If your matchboxes pass these two tests then they make excellent Matchbox Tangrams.

Fig. 4. Special Relationships

Magic Towers Here is a variation on the idea of magic squares. You will need 3 matchboxes for the first example. Mark the sides of the boxes and the ends of the trays as shown in figure 5. Write the 6's and 9's so that they are rotations of each other. The challenge is now to build a tower with the three boxes and trays so that sum on each side of the tower is equal and so is the sum of the numbers on each matchbox (some trays may need to be upside down).

de9

1,0 8

8

W9

80 8

6

5

6

4

05 009 Fig. 5. Magic Towers

A more challenging tower can be built from 4 boxes with the numbers marked as in figure 6. Now all sums, on sides and on boxes, should be equal.

1212

4

OIA 05 4

8

14 3 60 02 1

9 6

2

16

8 07 015 010 02 Fig. 6. More Magic Towers

3

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Matchbox Planes Imagining planes inside cuboids can be extremely hard. Here is a little activity which aims to introduce the idea of diagonal planes in a practical way.

Figure 7 shows various pieces of card that have been fitted inside the tray of a matchbox. Can you cut out pieces of card to fit as they do? Can you decide how on a practical way to find the length and width of each rectangle? Can you find a mathematical way?

Fig. 7. Matchbox Planes

Close the Box Puzzles Here is a family of puzzles which are easy to make but not always easy to solve. In the first example shown in figure 8 four matchbox trays have been glued to 4 matchbox covers. The aim of the puzle is to shut all the boxes. There are three different solutions two of which are shown in

figure 9.

Fig. 8. Matchbox Trays and Covers

Fig. 9. Close-the-Box Puzzles

A whole range of similar puzzles can be created becoming more and more difficult to solve as more boxes are used. Figure 10 shows two more puzzles.

It is quite easy to create your own unique puzzles of this type. All you have to do is to make an arrangement of matchboxes so that each cover is placed against the end of a matchbox tray. Then they are glued in pairs. There

Fig. 10. More Close-the-Box Puzzles

is often more than one solution as shown with the earlier

examples, so do not be surprised if the boxes can be shut in quite a different arrangement than the one you had planned.

Finally two matchbox investigations.

Matchbox Faces A single matchbox has six faces but when two matchboxes are joined the number of faces can vary considerably depending on the arrangement.

For example the shape made from 2 matchboxes in figure 11 has 8 faces.

Are there other ways of arranging the two boxes to give 8 faces?

Is it possible to arrange the two boxes so that there are 9 faces? How about 10 faces? Such questions can lead to an investigation into the maximum number of faces it is possible to create with two boxes.

Fig. 11. Matchbox Faces

Cutting open a Matchbox Tray How many different ways are there to cut open a matchbox tray so that it lies flat? Three possible ways are shown in figure 12 but there are many more so quite a lot of matchbox trays are needed if the investigation is to be carried out practically. Finding all the solutions leads on to other questions such as;

How can the different solutions be classified? Which needed the longest cut? Which has the longest perimeter?

Fig. 12. Cut open trays

4 Mathematics in School, September 1992

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