mathematics in a matchbox. part 1

Mathematics in a Matchbox. Part 1Author(s): William GibbsSource: Mathematics in School, Vol. 21, No. 3 (May, 1992), pp. 25-28Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30214881 .

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Part

Part

Part Part

Part 1

by William Gibbs, Oversea Education Unit, University of Leeds

Nearly all matchboxes throughout the world are of same shape and nearly always of the same size. Many of them including "Englands Glory" from Britain, "Kudu" from Zambia and "Kumi" from Kenya have sides with lengths which are very close to being in the ratio 1:2:3 (Fig. 1). This ratio means that the boxes can be used for several mathematical activities. For many of these it greatly helps to use a matchbox grid onto which the boxes will fit neatly (Fig. 2). The dimensions of the grid may vary a little depending on the boxes you are using but the one shown here in Fig. 2 is suitable for Bryant and May matchboxes and is based on 1.8 cm squares.

Fig. 1 Mathematics in a Matchbox (A)

Here then are some starting points for activities that exploit the special properties of matchboxes. Each sug- gested idea has the potential for exploration in depth but it is beyond the scope of this article to describe in detail where each journey may lead.

Fig. 2

Covering Boxes Ideas about area and surface area can be explored easily using the squares of the matchbox grid as units. Possible questions are;

* How many squares on the matchbox grid does the matchbox cover when placed on end, on its side, flat?

* Investigate how you could cut and fold the matchbox grid so that it wraps up the matchbox? How many different solutions are there?

* What is the surface area of the box in matchbox square units?

* Investigate what happens to the surface area when two boxes are joined? Are there different answers depending on how the boxes are joined? What is the greatest surface area? What is the smallest surface area?

Mathematics in School, May 1992 25



Now investigate joining more boxes. Can you generate any generalizations?

Fig. 3 Cuboids from 4 Matchboxes

One of the many interesting points that arises when using 4 boxes is that there are several ways of joining the boxes to give a cuboid of 4 x 2 x 3 units which has the minimum surface area (Fig. 3).

Box Edges The idea of the Total Edge Perimeter (T.E.P.) of the matchbox in matchbox units can also provide a starting point for an investigation. The T.E.P. for one box is 24 matchbox units. Here is a puzzle that might act as the stimulus to a more detailed exploration.

Perimeter puzzle Take 5 match boxes. * Can you build two cuboids one with two match-

boxes, the other with three matchboxes, so that the Total Edge Perimeters of the two cuboids are equal?

Are there other cuboids built with different numbers of boxes that have equal Total Edge Perimeters?

Rolling Boxes

A matchbox can be placed on the grid and then rolled to new positions. For example the matchbox shown in Fig. 4 can be rolled to cover all the shaded rectangles.

1

~

Fig. 4

Here is a little puzzle to start off an investigation into rolling boxes.

Rolling Box Puzzle 1 Colour a 4 by 5 rectangle on the matchbox grid. Place a matchbox on this rectangle and see if you can roll the box

so that all the coloured squares are covered by the box as you roll. * Can you find more than one solution? * Can you roll a matchbox over a 5 by 5 grid? A 5

by 7 grid? A 10 by 10 grid? (This last one is hard.)

These puzzles can lead on to a search for other rectangles and squares which can be covered by rolling. One way to start the search is to look first at the simple rectangles that can be covered by rolling the box in the same direction (Fig. 5).

A B C

_

ti I ( I

!II I I I i

I I I

Fig. 5 Rectangles from rolling boxes

Next we can investigate how these three movements can be combined. For example the solution to the first Rolling Box Puzzle in which a 4 by 5 rectangle was covered can be seen as the combination of two of these column rectangles (Fig. 6).

5

4

23

Fig. 6 How to roll a matchbox over a 4 x 5 grid

Extending and generalizing from this idea it is possible to find other combinations of rolling procedures that cover squares. For example by combining procedures A and C from Fig. 5 a square of 13 by 13 can be covered. Other combinations cover squares of 17 by 17 and 23 by 23. By combining all three column generating procedures a square of 60 by 60 can be rolled.

Rolling Boxes (2) Another way to play around with rolling boxes is to choose a starting point and to roll the box to a new position which is a translation of the original position. Here is a problem that can be used as an introductory challenge;

26 Mathematics in School, May 1992



Rolling Box Puzzle 2 Start with the matchbox in the top left hand corner of a 6 by 6 grid (position A in Fig. 7). Can you roll the matchbox so that it stays on the grid and finishes up in position B?

A ii iiii

Fig. 7 Rolling box puzzle

Starting in position A in Fig. 8 can you roll to position B?

AL

.. .. ...... . .. . . LI

I

Fig. 8 (2,2)

Further questions that arise from these puzzles might be; * What other positions can be reached if we enlarge

the grid? * Can all positions on the grid be reached? * How can rolls be recorded? e How can the resulting movement be described?

One possible notation that can be used is to describe the rolls in terms of the direction of the roll so that a roll to the right is denoted as (R), to the left (L), up (U) and down (D). The solution to the first Rolling Box Puzzle which gave a displacement of (3,-3) is the sequence RDLDRU and to the second (2,2) is LURDRU. An analysis of all the possible simple sequences that translate the box and either use 4 or 6 rolls can lead to an investigation of how sequences of rolls can be combined to create new displacements. For example combining (2,2) LUR- DRU and (1,-1) RURDLD will give (3,1) and (8,0) RRRR followed by (6, - 6) RDRDRD will give (14, - 6).

Another line for investigation is an analysis of how, for example, a roll that gives a translation of (2,2) can be modified to give translations of (2,-2), (-2,2) and (-2,-2). Also investigate what happens if a roll is reversed, e.g. LURDRU becomes URDRUL.

It is also possible to find and investigate rolls that rotate the box. For example LDR rotates the box 90 degrees about the bottom left hand corner.

Cuboid Puzzles

It has already been mentioned that 4 boxes can be joined in a variety of ways to make a cuboid. The ways 6 boxes can be combined into a cuboid are even more varied. One way of describing these arrangements is by the number of boxes each box touches. So cuboid A in Fig. 9 could be denoted as (1,1,2,2,2,2) which indicates that two of the boxes touch just one other and the other four each touch two. Cuboid B can be recorded as (2,2,3,3,4,4).

...

Fig. 9 Cuboids from 6 boxes

~_:::::::: r::j:::::i: ::;"-;

Building Investigation * Can you make a cuboid with 6 matchboxes which

would be recorded as (2,2,2,3,3,4)?

Find out how many different cuboids can be built with 6 boxes.

From these cuboids you can create 3 dimensional joining puzzles by gluing the matchboxes together in pairs. An example is shown in (Fig. 10). The three resulting pieces then have to be fitted together to make a cuboid.

II::::- i::

LB-~i:

-i:- * iii~i _:ii

---ii-;~_ii--

:::: i: :i: :: i"i :i::: ~i:;-: iii~: : :-::::::-:: ::~i

"":

:~i i:i:::- -::~--~---

1:: B

---,-

-isi

:: _:::

:: :

:__ ::::: _ : :- :-:_: ::::--:-

i~:li~l-::ll:ili:i-::: --iiiiii:i :ii_

Fig. 10

Cuboid puzzle. Can you make a cuboid from these 3 pieces?

Mathematics in School, May 1992 27



The question arises as to how many different puzzles it is possible to create using 6 boxes and once this has been investigated even more challenging puzzles can be created using 8 or more boxes.

The recording of the results of these investigations gives a splendid opportunity for practising isometeric drawing. The 1:2:3 property of the matchbox means that it is ideal for this type of representation (Fig.11).

.- , jsf--....-

- --.` ,

><I. >

" . . :

w

Fig. 11

In the same way the ratio property can be exploited and matchboxes used when dealing with plans and elevations. The 1:2:3 ratio of sides ensures that the boxes can be easily and unambiguously represented on squared paper. Students can be challenged to build with matchboxes the shapes represented by plans and elevations or build their own shapes and draw the appropriate plans and elevations on squared paper.

Building Puzzle Use three matchboxes and build the shapes shown by the plans in Fig. 12 (the illustration was made by placing matchboxes on a photocopier).

Fig. 12 Matchbox plans

Throwing The Matchbox How about using the matchbox in probability experiments? When a matchbox is thrown in the air can you predict which face it will land on most often? Does experimentation confirm your prediction? Does it make any difference if the box is half full of matches?

It is interesting to look out for, and collect, cuboids with other integral ratios of sides and to carry out the investigations outlined above with these different cuboids. For example what rectangles and squares can be rolled over with dominoes which have a ratio of sides 1:3: 6? What happens if we use bricks with ratio of sides 1:1:2? Can the findings be generalized for any cuboids with sides in the ratio a:b:c where a, b and c are integers?

Next issue; more ideas on mathematics in a matchbox.

Answers to some of the puzzles; Rolling Box Puzzles

5x5 1 2 1 2

5 4 3 OR3 6

5x7

3 2 1 4 3 10

2

4 10 9 8 OR 5 9

5 6 7 6 7 8

10 x 10

27 26 25 24 23 22 21

4 3 10 11 12 20

2

5 9 15 4 13819

61 7 8 16 17 18

Fig. 13 Solutions to rolling box puzzles

28 Mathematics in School, May 1992



mathematics in a matchbox. part 1

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