Rectangle Nightmare

Iceberg A task is the tip of a learning iceberg. There is always more to a task than is recorded on the card.

We have a conservation of area problem here which is part of a well-loved set of 'missing square' puzzles. In this case, the square has become a rectangle and Geoff supplies us with all the background in an article include with this cameo. The pieces can be fitted into the frame as shown. 5 pieces 6 pieces However, the photograph on the card suggests there is another way to fit the six shapes into the frame. How many ways can you find to fit the 5 pieces or the 6 pieces into the frame? Are the ways you find actually new ways, or just rotations or reflections of the pictures shown? At the first level of analysis there must be a very narrow space between the shapes in the first solution and the frame in which they fit. That space must be the same as the space taken up by the additional rectangle. Yes, but: The original 5 shapes must be very special. How were these shapes designed? Are there other shapes that could create a similar illusion?

Is it in Maths With Attitude? Maths With Attitude is a set of hands-on learning kits available from Years 3-10 which structure the use of tasks and whole class investigations into a week by week planner.

The Rectangle Nightmare task is an integral part of: MWA Chance & Measurement Years 9 & 10

Constructing Rectangle Nightmare

Rectangle Nightmare was designed by Geoff Giles, a well known Scottish mathematics educator who is perhaps most recognised for the challenging materials of the DIME Project. Geoff contributed several tasks to the Mathematics Task Centre Project and was also responsible for developing the educational use of Plastazote, the colourful, almost indestructible foam from which this task, and several others, are made.

Now deceased, Geoff, prepared the article below in the latter years of his retirement in response to our request for enlightenment about Rectangle Nightmare. It is reproduced here with his permission, albeit slightly modified.

Sleight of Hand in Geometry

Introduction

• If we wish to lead people astray in mathematics there is probably nowhere more suitable for doing this than in the general area of 'geometry'.
• Here we have so many opportunities to con, or bamboozle, or generally cause confusion that we often lose sight of the truth ourselves.
• This may be because geometrical truths so often seem to be so obvious and self-evident, that we are easily seduced into believing what we see. In such circumstances we may find it difficult to think logically about the situation at all.

First a Simple Example

Figure 1

Figure 2

But this means that, while the area of the rectangle is clearly 15 × 22 = 330 sq. cm., the area of the triangle is in fact ˝ × 24 × 27 = 324 sq. cm. So where has the extra area disappeared to?

The answer turns out to be very simple. If we look closely at Figure 2 we see that the sides of the triangle don't look quite straight.

Look at point A. We find that the sloping side rises vertically 5 as it moves horizontally 2, making a gradient of 2·5. But when it passes the horizontal line marked 10 -- 10, the gradient then becomes 22/10, or 2·2.

So clearly these two lines do not form exactly the straight line they seem to. The triangle is slightly fatter than it should be.

A More Difficult Case

Our second example takes a rectangle of size 3 x 8 (with an area equal to 24) and makes it appear to have the same area as a square of side 5 (with an area of 25).

Instead of presenting it as a finished puzzle, we now show how it was all worked out from the beginning so that you can see how the 'sleight of hand' was achieved.

• First of all the rectangle and square are drawn on a centimetre grid, as shown in Figure 3.

Figure 3a

Figure 3b

To make these very different shapes look as if they have the same area, we first draw them again on a grid of rectangles. By changing to a grid of rectangles each measuring 5mm by 8mm (a reduction of 80% horizontally and 50 % vertically), we get the first composite rectangle measuring 2·4cm by 4cm (Figure 4).

Figure 4

The corresponding 5 x 5 'square' now appears as the rectangle shown in Figure 5, which measures 2·5 cm by 4·0 cm. Comparing the two rectangles, we can see that the difference between the two areas is a strip measuring 40mm by 1mm wide, so its area is 40mm˛, which is of course the area of the additional red rectangle. Do you notice the connection with Rectangle Nightmare pieces?

Figure 5

Now suppose we have the blue rectangular frame made of spongy Plastazote, shown in Figure 6. If it is 25mm wide the shapes shown in Figure 4 will fit into it, but you will not find it possible to fit in the extra red rectangle as well. Even a brief glance is sufficient, isn't it, to convince you that there is not enough space to put in an extra small rectangle? But remember that your eye can be easily deceived, and you may not understand where you went wrong.

Figure 6

About the Task

Your students won't have the information above when they try the task. Just completing the two 'jigsaws' involved may provide headache enough for them, let alone explaining the paradox. Some questions you might use if students become frustrated are:

• If the students are trying Qu.3 and tell you they can't make the shapes fit even though the hint suggests they are the same orientation as the picture, ask them if the hint says anything about the frame being turned. It actually has to be in landscape orientation for the hint to work.
• If the students are trying Qu.4, you might again refer to the hint. The shapes didn't have to be turned in Qu.3. Perhaps they do have to be turned in Qu.4.

Explaining the paradox can be as straightforward as drawing attention to the thin space around the solution to Qu.3 which is isn't there in the solution to Qu.4. Or it can be as complex as Geoff's explanation above.

One final thing - because a mathematician is never finished with a problem. Could it be that there is something special about the 5 : 8 ratio that was chosen to distort the original 1cm squares above? 5 and 8 appear in the Fibonacci series:

1, 1, 2, 3, 5, 8, 13, 21, ...

• If you investigate further you might also find the 24 and 25 which are the width measurements of the two forms of the puzzle. And what about the 40? Can you find that?
• If you can, then try applying what you find to another consecutive pair from the series. Can you make another paradox puzzle?
• And what about the shapes that made up the original 3 x 8 rectangle (Figure 3a)? Why choose those? Surely the rectangle could be divided into five other '-omino' shapes? Are there any other '-omino' dissections of this rectangle that would also join with the single square to make a 5 x 5 square? If there are, what would the paradox puzzle look like then?
• Oh yes, and one more thing. There are at least two ways to place the six pieces in the frame!

Follow this link to Task Centre Home page.