Rectangle Nightmare

Task 84 ... Years 4 - 10

Summary

Rectangle Nightmare was designed by Geoff Giles, a well known Scottish maths educator. In essence the problem is:

We have a frame into which we seem to exactly fit a set of shapes. We remove one shape (a significant rectangle), rearrange and the remaining pieces still fit into the frame! Now you see it - now you don't. How can this paradox be explained?

This cameo has a From The Classroom section which provides Geoff's own paper on the mathematics behind the design of this intriguing puzzle and additional comments to assist with challenging students who are trying it.

Materials

1 set of shapes and their frame

Content

conservation of area
measuring area and length
scaling (non-proportional)
scaling (proportional)
ratio
Fibonacci Numbers

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 included in this cameo.

The pieces can be fitted into the frame as shown.

5 pieces

6 pieces

However, the way the material is displayed in the photograph above 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?

Whole Class Investigation

Tasks are an invitation for two students to work like a mathematician. Tasks can also be modified to become whole class investigations which model how a mathematician works.

There are at least two ways to convert this task to a whole class investigation so that all students have the opportunity to struggle with the spatial challenge.

For a few lessons prior to the investigation, use your task copy and pass it between pairs of students allowing say 10 minutes to find a solution before passing it on.
Use the 6 piece diagram above to prepare a scaled up copy for the students to cut out. You will also need a matching rectangle frame. The trick to creating the illusion is to make the borders of the frame thickish so that 6 pieces tightly touch the inside edge of the frame while the rearranged 5 pieces still appear to fit with 'acceptable, unnoticed' error.

Before taking either approach, read the article below and decide how you will structure the investigation.

At this stage, Rectangle Nightmare does not have a matching lesson on Maths300, however, Lesson 132, 64 = 65, is a companion missing square puzzle that is also related to Fibonacci Numbers. For more ideas and discussion about 64 = 65, open a new browser tab (or page) and visit Maths300.

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

From The Classroom

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

A rectangle 15cm by 22cm is divided up as shown in Figure 1.

Figure 1

Figure 2

But if we rearrange the pieces we can get a triangle as shown in Figure 2, with height 27cm and base 24cm.

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 ½ × 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.

Now if these shapes are cut out of a soft, spongy material like Plastazote, the small difference will not be noticed at all, and the deception will then be complete.

A More Difficult Case

The example we have been looking at involved "directions of lines". But some cases of deception are quite different.

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 40mm², 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

Geoff's article gives the background to how this puzzle was devised. Rectangle Nightmare, is the same principle, but the frame and the pieces have been scaled up by a factor of 3·2, so the inside width of the blue frame is 80mm. Now it is a good size for the desk top.

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!

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