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### World War II Window Blackout

```
Date: 10/21/2001 at 14:14:16
From: annonymous
Subject: Problem Solving

During the Second World War, families had to black out their windows.
Mr. Brown had a square window 120cm x 120cm, but the only material he
could find was a sheet of plywood 160cm x 90cm; same area, different
shape. He drew some lines and cut out just two congruent shapes, which
he joined to make a square of the correct size. How did he do it?

I have tried drawing sketches, scaled drawings, and cardboard cutouts,
but I can't solve it. I see no way of doing it. Please help!
```

```
Date: 10/23/2001 at 00:02:50
From: Doctor Ian
Subject: Re: Problem Solving

Hi,

My first thought was that this would end up being two L shapes, e.g.,

+-----------+
|           |
|       B   |
+-----+     |
|     |     |
|     |     |
|     |     |
|     |     |
|     |     |
|     +-----+
|   A       |
|           |
+-----------+

but that doesn't quite work.  You always end up leaving a hole in the
middle. Then I tried trapezoids,

+-----------+
|           |
|       B   |
+           |
| .         |
|   .       |
|     .     |
|       .   |
|         . |
|           +
|   A       |
|           |
+-----------+

but that doesn't quite work, either. You always end up leaving a hole
in the corner.

Here is the general idea:

+-----------+
|           |
|       B   |
+---+       |
|   |       |
|   |       |
|   +---+   |
|       |   |
|       |   |
|       +---+
|   A       |
|           |
+-----------+

By moving part B to the right and then down, you can make a square:

+---+-----------+
|   |           |
|   |       B   |
|   +---+       |
|       |       |
|       |       |
|       +---+   |
|   A       |   |
|           |   |
+-----------+---+

I'll leave it to you to work out the dimensions involved.

I finally came up with it by forgetting about the initial rectangle,
and focusing on the final square. I knew I'd have to shift one of
the pieces up and over to the left (do you see why?). I knew the base
of A would be 90cm wide, which meant that the tip of B would have to
be 30cm wide. The rest follows from the fact that the two pieces have
to be congruent.

This was a fun question to think about. Thanks for asking it!

- Doctor Ian, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Euclidean/Plane Geometry
High School Geometry
High School Puzzles
High School Triangles and Other Polygons
Middle School Geometry
Middle School Puzzles
Middle School Triangles and Other Polygons
Middle School Two-Dimensional Geometry

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