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/