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/
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