A Fibonacci Puzzle

Draw an 8 by 8 square and subdivide it into four pieces as shown below. This square has area 64. If you rearrange the four pieces you can build a rectangle whose sides are 13 and 5. This rectangle has area 65. Where did we gain an extra square? That is, how did we gain more area simply by rearranging the pieces. Your task in this assignment is to figure out what is going on.

The reason this puzzle is called a Fibonacci puzzle is because all of the lengths 3, 5, 8, and 13 mentioned above are Fibonacci numbers. In fact, given four consecutive Fibonacci numbers, you can build a puzzle in the same manner as described above. If you start with the Fibonacci numbers 5, 8, 13, and 21, draw the resulting two rectangles; one is a 13 by 13 square and the other is an 8 by 21 rectangle. Divide both into the appropriate four pieces.