A Fibonacci Jigsaw Puzzle

Date: 04/29/99 at 21:58:12
From: Laik
Subject: Geometry: area

For a homework problem in math, our geometry teacher showed us a 
problem dealing with the area of a square and making it equal to a 
rectangle. The area of the square was 64 but by the time we got to 
the rectangle (using Area formula A = b(h)) it equaled 65.

The question is where did the mistake occur?

He had an 8*8 square, which he cut into 2 pieces; the areas of these 
were 24 and 40. Then he cut the rectangular 24 piece into a right 
triangle, and the 40 piece into a trapezoid. Each time both pieces 
were equal. (There were 2 equal right triangles and 2 equal 
trapezoids.) Each trapezoid and triangle was then matched, making a 
proportionally bigger right triangle. These two triangles were then 
put together to make a rectangle with a width of 13 and height of 5. 
This area then equalled 65, but it is physically impossible for this 
to happen. What am I missing?

The figure looked like this:

_ _ _ _ _ a/e _             line ac       
|         |    |                 entire area =65cm^2
|  40     | 24 |     A_ _ _ _ _ _ _ _ _ _ _ _ _ D              
|f        |    |     |                |        |
|         |    |     |                |
|         c    |     |       e|      f|        |
|         |    |     |        |                |
|         |    |     |        |                |
|         |    |     - - - - - - - - - - - - -                    
- - - - - - - - c/f  B                         C

Does this make sense to you?

Laik

Date: 04/29/99 at 23:42:27
From: Doctor Peterson
Subject: Re: Geometry: area

Hi, Laik. My father-in-law made a copy of this puzzle out of wood, so 
I'm very familiar with it.

 _ _ _ _ _ a/e _             line ac       
 |         |    |                 entire area =65cm^2
 |  40     | 24 |     A_ _ _ _ _ _ _ _ _ _ _ _ _ D              
 |f        |    |     |                |        |
 |         |    |     |                |
 |         c    |     |       e|      f|        |
 |         |    |     |        |                |
 |         |    |     |        |                |
 |         |    |     - - - - - - - - - - - - -                    
 - - - - - - - - c/f  B                         C

Here's a site by Ron Knott that explains the puzzle, but doesn't quite 
give the actual answer:

http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/   

Scroll down the page and look for The Harder Fibonacci Puzzles page 
under The Puzzling World of Fibonacci Numbers. On the Harder Puzzles 
page, look for "A Fibonacci Jigsaw puzzle or How to Prove 64=65!"

The trick is that in the rectangle, the diagonal line isn't quite 
exact: calculate the slopes of the segments Ae and eC, and you'll find 
that they are not the same; AeCf is really a very narrow parallelogram 
with area 1. See if you can calculate the area and prove me right.

This is related to the Fibonacci sequence, as explained by Ron Knott, 
because the ratios of successive Fibonacci numbers, in this case 5:8 
and 8:13, are close, approaching the golden ratio as you use larger 
numbers; that allows the slopes to fool you.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/