Fibonacci Riddle

Date: 11/21/2001 at 21:02:40
From: Daniel Arnold 
Subject: Fibonacci's 64 = 65

An 8x8 square has an area of 64. We can cut the square into four 
pieces and reassemble to get a 5x13 rectangle with an area of 65.  
Where does the extra 1x1 square come from?

Date: 11/21/2001 at 21:56:03
From: Doctor Tom
Subject: Re: Fibonacci's 64 = 65

The pieces don't fit EXACTLY. I don't know what version of this 
problem you're working on, but you'll probably find that the triangles 
are not perfectly aligned, but it's hard to see the error, since it's 
spread out over a long line.

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

Date: 11/21/2001 at 22:15:38
From: Doctor Rob
Subject: Re: Fibonacci's 64 = 65

Thanks for writing to Ask Dr. Math, Daniel.

If you check out the 5 x 13 rectangle, you will find that the pieces
do not exactly cover the rectangle. There is a very narrow 
parallelogram with two opposite corners at opposite corners of that
rectangle whose area is 1 square unit. That is because the slopes
of the diagonal edges of the two kinds of pieces are not exactly the
same, although they are close.

   o---+---+---+---+---o---+---+---o
   |                   |          /|
   +                   +         / +
   |                   |        /  |
   +                   +           +
   |                   |       /   |
   o.__                +      /    +
   |   `-.__ CD        |     /     |
   +        `-._       +  AC       +
   |        AB  `--._  |    /      |
   +                 `-o     BD    +
   |                   |   /       |
   +                   +  /        +
   |                   | /         |
   +                   +           +
   |                   |/          |
   o---+---+---+---+---o---+---+---o

  A
   o---+---+---+---+---+---+---+---o---+---+---+---+---o
   | `--.__                        |                   |
   +       `--.__                  +                   +
   |             `--.__            |                   |
   +                   o--._       +                   +
   |                 B |    `--._  | C                 |
   +                   +         `-o._                 +
   |                   |              `--._            |
   +                   +                   `--.__      +
   |                   |                         `--._ |
   o---+---+---+---+---o---+---+---+---+---+---+---+---o
                                                         D

The slope of AC is -3/8 = -0.375, and the slope of AB is 
-2/5 = -0.400.  Thus <BAC is not zero, and B is slightly below
line segment AC. Similarly, <BDC is not zero, and C is slightly
above line segment BD. The narrow parallelogram is ABDC, whose
area is 1 square unit.

Another way to see this is that

   AB = CD = sqrt(5^2+2^2) = sqrt(29) = 5.385165...,
   BD = AC = sqrt(8^2+3^2) = sqrt(73) = 8.544004...,
   AD = sqrt(13^2+5^2) = sqrt(194) = 13.928388...,
   AB + BD = AC + CD,
           = 5.385165... + 8.544004...,
           = 13.929169...,
           > 13.928388...,
             = AD.

This proves that neither B nor C lies on line segment AD.

For an animated illustration, see:

   Fibonacci or 64 = 65 - Bernard Langellier
   http://perso.wanadoo.fr/bernard.langellier/english/fiboswf.html   

Feel free to write again if I can help further.

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