64 = 65? Still Not Convinced? But this can not be true what is the ...

64 = 65?
The 8 – by – 8 square in the diagram above can be cut into four pieces along the
thick lines. These pieces can then be rearranged to make a 5 – by – 13 rectangle. But the
square contains 8x8 = 64 area units and the rectangle contains 5x13 = 65 area units.
Where has the extra square come from?
Still Not Convinced?
Good, you should not be. Try to cutting the pieces as shown and rearranging
them yourself if you are not sure the puzzle is correct.
But this can not be true what is the explanation?
• Take a careful look at one of the puzzles. Is it really what it seems?
• Maybe things are not as straight forward as they look.
• An old Greek might lead you on the right path.
Assignment:
It is obvious that 64 can not be equal to 65. Give me a convincing explanation using
math telling me what is wrong with the figure. Also try to explain where the extra square
came from.

Answer:
Ok, the 8x8 figure is correct. The right figure (the red one) is built on three consecutive
Fibonacci numbers, but they can figure what is wrong with it by using the Pythagorean
Theorem.
First, if they cut and rearrange this small figure they will not discover that anything is
wrong as one area unit is so small. So that is a red herring.
Now to the solution,
Calculate how long the diagonal should be in the right figure (the red one):
2 2
Diagonal: 13 + 5 = 13.92839
Next calculate the length of the sides of the two figures on the bottom (the triangle and
the top of the trapezoid).
2 2
Triangle: 8 + 3 = 8.544004
Trapezoid:
2 2
5 + 2 = 5.385165
Sum: 13.92917
So the sum of the length of the two figures is slightly longer than the diagonal.
Conclusion: When rearranging there is not a straight line corner to corner, the line is
bowing slightly inward which creates that extra square between the figures along the
diagonal.