64 = 65 ??

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Wow they proved it! Obviously there's a flaw somewhere that escaped my eye.

Cheers to the first one here to point it out...
 
Maybe I'm just stupid and I'm thinking far too simple here, but did they just take an 8X8 square and turn the same area into a 5X13 rectangle? Where did that extra area come from?
 
lol.. saw this ages back! ..was really freaked to see sucha wierd geometric paradox at first, but then as every such unreal equations (illusions!) does have a virtually impractical trick behind it (remember 1=2 thing during school times!? :P) so is the same with this!

The following may elaborate it in a significantly better way:

However, a closer look at the slanted sides of the trapezoidal and triangular pieces shows that they cannot be aligned as implied in the above fallacious illustrations. In fact, they are the diagonals of two dissimilar rectangles of sizes 2x5 and 3x8, respectively, and hence have distinct slopes. But the difference of the ratios (2/5==0.4 versus 3/8==0.375) is too small to be perceived by the eye.

Note that the dissection cuts the sides of the 8x8 squares according to the proportion 5:3. The illusion becomes even more effective if the numbers 3, 5, 8 are replaced by a triple of higher consecutive Fibonacci numbers.

Dissection Fallacy -- from Wolfram MathWorld
Click to expand...
 
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