I have a little problem with this mathematical equation below. Can any of you help me to solve this paradox?
That equation seems to be true, but actually there is a logical explanation about this mathematical paradox.
The 64 = 65 paradox arises from the fact that the edges of the four pieces, which lie along the diagonal of the formed rectangle, do not coincide exactly in direction. This diagonal is not a straight segment line but a small lozenge (diamond-shaped figure), whose acute angle is arctan 2/3 - arctan 3/8 = arctan 1/46 which is less than 1 degree 15'.
Only a very precise drawing can enable us to distinguish such a small angle. Using analytic geometry or trigonometry, we can easily prove that the area of the "hidden" lozenge is equal to that of a small square of the chessboard.
Okay, sorry, I didn't even understand myself about that answer that I wrote up there. To make it simple: there is a "hole" in that picture, try to redraw that equation and you'll realize that those just a nice optical illusion.
Fun or Not: