This is a very famous puzzle. Here is motivation:
Now consider following variation.
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
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.
Also there is an interesting video illustrating this in real life: