The chessboard paradox (also called paradox of Loyd and Schlömilch) is based on an optical illusion.

A chessboard or a square with a side length of 8 units is cut into four pieces.

Those four pieces are used to form a rectangle with side lengths of 13 and 5 units. Hence the combined area of all four pieces is 64 area units in the square but 65 area units in the rectangle, and 63 with another arrangement.

This paradoxical results are generated by an optical illusion, as the four pieces don’t fit exactly in the rectangle, but leave a small barely visible gap around the rectangle’s diagonal. 

Overlaying the hypotenuses from the figures results in a very thin parallelogram  with an area of exactly one grid square, so the “missing” area.