[Illustration]
The explanation of this little fallacy is as follows. The error lies in
assuming that the little triangular piece, marked C, is exactly the same
height as one of the little squares of the board. As a matter of fact,
its height (if we make the sixty-four squares each a square inch) will
be 1+1/7 in. Consequently the rectangle is really 9+1/7 in. by 7 in., so
that the area is sixty-four square inches in either case. Now, although
the pieces do fit together exactly to form the perfect rectangle, yet
the directions of the horizontal lines in the pieces will not coincide.
The new diagram above will make everything quite clear to the reader.