(related to Problem: A Chessboard Fallacy)
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 $1\frac{1}{7}$ in. Consequently, the rectangle is really $9\frac{1}{7}$ 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.
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