| Take a chessboard and cut it into four pieces as shown below. Now, reassemble the pieces to create a rectangle with larger area. Here is the original square:  Rearranged, we have:  The area of the square is 64, while the area of the rectangle is 65. How is this possible? |
Solution:
| Look at the main diagonal in the rectangle. Observe that the slope is not constant. On the ends, the slope is -2/5, but in the middle, it is -1/3. Clearly something is amiss. Well, the simple answer is that we cheated. What actually happens is that the part of the rectangle that’s not covered by the four pieces forms a very acute (and nearly undetectable) parallelogram of area 1. The missing area is hidden under the thick lines that we drew. As you might imagine, as we increase the size of the original square or the thickness of the lines, we can hide the area. To see where the missing area went, let us consider a smaller example. Consider a 3 by 3 square board, partitioned as below. We shall rearrange the pieces in a similar manner to above. In this case, notice that we did not use thick lines. As a result, the missing area is revealed. (Also, you can see that the slopes are not equal.)  Rearranged, we have:  The thin parallelogram in the middle is the missing unit of area. |
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