Increasing and Decreasing

Choose any N numbers from the set { 1, 2, 3, ..., 2N }, and write them in a row in increasing order.

Then write the remaining numbers in a second row below, in decreasing order.

Then, in a third row, write the absolute values of the differences between the corresponding numbers in row 1 and row2.

Finally, add up the numbers in the third row.

Show that the result is N2.

For example, let's try an example with N = 5:

236810
97541
74149

and 7 + 4 + 1 + 4 + 9 = 25 = 52.

Solution