A Magic Square

Think of a number between 1 and 26.
Look at the following table of six squares, one square at a time.

+--------------+--------------+---------------+
|   1   4   7  |  2   5   8  |  3   4   5  |
| 10 13 16   | 11 14 17  | 12 13 14  |
| 19 22 25   | 20 23 26  | 21 22 23  |
+---------------+--------------+---------------+
|   6   7   8  |   9 10 11  | 18 19 20  |
|  15 16 17  | 12 13 14  | 21 22 23  |
|  24 25 26  | 15 16 17  | 24 25 26  |
+---------------+--------------+---------------+


For example, say you pick 16,
16 is in the 1st, 4th and 5th squares.
If you add the first numbers of each of these squares, you get

1 + 6 + 9 = 16

Say you pick the number 13.
13 is in the 1st, 3rd and 5th squares
By adding the first numbers of each of these squares, you get

1 + 3 + 9 = 13

etc.

Can you explain this result?