Now consider the symmetrical with odd order n=5. This means that there is no real center point as the center of symmetry, but a (magenta coloured) center cell.
Since there can't exists a complementary number, this number must be the average number of all numbers in this square. In this case, we will find the number 13 in the center cell. For an arbitrary order n this number z is given by:
3 | 16 | 9 | 22 | 15 |
20 | 8 | 21 | 14 | 2 |
7 | 25 | 13 | 1 | 19 |
24 | 12 | 5 | 18 | 6 |
11 | 4 | 17 | 10 | 23 |
3 | 16 | 9 | 22 | 15 |
20 | 8 | 21 | 14 | 2 |
7 | 25 | 13 | 1 | 19 |
24 | 12 | 5 | 18 | 6 |
11 | 4 | 17 | 10 | 23 |
It is easy to see that also in this case not only the coloured cells, but all symmetrically opposite cells sum to a constant number. This number is always equal twice the central number plus 1.
n^{2} + 1=5^{2} + 1 =26
The following figures show the location of all complementary pairs of numbers, which are found on their symmetrical places to the center cell. To be not too confusing, the presentation is divided into two squares.