Nowadays, we know that there can be no bimagic square, whose order is less than 8 (take a look at the multimagic pages of Christian Boyer). All bimagic squares with a lower order are not composed of consecutive numbers beginning with 1, such as the following pandiagonal bimagic square sixth order of David M. Collison. (left fig.)
Other squares are indeed composed of the numbers from 1 to 36, but not all rows, columns and diagonals have the same sum. Although the following magic square, created by Pfeffermann in 1894, possesses magical columns, rows and diagonals fail. (right fig.)
15 | 13 | 47 | 45 | 9 | 21 |
12 | 43 | 20 | 16 | 49 | 10 |
48 | 19 | 8 | 14 | 17 | 44 |
5 | 41 | 29 | 35 | 37 | 3 |
34 | 1 | 40 | 38 | 7 | 30 |
36 | 33 | 6 | 2 | 31 | 42 |
1 | 30 | 9 | 26 | 23 | 22 |
29 | 18 | 31 | 2 | 27 | 4 |
17 | 5 | 12 | 24 | 32 | 21 |
16 | 34 | 13 | 25 | 3 | 20 |
33 | 10 | 35 | 6 | 19 | 8 |
15 | 14 | 11 | 28 | 7 | 36 |
There are several methods to construct bimagic squares of orders 8, 9 and 16, where each of them generates a great variety of different squares. But for different orders only a few squares are known and some of them are listed in the gallery.
For orders 8, 9 and 16 bimagic squares can be created that have additional properties. The following bimagic square of Brutus Portier for example, is symmetric and self-complementary. Additionally, you can also split the square into two Euler squares.
66 | 79 | 59 | 13 | 20 | 9 | 44 | 51 | 28 |
17 | 24 | 1 | 39 | 52 | 32 | 67 | 74 | 63 |
40 | 47 | 36 | 71 | 78 | 55 | 12 | 25 | 5 |
61 | 68 | 75 | 2 | 18 | 22 | 33 | 37 | 53 |
6 | 10 | 26 | 34 | 41 | 48 | 56 | 72 | 76 |
29 | 45 | 49 | 60 | 64 | 80 | 7 | 14 | 21 |
77 | 57 | 70 | 27 | 4 | 11 | 46 | 35 | 42 |
19 | 8 | 15 | 50 | 30 | 43 | 81 | 58 | 65 |
54 | 31 | 38 | 73 | 62 | 69 | 23 | 3 | 16 |