Number of magic squares

Realizing the great number of different magic squares, it was nearly impossible to recognize differences on existing magic squares. But this changed with fast computer systems, because they were able to distinguish magic squares by reflections and rotations. As a result it was found, for example, that there are an incredible 275 305 224 magic squares of 5th order (in words: 275 millions 305 thousands 224 magic squares).

The current state of knowledge is given in the following table.

Order N(n) P(n)
2 0 0
3 8 1
4 7040 880
5 2 202 441 792 275 305 224
6 ? ?

Walter Trump has dealt extensively with the number of different magic squares. His results are summarized in the following table.

Order semi-magic normal symmetrical pandiagonal ultramagic
3 9 1 1 0 0
4 68 688 880 48 48 0
5 579 043 051 200 275 305 224 48 544 3 600 16
6 9.4597 ·1022 1.775399 ·1019 0 0 0
7 4.2848 ·1038 3.79809 ·1034 1.12515 ·1018 1.21 ·1017 20 190 684
8 1.0806 ·1059 5.2225 ·1054 2.5228 ·1027 ? 4.677 ·1015
9 2.9008 ·1084 7.8448 ·1079 7.2800 ·1040 ? 1.363 ·1024
10 1.4626 ·10115 2.4149 ·10110 0 0 0

Further explanation and information about the values of this table can be found on the web pages of Walter Trump.