Number of magic squares

Number of classic magic squares

To determine the numbers of magic squares following methods were used:
Exhaustive search by Standard Backtracking: orders 4 and 5
Approximation by Monte Carlo Backtracking: orders 6 to 20 (results for 18 to 20 are not reliable)
Estimation by statistical considerations on magic series combined with extrapolations of known approximations: orders greater than 20
For more details and results of the last method see: Estimates of the number of magic squares

Order	Number of magic squares	Max. expected relative error
3	1	0 %
4	880	0 %
5	275 305 224	0 %
6	(1.775399 ± 0.000042) · 10¹⁹	0.0024 %
7	(3.79809 ± 0.00050) · 10³⁴	0.014 %
8	(5.2225 ± 0.0018) · 10⁵⁴	0.035 %
9	(7.8448 ± 0.0038) · 10⁷⁹	0.049 %
10	(2.4149 ± 0.0012) · 10¹¹⁰	0.049 %
11	(2.3358 ± 0.0014) · 10¹⁴⁶	0.059 %
12	(1.1424 ± 0.0010) · 10¹⁸⁸	0.087 %
13	(4.0333 ± 0.0054) · 10²³⁵	0.14 %
14	(1.5057 ± 0.0024) · 10²⁸⁹	0.16 %
15	(8.052 ± 0.022) · 10³⁴⁸	0.27 %
16	(8.509 ± 0.027) · 10⁴¹⁴	0.31 %
17	(2.314 ± 0.009) · 10⁴⁸⁷	0.39 %
18	(2.047 ± 0.008) · 10⁵⁶⁶	(?) 0.40 %
19	(8.110 ± 0.035) · 10⁶⁵¹	(?) 0.44 %
20	(1.810 ± 0.008) · 10⁷⁴⁴	(?) 0.44 %
21	2.6 · 10⁸⁴³	~ 15 %
22	3.2 · 10⁹⁴⁹	~ 20 %
23	3.9 · 10¹⁰⁶²	~ 25 %
24	5.9 · 10¹¹⁸²	~ 30 %
25	1.3 · 10¹³¹⁰	~ 35 %
26	4.9 · 10¹⁴⁴⁴	~ 35 %
27	3.8 · 10¹⁵⁸⁶	~ 40 %
28	7.2 · 10¹⁷³⁵	~ 40 %
29	3.8 · 10¹⁸⁹²	~ 45 %
30	6.5 · 10²⁰⁵⁶	~ 45 %

For higher orders see: Estimates of the number of normal magic squares

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