|How many magic squares are there?|
|Results of historical and computer enumeration|
|5||579 043 051 200||275 305 224||48 544||3 600||16|
|6||94 590 660 245 399 996 601 600||1.775399 (42) ·1019||0||0||0|
|7||4.2848 (17) ·1038||3.79809 (50) ·1034||1 125 154 039 419 854 784||1.21 (12) ·1017||20 190 684|
|8||1.0806 (12) ·1059||5.2225 (18) ·1054||2.5228 (14) ·1027||C8 + ?||4.677 (17) ·1015|
|9||2.9008 (22) ·1084||7.8448 (38) ·1079||7.28 (15) ·1040||81·E9 + ?||1.363 (21) ·1024|
|10||1.4626 (16) ·10115||2.4149 (12) ·10110||0||0||0|
|Variants of a square by means of rotations and reflections are not counted.|
Statistical notation: 1.2345 (25) ·109
means that the number is not known precisely but is in the interval
Ultramagic squares are associative (centrally symmetrical) and pandiagonal magic.|
|B3 = 1: the Lo Shu (as it is known) is unique.|
|B4 was found by the Frenchman Bernard Frénicle de Bessy in 1693. First analytical proof by Kathleen Ollerenshaw and Herman Bondi (1982).|
|A4, C5 and E5 could be found on the former website of Mutsumi Suzuki.|
|B5 was calculated in 1973 by Richard Schroeppel (computer program), published in Scientific American in January 1976 in.|
|A5 was calculated by myself in March 2000 using a common PC. Suzuki published the result on his website. I was able to confirm the result by using other methods.|
|D5 is equal to the number of regular panmagic squares. They can be generated using Latin squares, as Leonhard Euler pointed out in the 18th century.|
was calculated by Artem Ripatti (Russia) until April 2018. This is a new milestone in magic square enumeration.|
Read his paper at arxiv.org/abs/1807.02983 (external link). Download the folder alldata.zip (127 MB) in order to get all data about the number of semi-magic squares over the 9366138 classes.
Artem told me details and results of his calculation already in 2017. I can confirm the correctness of his method.
D6 and D10 were proved by A.H. Frost (1878) and more elegantly by C. Planck (1919).|
C6 and C10 are also equal to 0, because each associative (symmetrical) magic square of even order can be transformed into a pandiagonal magic square.
B6 was first estimated (with proper result) by Klaus Pinn and Christian Wieczerkowski and published in May 1998.
They used a method called 'parallel tempering Monte Carlo' and got
They also estimated B7 with lower accuracy:
My own estimates (see table) match these results. I used another method that could be called 'Monte Carlo backtracking'.
|All estimates in the columns B, C and D are found with the same method, that is more like the approach of Schroeppel than the one of Pinn and Wieczerkowski. There should be no systematic error, because the method was checked by Prof. Peter Loly (University of Manitoba, Canada) and all results could be confirmed by different programs. For higher orders see: Numbers of classic magic squares|
|C7 (NEW) was calculated by Go Kato (Japan) and first published in November 2018 at OEIS A081262 (external link). His approach is based on Ripatti's method. A short description can be found on OEIS. Read about all details in a paper written by Go Kato and Shin-ichi Minato at arxiv.org/abs/1906.07461 (external link). Kato's approach is definitely correct. There is no error in his calculation as I found exactly the same result with an own program based on his method. This new milestone also is a confirmation of my approximation method because Kato's result is very close to the estimate 1.125151(51)·1018 which was shown in the table before.|
|E7 was calculated by myself in May 2001. Special transformations made it possible to consider only two positions of the integers 1, 25 and 49. With advanced equations and a heuristic backtracking algorithm the calculation time could be reduced to a few days. All ultramagic squares of order 7 have been saved and are available for further research. For more details see: Ultramagic Squares of Order 7|
D7 (November 2001) was a big surprise. There are |
|D8 is greater than C8 because each associative magic square of order 8 can be transformed into a pandiagonal one and there are examples of additional pandiagonal squares that could not be derived from an associative square.|
|E8 and E9 were estimated in March 2002. In the case of E8 I could find 64 transformations and several equations with only 4 variables.|
|D9 is greater than 81·E9, because each ultramagic square of order 9 can be transformed by cyclic permutation of rows and columns into 80 other pandiagonal magic squares that are not associative.|