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An associative magic square is a magic square for which every pair of numbers symmetrically opposite to the center sum up to the same value.
3 × 3
The Lo Shu Square, the unique 3 × 3 normal magic square, is associative, and as such is the only 3 × 3 associative magic square. The square is shown below in Frénicle standard form.[citation needed] All rows, columns and diagonals sum to 15 and all pairs symmetrically opposite the center sum to 10.
2
7
6
9
5
1
4
3
8
4 × 4
There are 48 order four associative magic squares. [1] Dürer's associative magic square is shown below.[2]
16
3
2
13
5
10
11
8
9
6
7
12
4
15
14
1
5 × 5
There are 48,544 order 5 × 5 associative magic squares.[3]
25
22
5
10
3
7
12
11
17
18
2
6
13
20
24
8
9
15
14
19
23
16
21
4
1
7 x 7
There are 1125154039419854784 order 7 x 7 associative magic squares.[3]
26
28
23
34
9
15
40
29
12
33
8
37
45
11
32
6
49
3
48
7
30
19
46
14
25
36
4
31
20
43
2
47
1
44
18
39
5
13
42
17
38
21
10
35
41
16
27
22
24
Larger sizes
Estimates for larger order associative magic squares are given below.[4]
square size
magic square
associative magic square
3×3
1
1
4×4
880
48
5×5
275,305,224
48,544
6×6
~1.77 × 1019
0
7×7
~3.79 × 1034
~1.12 × 1018
8×8
~5.22 × 1054
~2.52 × 1027
9×9
~7.84 × 1079
~7.28 × 1040
10×10
~2.41 × 10110
0
Physical Properties
The image below shows areas completely surrounded by larger numbers with a blue background. A water retention topographical model is one example of the physical properties of magic squares. The water retention model progressed from the specific case of the magic square to a more generalized system of random levels. A quite interesting counter-intuitive finding that a random two-level system will retain more water than a random three-level system when the size of the square is greater than 51 X 51 was discovered. Percolation theory explains the counter-intuitive retention. This was reported in the Physical Review Letters in 2012 and referenced in the Nature article in 2018.[5][6][7]
This 16 x 16 associative magic square has five additional properties. [8]
The "Get Type" utility evaluates a square for over twenty different characteristics. [9]
