Associative magic square
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.[1] All rows, columns and diagonals sum to 15 and all pairs symmetrically opposite the center sum to 10.
4 × 4
There are, up to rotation and reflection, 48 distinct 4 × 4 associative magic squares, four of which are given here.[2]
1 | 14 | 15 | 4 |
8 | 11 | 10 | 5 |
12 | 7 | 6 | 9 |
13 | 2 | 3 | 16 |
|
1 | 15 | 14 | 4 |
8 | 10 | 11 | 5 |
12 | 6 | 7 | 9 |
13 | 3 | 2 | 16 |
|
2 | 13 | 16 | 3 |
7 | 12 | 9 | 6 |
11 | 8 | 5 | 10 |
14 | 1 | 4 | 15 |
|
2 | 16 | 13 | 3 |
7 | 9 | 12 | 6 |
11 | 5 | 8 | 10 |
14 | 4 | 1 | 15 |
|
5 × 5
There are 48,544 distinct 5 × 5 associative magic squares, up to rotation and reflection, such as the two below.[3]
15 | 22 | 5 | 10 | 3 |
7 | 12 | 11 | 17 | 18 |
2 | 6 | 13 | 20 | 24 |
8 | 9 | 15 | 14 | 19 |
23 | 16 | 21 | 4 | 1 |
|
19 | 6 | 9 | 16 | 15 |
5 | 22 | 23 | 1 | 14 |
18 | 2 | 13 | 24 | 8 |
12 | 25 | 3 | 4 | 21 |
11 | 10 | 17 | 20 | 7 |
|
Larger sizes
There are no 6 × 6[3] (or 10 × 10) associative magic squares. Other than when there are none the precise number of associative magic squares at sizes bigger than 5 × 5 is unknown. The following table, due to Walter Trump, gives the sizes up to 10, followed by examples of 7 × 7 and 8 × 8 tables.
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 |
7 × 7 [4]
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 |
|
8 × 8
1 | 32 | 34 | 63 | 37 | 60 | 6 | 27 |
48 | 49 | 15 | 18 | 12 | 21 | 43 | 54 |
19 | 14 | 52 | 45 | 55 | 42 | 24 | 9 |
62 | 35 | 29 | 4 | 26 | 7 | 57 | 40 |
25 | 8 | 58 | 39 | 61 | 36 | 30 | 3 |
56 | 41 | 23 | 10 | 20 | 13 | 51 | 46 |
11 | 22 | 44 | 53 | 47 | 50 | 16 | 17 |
38 | 59 | 5 | 28 | 2 | 31 | 33 | 64 |
|
References
External links