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.
2
7
6
9
5
1
4
3
8
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]
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
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.
^Robert Dickter "Archived copy". Archived from the original on 2012-01-06. Retrieved 2012-02-21. {{cite web}}: Unknown parameter |deadurl= ignored (|url-status= suggested) (help)CS1 maint: archived copy as title (link)
