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. 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. Template:Multicol

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. Template:Multicol

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 (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

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

External links

• Associative magic square, MathWorld
• Greg Ross, Futility Closet