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Seki

The Japanese mathematician Takakazu Seki (1642 - 1708) designed a method to create multi-bordered magic squares of odd order. The method will be explained using the order n=9. With n=2k+1 this results in k=4.

The entire outer border comprises a total of 4 · (n − 1) numbers. In this example, this results in 2n − 2=16 numbers and their complements, which must first be suitably placed in the border. Seki uses the following steps:

The number 1 is always placed directly to the left of the upper right corner.
This number is followed by k − 1 numbers, which are entered from the lower right corner upwards.
Further k − 1 numbers are placed in the upper row to the left of the 1, continuously move to the left.
The next single number follows the vertical sequence already entered in the right column.
Now k numbers are entered in the left column from the center cell upwards.
Finally, the next k numbers are placed in the bottom row from the left neighbor of the center cell to the left.

7 6 5 1
8
4
3
2
7 6 5 1
12
11
10
9
8
4
3
16 15 14 13 2

The border is thus clearly defined, and the cells that are still empty are filled with the complementary values to the numbers already entered. As usual, the number complementary to a number x is calculated with n² + 1 − x. This means:

Numbers in the corners are complementary to the numbers in the diagonally opposite corners.
Numbers in rows or columns are complementary to the number directly opposite in the row or column.

80	67	68	69	7	6	5	1	66
12								70
11								71
10								72
9								73
74								8
78								4
79								3
16	15	14	13	76	77	78	81	2

This method can now be continued from the outer border towards the center. The inner square of order 7, which has not yet been filled, contains a total of 4 · (7 − 1)=24 cells, half of which are filled with complementary numbers. So 12 numbers in the range 17 … 28 must be placed together with their complements.

80 67 68 69 7 6 5 1 66
12 21 20 17 70
11 25 71
10 24 72
9 23 73
74 22 8
78 19 4
79 28 27 26 18 3
16 15 14 13 75 76 77 81 2
80 67 68 69 7 6 5 1 66
12 64 55 56 21 20 17 54 70
11 25 57 71
10 24 58 72
9 23 59 73
74 60 22 8
78 63 19 4
79 28 27 26 61 62 65 18 3
16 15 14 13 75 76 77 81 2

Do the same with the remaining squares until you finally get the multi-bordered magic square from Seki.

80 67 68 69 7 6 5 1 66
12 64 55 56 21 20 17 54 70
11 25 52 47 31 29 46 57 71
10 24 34 44 37 42 48 58 72
9 23 33 39 41 43 49 59 73
74 60 50 40 45 38 32 22 8
78 63 36 35 51 53 30 19 4
79 28 27 26 61 62 65 18 3
16 15 14 13 75 76 77 81 2
80 67 68 69 7 6 5 1 66
12 64 55 56 21 20 17 54 70
11 25 52 47 31 29 46 57 71
10 24 34 44 37 42 48 58 72
9 23 33 39 41 43 49 59 73
74 60 50 40 45 38 32 22 8
78 63 36 35 51 53 30 19 4
79 28 27 26 61 62 65 18 3
16 15 14 13 75 76 77 81 2

With this method from Seki, bordered magic squares can be created for all odd orders. Two other examples for n=5 and n=7 are shown in the following figure.

24 19 3 1 18
6 16 9 14 20
5 11 13 15 21
22 12 17 10 4
8 7 23 25 2
48 39 40 5 4 1 38
9 36 31 15 13 30 41
8 18 28 21 26 32 42
7 17 23 25 27 33 43
44 34 24 29 22 16 6
47 20 19 35 37 14 3
12 11 10 45 46 49 2

80	67	68	69	7	6	5	1	66
12				21	20	17		70
11	25							71
10	24							72
9	23							73
74							22	8
78							19	4
79	28	27	26				18	3
16	15	14	13	75	76	77	81	2

80	67	68	69	7	6	5	1	66
12	64	55	56	21	20	17	54	70
11	25						57	71
10	24						58	72
9	23						59	73
74	60						22	8
78	63						19	4
79	28	27	26	61	62	65	18	3
16	15	14	13	75	76	77	81	2

80	67	68	69	7	6	5	1	66
12	64	55	56	21	20	17	54	70
11	25	52	47	31	29	46	57	71
10	24	34	44	37	42	48	58	72
9	23	33	39	41	43	49	59	73
74	60	50	40	45	38	32	22	8
78	63	36	35	51	53	30	19	4
79	28	27	26	61	62	65	18	3
16	15	14	13	75	76	77	81	2

80	67	68	69	7	6	5	1	66
12	64	55	56	21	20	17	54	70
11	25	52	47	31	29	46	57	71
10	24	34	44	37	42	48	58	72
9	23	33	39	41	43	49	59	73
74	60	50	40	45	38	32	22	8
78	63	36	35	51	53	30	19	4
79	28	27	26	61	62	65	18	3
16	15	14	13	75	76	77	81	2

48	39	40	5	4	1	38
9	36	31	15	13	30	41
8	18	28	21	26	32	42
7	17	23	25	27	33	43
44	34	24	29	22	16	6
47	20	19	35	37	14	3
12	11	10	45	46	49	2

80	67	68	69	7	6	5	1	66
12				21	20	17		70
11	25							71
10	24							72
9	23							73
74							22	8
78							19	4
79	28	27	26				18	3
16	15	14	13	75	76	77	81	2

80	67	68	69	7	6	5	1	66
12	64	55	56	21	20	17	54	70
11	25						57	71
10	24						58	72
9	23						59	73
74	60						22	8
78	63						19	4
79	28	27	26	61	62	65	18	3
16	15	14	13	75	76	77	81	2

80	67	68	69	7	6	5	1	66
12	64	55	56	21	20	17	54	70
11	25	52	47	31	29	46	57	71
10	24	34	44	37	42	48	58	72
9	23	33	39	41	43	49	59	73
74	60	50	40	45	38	32	22	8
78	63	36	35	51	53	30	19	4
79	28	27	26	61	62	65	18	3
16	15	14	13	75	76	77	81	2

80	67	68	69	7	6	5	1	66
12	64	55	56	21	20	17	54	70
11	25	52	47	31	29	46	57	71
10	24	34	44	37	42	48	58	72
9	23	33	39	41	43	49	59	73
74	60	50	40	45	38	32	22	8
78	63	36	35	51	53	30	19	4
79	28	27	26	61	62	65	18	3
16	15	14	13	75	76	77	81	2

48	39	40	5	4	1	38
9	36	31	15	13	30	41
8	18	28	21	26	32	42
7	17	23	25	27	33	43
44	34	24	29	22	16	6
47	20	19	35	37	14	3
12	11	10	45	46	49	2

80	67	68	69	7	6	5	1	66
12				21	20	17		70
11	25							71
10	24							72
9	23							73
74							22	8
78							19	4
79	28	27	26				18	3
16	15	14	13	75	76	77	81	2

80	67	68	69	7	6	5	1	66
12	64	55	56	21	20	17	54	70
11	25						57	71
10	24						58	72
9	23						59	73
74	60						22	8
78	63						19	4
79	28	27	26	61	62	65	18	3
16	15	14	13	75	76	77	81	2

80	67	68	69	7	6	5	1	66
12	64	55	56	21	20	17	54	70
11	25	52	47	31	29	46	57	71
10	24	34	44	37	42	48	58	72
9	23	33	39	41	43	49	59	73
74	60	50	40	45	38	32	22	8
78	63	36	35	51	53	30	19	4
79	28	27	26	61	62	65	18	3
16	15	14	13	75	76	77	81	2

80	67	68	69	7	6	5	1	66
12	64	55	56	21	20	17	54	70
11	25	52	47	31	29	46	57	71
10	24	34	44	37	42	48	58	72
9	23	33	39	41	43	49	59	73
74	60	50	40	45	38	32	22	8
78	63	36	35	51	53	30	19	4
79	28	27	26	61	62	65	18	3
16	15	14	13	75	76	77	81	2

48	39	40	5	4	1	38
9	36	31	15	13	30	41
8	18	28	21	26	32	42
7	17	23	25	27	33	43
44	34	24	29	22	16	6
47	20	19	35	37	14	3
12	11	10	45	46	49	2