Bordered and concentric magic squares

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Bordered magic squares

A bordered magic square is a magic square, which remains magic when its borders are removed. Let us take the depicted magic square of order 8. When we remove its border, we will get a square of order 6.

64	4	9	54	63	3	10	53
60	15	16	47	48	49	20	5
7	44	22	42	41	25	21	58
51	33	37	29	30	28	38	14
6	32	34	35	36	31	27	59
8	26	40	24	23	43	39	57
52	45	46	18	17	19	50	13
12	61	56	11	2	62	55	1

15	16	47	48	49	20
44	22	42	41	25	21
33	37	29	30	28	38
32	34	35	36	31	27
26	40	24	23	43	39
45	46	18	17	19	50

This square isn't normalized anymore, because it doesn't contain the numbers 1, 2, … , n² anymore. But still each row, each column and both diagonals sum S=195.

A second condition for bordered magic squares demands that the numbers of the border enclose the numbers of the inner square. Satisfiying this condition means that the numbers 1, … ,2(n−1) and n²−2(n−1) + 1 , … , n² must form the border. All other numbers must be elements of the inner square.

Let's take a bordered magic square of order n=6 as an example. The inner square is of order n=4, and must be formed of 4²=16 numbers. On the other side, there are

6² − 4² = 36 − 16 = 20

numbers for the border, which have to enclose the inner elements. So, ten numbers are less and ten numbers are greater than the inner numbers.

lower numbers of the border:	1 … 10
inner numbers:	11 … 26
upper numbers of the border:	27 … 36

One example of such an arrangement is shown in the following magic square of order 6:

36	2	3	7	32	31
4	26	13	12	23	33
9	15	20	21	18	28
27	19	16	17	22	10
29	14	25	24	11	8
6	35	34	30	5	1

36	2	3	7	32	31
4	26	13	12	23	33
9	15	20	21	18	28
27	19	16	17	22	10
29	14	25	24	11	8
6	35	34	30	5	1

36	2	3	7	32	31
4	26	13	12	23	33
9	15	20	21	18	28
27	19	16	17	22	10
29	14	25	24	11	8
6	35	34	30	5	1

36	2	3	7	32	31
4	26	13	12	23	33
9	15	20	21	18	28
27	19	16	17	22	10
29	14	25	24	11	8
6	35	34	30	5	1