JavaScript-Error

Note: JavaScript is absolutely required for the functionality of these interactive pages and for the representation of all menus.

Please note also that you can deactivate JavaScript in general, but automatically activate it easily and without any risk for some special web pages like these. Here are the instructions how to enable JavaScript in your web browser

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

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

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

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

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

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

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