Magic Squares: bordered/concentric

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Multi-bordered magic squares

The following bordered magic square was created by Michael Stifel in 1544. A closer examination shows that it is not only bordered, but nested bordered or multi-bordered. You can also remove the border of the inner magic square of order 5 and will get another magic square of order 3. So continuous removing of the borders will always create new magic squares.

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

Another wonderful example of multi-concentric magic squares of order 12 is given by Kraitchik, which he shows in his book Mathematical Recreations on page 167. Four borders can be removed and each new inner square is magic again.

1	142	141	140	139	138	129	11	10	9	8	2
12	23	120	119	118	117	112	29	31	32	24	133
15	39	41	102	101	100	99	47	48	42	106	130
18	36	49	55	88	87	86	63	56	96	109	127
19	40	52	83	65	72	74	79	62	93	105	126
22	30	54	84	76	77	67	70	61	91	115	123
132	110	95	60	71	66	80	73	85	50	35	13
131	107	94	64	78	75	69	68	81	51	38	14
128	111	92	89	57	58	59	82	90	53	34	17
125	108	103	43	44	45	46	98	97	104	37	20
124	121	25	26	27	28	33	116	114	113	122	21
143	3	4	5	6	7	16	134	135	136	137	144

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

1	142	141	140	139	138	129	11	10	9	8	2
12	23	120	119	118	117	112	29	31	32	24	133
15	39	41	102	101	100	99	47	48	42	106	130
18	36	49	55	88	87	86	63	56	96	109	127
19	40	52	83	65	72	74	79	62	93	105	126
22	30	54	84	76	77	67	70	61	91	115	123
132	110	95	60	71	66	80	73	85	50	35	13
131	107	94	64	78	75	69	68	81	51	38	14
128	111	92	89	57	58	59	82	90	53	34	17
125	108	103	43	44	45	46	98	97	104	37	20
124	121	25	26	27	28	33	116	114	113	122	21
143	3	4	5	6	7	16	134	135	136	137	144

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

1	142	141	140	139	138	129	11	10	9	8	2
12	23	120	119	118	117	112	29	31	32	24	133
15	39	41	102	101	100	99	47	48	42	106	130
18	36	49	55	88	87	86	63	56	96	109	127
19	40	52	83	65	72	74	79	62	93	105	126
22	30	54	84	76	77	67	70	61	91	115	123
132	110	95	60	71	66	80	73	85	50	35	13
131	107	94	64	78	75	69	68	81	51	38	14
128	111	92	89	57	58	59	82	90	53	34	17
125	108	103	43	44	45	46	98	97	104	37	20
124	121	25	26	27	28	33	116	114	113	122	21
143	3	4	5	6	7	16	134	135	136	137	144

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

1	142	141	140	139	138	129	11	10	9	8	2
12	23	120	119	118	117	112	29	31	32	24	133
15	39	41	102	101	100	99	47	48	42	106	130
18	36	49	55	88	87	86	63	56	96	109	127
19	40	52	83	65	72	74	79	62	93	105	126
22	30	54	84	76	77	67	70	61	91	115	123
132	110	95	60	71	66	80	73	85	50	35	13
131	107	94	64	78	75	69	68	81	51	38	14
128	111	92	89	57	58	59	82	90	53	34	17
125	108	103	43	44	45	46	98	97	104	37	20
124	121	25	26	27	28	33	116	114	113	122	21
143	3	4	5	6	7	16	134	135	136	137	144