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

Composite magic squares

A magic square of order m · n is called composite, when it can be decomposed into m² magic subsquares, each of order n.

It is to be noted that the minimal composite magic square must be of order n=9. This is the minimal number with two divisors, for which magic squares exist.

This means that the fundamental magic square can be decomposed into m²=3²=9 magic subsquares, each of them with order n=. Of course it is impossible that all subsquares are normalized, i.e. are composed of consecutive integers 1, 2, … , n².

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

The next composite magic square is found for order m · n=12. This order can be divided in the following manner:

This means that the composite square can be decomposed in m²=3²=9 magic subsquares of order n=4.

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

But, also another decomposition is possible.

This composite magic square is decomposed into m²=4²=16 magic subsquares, each of order n=3.

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

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

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

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

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

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

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

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

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

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