MULTIMAGIE.COM - Bimagic squares, 10th and 11th-order

Bimagic squares, 10th and 11th-order

Fredrik Jansson (Turku 1982 - )

Numerous 8th and 9th-order bimagic squares have been constructed since Pfeffermann, a century ago, but none 10th or 11th-order have been successfully constructed.

The first known 10th-order bimagic square was constructed in January 2004 by Fredrik Jansson, Finland. Fredrik is a young student, currently a second year physics student at the Åbo Akademi University, in Turku. He also studies mathematics and computer science.

First known bimagic square of order 10,
by Fredrik Jansson, January 2004
2	19	70	1	66	74	73	60	68	72
58	77	15	3	65	4	67	69	71	76
62	63	82	75	61	59	79	6	5	13
49	18	14	78	98	40	25	96	43	44
94	41	27	42	35	91	21	95	37	22
93	39	23	38	31	90	33	30	29	99
34	100	36	83	45	24	26	28	97	32
8	85	64	57	7	56	80	48	16	84
54	11	86	47	87	12	92	20	50	46
51	52	88	81	10	55	9	53	89	17

He used the fact that there are so many series of order 10, as mentioned in the old version of this page: if we list the series of 10 different integers, from 1 to 100, having for magic sums S1 = 505, and S2 = 33,835, we find 24,643,236 series. And if we limit ourselves to series having the number 100 among their 10 integers, there are still 2,240,776 series! Thinking of these enormous numbers, the probability was very high that 10th-order bimagic squares existed... but nobody had succeeded. Congratulations to Fredrik!

See the story of the first 10th-order bimagic square, text written by Fredrik Jansson.

Here are other bimagic squares of order 10, constructed later, in 2006 and 2007:

Bimagic square of order 10,
by Christian Boyer, October 2006
81	44	41	63	88	3	49	53	1	82
26	38	92	90	25	45	42	62	2	83
96	97	31	46	68	8	22	24	57	56
16	100	9	75	11	71	43	54	65	61
28	48	7	51	34	91	95	59	77	15
13	27	87	14	60	89	55	64	79	17
72	36	52	18	86	47	23	6	66	99
58	10	74	30	84	50	5	94	67	33
80	76	39	98	37	32	78	4	21	40
35	29	73	20	12	69	93	85	70	19

Bimagic square of order 10,
by Pan Fengchu, September 2007
89	51	52	88	53	55	10	9	17	81
59	82	62	61	13	6	79	75	63	5
1	2	66	68	72	74	70	73	60	19
42	41	27	22	91	21	35	37	95	94
57	80	64	8	16	85	56	48	7	84
54	20	86	92	11	50	12	87	46	47
78	98	14	40	43	18	44	96	25	49
3	65	4	77	71	58	76	15	67	69
39	30	33	23	90	38	99	31	93	29
83	36	97	26	45	100	24	34	32	28

G. Pfeffermann, again him, and again in Les Tablettes du Chercheur, had published 3 non-normal bimagic squares of order 10, in 1894 and 1896. Here is the last published, which uses non-consecutive integers between 1 and 137.

Non-normal bimagic square of order 10,
by G. Pfeffermann, March 1st 1896
24	133	108	51	129	19	71	42	13	100
130	28	35	134	72	52	14	111	94	20
1	7	84	113	106	40	37	64	117	121
107	65	2	18	23	99	128	122	85	41
83	112	116	29	15	135	95	6	36	63
75	102	132	43	3	123	109	22	26	55
97	53	16	10	39	115	120	136	73	31
17	21	74	101	98	32	25	54	131	137
118	44	27	124	86	66	4	103	110	8
38	125	96	67	119	9	87	30	5	114

11th-order bimagic squares

Also in January 2004, only 18 days later than his 10th-order bimagic square, Fredrik Jansson constructed the first 11th-order bimagic square, using similar methods (combining bimagic series).

First known bimagic square of order 11,
by Fredrik Jansson, January 2004
84	80	88	2	82	10	81	74	1	86	83
53	114	118	35	47	26	27	55	58	113	25
119	45	40	51	116	38	42	29	33	117	41
21	109	20	66	60	37	115	111	54	59	19
69	87	85	4	79	89	94	8	3	75	78
39	15	14	105	96	64	103	61	98	63	13
121	34	44	57	46	120	30	48	108	31	32
73	91	90	71	7	92	95	5	76	6	65
52	36	17	107	16	104	18	77	70	62	112
12	11	99	72	100	67	43	97	68	9	93
28	49	56	101	22	24	23	106	102	50	110

Another 11th-order bimagic square was constructed later by Chen Mutian, China, in May 2005. This square is symmetrical around its centre, meaning that two cells symmetrical (around the centre) have always the same sum, here 122. An interesting consequence is that the 4 lines of 11 numbers going through the centre are trimagic: the central row, the central column, and the 2 diagonals.

Symmetrical bimagic square of order 11,
by Chen Mutian, May 2005.
The two diagonals, the central row and the central column are trimagic.
9	19	65	30	72	76	106	121	93	47	33
101	97	88	20	56	4	27	74	70	108	26
86	87	51	109	112	41	12	54	3	37	79
84	11	58	94	8	91	40	78	120	24	63
6	104	23	115	22	105	69	60	55	73	39
15	32	117	80	45	61	77	42	5	90	107
83	49	67	62	53	17	100	7	99	18	116
59	98	2	44	82	31	114	28	64	111	38
43	85	119	68	110	81	10	13	71	35	36
96	14	52	48	95	118	66	102	34	25	21
89	75	29	1	16	46	50	92	57	103	113

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2	19	70	1	66	74	73	60	68	72
58	77	15	3	65	4	67	69	71	76
62	63	82	75	61	59	79	6	5	13
49	18	14	78	98	40	25	96	43	44
94	41	27	42	35	91	21	95	37	22
93	39	23	38	31	90	33	30	29	99
34	100	36	83	45	24	26	28	97	32
8	85	64	57	7	56	80	48	16	84
54	11	86	47	87	12	92	20	50	46
51	52	88	81	10	55	9	53	89	17

81	44	41	63	88	3	49	53	1	82
26	38	92	90	25	45	42	62	2	83
96	97	31	46	68	8	22	24	57	56
16	100	9	75	11	71	43	54	65	61
28	48	7	51	34	91	95	59	77	15
13	27	87	14	60	89	55	64	79	17
72	36	52	18	86	47	23	6	66	99
58	10	74	30	84	50	5	94	67	33
80	76	39	98	37	32	78	4	21	40
35	29	73	20	12	69	93	85	70	19

89	51	52	88	53	55	10	9	17	81
59	82	62	61	13	6	79	75	63	5
1	2	66	68	72	74	70	73	60	19
42	41	27	22	91	21	35	37	95	94
57	80	64	8	16	85	56	48	7	84
54	20	86	92	11	50	12	87	46	47
78	98	14	40	43	18	44	96	25	49
3	65	4	77	71	58	76	15	67	69
39	30	33	23	90	38	99	31	93	29
83	36	97	26	45	100	24	34	32	28

24	133	108	51	129	19	71	42	13	100
130	28	35	134	72	52	14	111	94	20
1	7	84	113	106	40	37	64	117	121
107	65	2	18	23	99	128	122	85	41
83	112	116	29	15	135	95	6	36	63
75	102	132	43	3	123	109	22	26	55
97	53	16	10	39	115	120	136	73	31
17	21	74	101	98	32	25	54	131	137
118	44	27	124	86	66	4	103	110	8
38	125	96	67	119	9	87	30	5	114

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

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

2	19	70	1	66	74	73	60	68	72
58	77	15	3	65	4	67	69	71	76
62	63	82	75	61	59	79	6	5	13
49	18	14	78	98	40	25	96	43	44
94	41	27	42	35	91	21	95	37	22
93	39	23	38	31	90	33	30	29	99
34	100	36	83	45	24	26	28	97	32
8	85	64	57	7	56	80	48	16	84
54	11	86	47	87	12	92	20	50	46
51	52	88	81	10	55	9	53	89	17

81	44	41	63	88	3	49	53	1	82
26	38	92	90	25	45	42	62	2	83
96	97	31	46	68	8	22	24	57	56
16	100	9	75	11	71	43	54	65	61
28	48	7	51	34	91	95	59	77	15
13	27	87	14	60	89	55	64	79	17
72	36	52	18	86	47	23	6	66	99
58	10	74	30	84	50	5	94	67	33
80	76	39	98	37	32	78	4	21	40
35	29	73	20	12	69	93	85	70	19

89	51	52	88	53	55	10	9	17	81
59	82	62	61	13	6	79	75	63	5
1	2	66	68	72	74	70	73	60	19
42	41	27	22	91	21	35	37	95	94
57	80	64	8	16	85	56	48	7	84
54	20	86	92	11	50	12	87	46	47
78	98	14	40	43	18	44	96	25	49
3	65	4	77	71	58	76	15	67	69
39	30	33	23	90	38	99	31	93	29
83	36	97	26	45	100	24	34	32	28

24	133	108	51	129	19	71	42	13	100
130	28	35	134	72	52	14	111	94	20
1	7	84	113	106	40	37	64	117	121
107	65	2	18	23	99	128	122	85	41
83	112	116	29	15	135	95	6	36	63
75	102	132	43	3	123	109	22	26	55
97	53	16	10	39	115	120	136	73	31
17	21	74	101	98	32	25	54	131	137
118	44	27	124	86	66	4	103	110	8
38	125	96	67	119	9	87	30	5	114

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

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

2	19	70	1	66	74	73	60	68	72
58	77	15	3	65	4	67	69	71	76
62	63	82	75	61	59	79	6	5	13
49	18	14	78	98	40	25	96	43	44
94	41	27	42	35	91	21	95	37	22
93	39	23	38	31	90	33	30	29	99
34	100	36	83	45	24	26	28	97	32
8	85	64	57	7	56	80	48	16	84
54	11	86	47	87	12	92	20	50	46
51	52	88	81	10	55	9	53	89	17

81	44	41	63	88	3	49	53	1	82
26	38	92	90	25	45	42	62	2	83
96	97	31	46	68	8	22	24	57	56
16	100	9	75	11	71	43	54	65	61
28	48	7	51	34	91	95	59	77	15
13	27	87	14	60	89	55	64	79	17
72	36	52	18	86	47	23	6	66	99
58	10	74	30	84	50	5	94	67	33
80	76	39	98	37	32	78	4	21	40
35	29	73	20	12	69	93	85	70	19

89	51	52	88	53	55	10	9	17	81
59	82	62	61	13	6	79	75	63	5
1	2	66	68	72	74	70	73	60	19
42	41	27	22	91	21	35	37	95	94
57	80	64	8	16	85	56	48	7	84
54	20	86	92	11	50	12	87	46	47
78	98	14	40	43	18	44	96	25	49
3	65	4	77	71	58	76	15	67	69
39	30	33	23	90	38	99	31	93	29
83	36	97	26	45	100	24	34	32	28

24	133	108	51	129	19	71	42	13	100
130	28	35	134	72	52	14	111	94	20
1	7	84	113	106	40	37	64	117	121
107	65	2	18	23	99	128	122	85	41
83	112	116	29	15	135	95	6	36	63
75	102	132	43	3	123	109	22	26	55
97	53	16	10	39	115	120	136	73	31
17	21	74	101	98	32	25	54	131	137
118	44	27	124	86	66	4	103	110	8
38	125	96	67	119	9	87	30	5	114

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

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