Bimagic and trimagic squares, from 12th to 16th-order

Table of first known bimagic and trimagic squares

Order	Bimagic	Trimagic
£ 7	Impossible (C. Boyer - W. Trump, 2002)	Impossible (C. Boyer - W. Trump, 2002)
8	G. Pfeffermann, France, 1890
9	G. Pfeffermann, France, 1891
10	Fredrik Jansson, Finland, 2004
11	Fredrik Jansson, Finland, 2004
12	Walter Trump, Germany, 2002
13	Chen Qinwu - Chen Mutian, China, 2006 (3 days later, indep., Jacques Guéron, France)	Unknown
14	Chen Qinwu - J. Guéron, China - France, 2006	Impossible
15	Chen Qinwu, China, 2006	Unknown
16	Gaston Tarry, France, 1900	Chen Mutian - Chen Qinwu, China, 2005
³ 17	See the page on 17th to 64th-order

12th-order bimagic and trimagic squares?

Yes, KNOWN!

The 12th-order trimagic square constructed in 2002 by Walter Trump, Germany, is of course also a bimagic square. No 12th-order bimagic square was known before.

It means that this Trump square is both:

• the first known 12th-order trimagic square
• AND the first known 12th-order bimagic square!

In September 2007, Pan Fengchu, China, constructed several 12th-order bimagic squares (but not trimagic). For example this one:

2007, September: 12th-order bimagic square,
by Pan Fengchu. S1 = 870, S2 = 83810

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

13th-order bimagic and trimagic squares?

Bimagic known. Trimagic unknown.

Chen Qinwu and Chen Mutian

2006, February 6th: the first known 13th-order bimagic square is due to Chen Qinwu and Chen Mutian, professors at the Computer Science Department of the Shantou University, Guangdong Province, China (later published in The American Mathematical Monthly, October 2007, page 702).

2006, Feb 6th: 13th-order bimagic square,
by Chen Qinwu and Chen Mutian. S1 = 1 105, S2 = 124 865.

126	49	10	60	149	140	90	8	50	123	86	157	57
143	103	117	4	105	166	41	128	39	5	78	75	101
35	33	73	94	87	37	9	139	136	168	146	51	97
107	141	76	40	55	12	148	98	129	7	162	72	58
138	121	142	79	13	102	74	6	108	69	2	106	145
167	144	68	104	23	31	135	19	119	77	59	127	32
36	122	34	30	130	47	85	96	1	99	115	154	156
14	3	80	95	150	91	116	153	124	133	66	62	18
82	25	53	89	67	113	165	84	20	26	155	161	65
52	125	114	151	24	92	11	134	132	137	70	48	15
42	100	158	38	160	164	93	45	44	63	111	16	71
43	21	163	152	81	64	29	83	147	110	27	54	131
120	118	17	169	61	46	109	112	56	88	28	22	159

Incredible: only 3 days later, and completely independently, Jacques Guéron, French teacher of mathematics, constructed another 13th-order bimagic square!

2006, Feb 9th: 13th-order bimagic square,
by Jacques Guéron. S1 = 1 105, S2 = 124 865.

109	62	75	149	121	137	43	163	101	7	14	36	88
71	93	54	39	80	155	46	13	105	119	141	169	20
134	95	6	151	66	83	167	32	110	52	55	128	26
96	162	61	42	143	35	126	72	87	9	144	112	16
85	44	150	130	139	157	59	11	17	33	97	116	67
78	48	158	27	2	56	123	89	145	104	22	138	115
64	156	84	77	41	100	159	122	8	114	34	15	131
4	53	70	50	153	91	30	103	135	160	124	108	24
40	148	166	98	60	18	37	111	10	86	129	69	133
99	28	125	90	47	1	117	146	73	132	21	65	161
3	29	79	76	147	127	49	23	165	63	136	106	102
168	45	19	12	81	94	118	113	57	152	68	38	140
154	142	58	164	25	51	31	107	92	74	120	5	82

In his Excel file, he explains his construction secrets.

• Download the 13th-order bimagic square of Jacques Guéron, with construction method (in French), Excel file 70Kb.

Then, one month later, Jacques Guéron constructed the first 17th and 19th-order bimagic squares...

Nobody has successfully constructed a 13th-order trimagic square. But Fredrik Jansson, Finland, the author of the first known 10th and 11th-order bimagic squares, has done some computing research, using a method I sent him. His best result in May 2006 was this square. It proves at least that it is possible to arrange the 169 numbers in 13 trimagic rows, which was not obvious before.

2006: 13th-order square by Fredrik Jansson,
13 trimagic rows, 9 trimagic columns.
S1 = 1 105,  S2 = 124 865,  S3 = 15 873 325.

1	2	71	78	129	83	69	166	167	61	81	86	111
90	30	70	122	5	76	164	134	114	18	54	74	154
25	95	125	162	97	147	7	46	127	37	49	58	130
113	103	105	38	55	9	123	144	143	34	35	44	159
118	138	150	68	72	104	128	60	10	21	28	50	158
31	153	39	117	107	155	77	14	51	19	88	108	146
169	41	3	4	168	101	87	92	99	59	84	89	109
80	165	56	36	140	6	94	48	100	16	96	116	152
145	73	43	124	75	163	23	8	45	40	112	121	133
57	115	27	26	67	47	161	132	65	11	126	135	136
52	98	160	110	32	42	66	102	20	12	120	142	149
139	63	119	156	17	93	15	53	131	24	62	82	151
85	29	137	64	141	79	91	106	33	13	22	148	157

Simply reordering the rows (first column becoming 85 90 145 118 80 31 139 52 25 57 1 169 113), Nicolas Rouanet remarked in March 2018 that the above Jansson's square has a trimagic diagonal! (85 30 43 68 140 155 15 102 127 11 81 89 159). And he constructed a better square with two more trimagic columns, and again one trimagic diagonal. This square is associative (every pair of numbers symmetrically opposite to the center sum up to the same value, here 170). Who will obtain a square with 13 trimagic columns? If his computing is correct, there is no 13th-order associative trimagic square.

2018: 13th-order square by Nicolas Rouanet,
13 trimagic rows, 11 trimagic columns, 1 trimagic diagonal.
S1 = 1 105,  S2 = 124 865,  S3 = 15 873 325.

117	38	130	114	20	70	168	104	120	30	26	32	136
112	68	155	29	159	33	77	139	121	27	81	8	96
106	6	153	65	9	51	47	133	79	98	167	69	122
28	52	43	157	99	1	73	147	41	109	151	80	124
82	126	113	5	35	39	125	163	55	95	87	156	24
4	94	60	78	152	42	158	86	36	149	116	108	22
160	154	59	145	63	67	85	103	107	25	111	16	10
148	62	54	21	134	84	12	128	18	92	110	76	166
146	14	83	75	115	7	45	131	135	165	57	44	88
46	90	19	61	129	23	97	169	71	13	127	118	142
48	101	3	72	91	37	123	119	161	105	17	164	64
74	162	89	143	49	31	93	137	11	141	15	102	58
34	138	144	140	50	66	2	100	150	56	40	132	53

14th-order bimagic and trimagic squares?

Bimagic known. Trimagic impossible.

G. Pfeffermann, France, has constructed (published in 1894 by Commandant Coccoz, AFAS) this 14th-order non-normal bimagic square, "non-normal" meaning using non-consecutive numbers.

1894: 14th-order non-normal bimagic square,
using non-consecutive integers from 1 to 225,
by G. Pfeffermann. S1 = 1 582, S2 = 242 046

163	166	182	7	25	101	147	139	95	21	9	194	180	153
37	220	206	132	88	61	47	59	75	78	124	200	216	39
207	133	76	62	52	40	221	215	36	54	74	90	123	199
77	67	55	41	222	208	121	135	198	214	35	51	69	89
56	42	223	196	122	82	70	66	84	134	210	213	34	50
211	197	127	85	71	57	43	33	49	65	81	129	209	225
130	86	72	58	31	212	202	204	224	45	48	64	80	126
100	146	162	178	181	2	22	24	14	195	168	154	140	96
1	17	97	145	161	177	193	183	169	155	141	99	29	15
176	192	13	16	92	142	160	156	144	104	30	3	184	170
137	157	175	191	12	28	91	105	18	4	185	171	159	149
27	103	136	152	172	190	11	5	186	174	164	150	93	19
187	10	26	102	148	151	167	179	165	138	94	20	6	189
73	46	32	217	205	131	87	79	125	201	219	44	60	63

In December 2005, Jacques Guéron, constructed this 14th-order nearly bimagic square (using consecutive integers): both diagonals are magic, but only one is bimagic.

2005: 14th-order nearly bimagic square,
using consecutive integers from 1 to 196,
by Jacques Guéron. S1 = 1 379, S2 = 180 649.
14 bimagic rows, 14 bimagic columns,
1 bimagic diagonal, the other one being only magic.

160	179	11	73	124	76	92	13	85	137	44	153	185	47
163	148	25	128	191	96	7	34	67	77	107	118	175	43
10	188	152	19	64	75	108	48	161	127	178	88	38	123
103	1	84	56	95	30	190	168	142	119	28	170	135	58
9	61	93	187	181	50	138	159	81	141	116	32	111	20
122	139	192	36	174	23	72	109	6	51	155	94	60	146
15	3	37	98	55	176	83	130	143	115	195	102	157	70
162	97	173	129	79	112	114	145	29	52	65	193	5	24
46	90	63	12	149	182	133	120	105	18	71	164	42	184
156	147	41	196	45	113	14	78	104	16	91	132	66	180
82	89	101	136	54	169	144	4	194	166	33	22	117	68
140	87	167	154	8	125	74	183	59	171	53	27	31	100
131	40	106	69	121	150	21	62	177	17	186	35	99	165
80	110	134	86	39	2	189	126	26	172	57	49	158	151

In January 2006, using the above square from this webpage, using its good diagonal and its 14 columns (for example, look at the same numbers used in the blue column), but reorganizing cells to get different rows, Chen Qinwu succeeded in constructing the first known 14th-order bimagic square. We can attribute this new square, and they agree, to both Chen Qinwu and Jacques Guéron!

2006: 14th-order bimagic square,
by Chen Qinwu and Jacques Guéron. S1 = 1 379, S2 = 180 649.

160	188	93	128	45	30	92	168	6	141	28	118	135	47
82	148	192	19	64	50	72	48	67	119	178	32	185	123
163	139	152	187	55	76	7	13	85	137	116	153	38	58
162	61	37	56	124	182	190	34	143	52	44	94	157	43
46	3	25	129	181	112	133	120	81	77	107	170	175	20
80	1	84	196	191	23	114	130	142	51	91	88	42	146
15	87	41	154	39	176	83	159	104	127	155	164	5	70
140	40	63	69	174	150	108	145	177	17	65	35	31	165
131	90	101	86	149	96	138	4	105	18	195	22	60	184
156	89	167	12	95	113	74	62	194	16	186	49	66	100
122	147	106	73	54	75	14	183	29	171	33	193	111	68
103	110	11	98	8	125	189	109	59	172	71	27	117	180
9	179	134	36	121	169	21	126	161	115	53	132	99	24
10	97	173	136	79	2	144	78	26	166	57	102	158	151

14th-order trimagic series cannot exist, implying that 14th-order trimagic squares are impossible. See the Word document written in 2002 by Walter Trump.

15th-order bimagic and trimagic squares?

Bimagic known. Trimagic unknown.

Gaston Tarry, France, published three different 15th-order "nearly" bimagic squares using consecutive numbers:

• in Nouvelles Annales de Mathématiques, 1900
• in Compte-Rendu de l'AFAS, 1903
• in Sphinx-Oedipe, 1912

His best one is the last one: the 15 rows are bimagic, the 15 columns are bimagic, but the 2 diagonals are "only" magic.

1912: 15th-order "nearly" bimagic square,
by Gaston Tarry. S1 = 1 695, S2 = 254 815.
The 2 diagonals are only magic.

174	150	1	22	73	109	160	206	212	83	59	35	126	102	183
65	111	162	198	219	90	46	37	133	94	190	176	137	8	29
221	77	53	44	125	96	192	168	144	15	16	67	118	154	205
127	103	184	175	146	2	23	74	110	156	207	213	84	60	31
138	9	30	61	112	163	199	220	86	47	38	134	95	186	177
119	155	201	222	78	54	45	121	97	193	169	145	11	17	68
85	56	32	128	104	185	171	147	3	24	75	106	157	208	214
91	187	178	139	10	26	62	113	164	200	216	87	48	39	135
12	18	69	120	151	202	223	79	55	41	122	98	194	170	141
158	209	215	81	57	33	129	105	181	172	148	4	25	71	107
49	40	131	92	188	179	140	6	27	63	114	165	196	217	88
195	166	142	13	19	70	116	152	203	224	80	51	42	123	99
21	72	108	159	210	211	82	58	34	130	101	182	173	149	5
197	218	89	50	36	132	93	189	180	136	7	28	64	115	161
43	124	100	191	167	143	14	20	66	117	153	204	225	76	52

In 2003, I was pleased to improve his square and to obtain the following Tarry-Boyer square: one diagonal is magic, but the other one is now bimagic.

2003: 15th-order "VERY nearly" bimagic square,
by Gaston Tarry and Christian Boyer.
S1 = 1 695, S2 = 254 815.
15 bimagic rows, 15 bimagic columns,
1 bimagic diagonal, the other one being only magic

73	35	102	160	174	1	83	206	109	126	183	212	150	22	59
219	176	8	46	65	162	94	37	90	137	29	133	111	198	190
125	67	154	192	221	53	15	168	96	118	205	144	77	44	16
146	213	60	23	127	184	156	74	2	84	31	110	103	175	207
112	134	186	199	138	30	47	220	163	95	177	86	9	61	38
78	145	17	45	119	201	193	121	54	11	68	97	155	222	169
104	106	208	171	85	32	24	147	185	157	214	3	56	128	75
10	87	39	62	91	178	200	113	26	48	135	164	187	139	216
151	98	170	223	12	69	41	79	202	194	141	55	18	120	122
57	4	71	129	158	215	172	105	33	25	107	181	209	81	148
188	165	217	140	49	131	63	6	179	196	88	27	40	92	114
19	51	123	116	195	142	224	152	70	42	99	203	166	13	80
210	182	149	82	21	108	130	58	211	173	5	34	72	159	101
36	28	115	93	197	89	136	189	132	64	161	180	218	50	7
167	204	76	14	43	100	117	20	143	225	52	66	124	191	153

And in January 2006, Chen Qinwu constructed the first known 15th-order bimagic square.

2006: 15th-order bimagic square, by Chen Qinwu.
S1 = 1 695, S2 = 254 815.

160	28	194	146	141	58	102	217	16	101	43	31	187	92	179
169	63	214	129	132	38	144	93	216	84	23	48	83	44	215
185	46	106	107	99	18	52	164	53	126	177	189	220	5	148
202	204	155	59	65	1	86	150	15	67	61	156	170	114	190
224	8	172	96	40	75	111	200	191	25	60	68	121	147	157
135	45	88	87	77	212	122	184	73	117	3	13	209	131	199
98	162	205	4	116	176	19	175	33	137	197	30	154	81	108
136	171	123	20	7	192	74	113	152	34	219	206	103	55	90
118	145	72	196	29	89	193	51	207	50	110	222	21	64	128
27	95	17	213	223	109	153	42	104	14	149	139	138	181	91
69	79	105	158	166	201	35	26	115	151	186	130	54	218	2
36	112	56	70	165	159	211	76	140	225	161	167	71	22	24
78	221	6	37	49	100	173	62	174	208	127	119	120	180	41
11	182	143	178	203	142	10	133	82	188	94	97	12	163	57
47	134	39	195	183	125	210	9	124	168	85	80	32	198	66

16th-order bimagic and trimagic squares?

Yes, KNOWN!

The first known 16th-order bimagic square seems to be the following one, published by Gaston Tarry, France, in the Revue Scientifique, 1903. And it is a partial trimagic square: the 16 rows are trimagic.

1903: 16th-order bimagic square, by Gaston Tarry.
S1 = 2 056, S2 = 351 576.
The 16 columns and 2 diagonals are bimagic,
the 16 rows are trimagic with S3 = 67 634 176.

1	52	86	103	16	61	91	106	241	196	166	151	256	205	171	154
102	87	49	4	107	90	64	13	150	167	193	244	155	170	208	253
55	6	100	81	58	11	109	96	199	246	148	161	202	251	157	176
84	97	7	54	93	112	10	59	164	145	247	198	173	160	250	203
249	204	174	159	248	197	163	146	9	60	94	111	8	53	83	98
158	175	201	252	147	162	200	245	110	95	57	12	99	82	56	5
207	254	156	169	194	243	149	168	63	14	108	89	50	3	101	88
172	153	255	206	165	152	242	195	92	105	15	62	85	104	2	51
128	77	43	26	113	68	38	23	144	189	219	234	129	180	214	231
27	42	80	125	22	39	65	116	235	218	192	141	230	215	177	132
74	123	29	48	71	118	20	33	186	139	237	224	183	134	228	209
45	32	122	75	36	17	119	70	221	240	138	187	212	225	135	182
136	181	211	226	137	188	222	239	120	69	35	18	121	76	46	31
227	210	184	133	238	223	185	140	19	34	72	117	30	47	73	124
178	131	229	216	191	142	236	217	66	115	21	40	79	126	28	41
213	232	130	179	220	233	143	190	37	24	114	67	44	25	127	78

David Collison, USA, in 1991, and Jacques Guéron, France, in 2002, have succeeded in constructing 16th-order non-normal trimagic squares.

The first known 16th-order normal trimagic square is due to Chen Mutian and Chen Qinwu. The square was found in May 2005. It is "symmetrical" left/right: the i^th element of a row + the (17-i)^th element of this same row = 16² + 1 = 257. In March 2008, Li Wen constructed another 16th-order trimagic square.

2005: 16th-order trimagic square, by Chen Mutian and Chen Qinwu.
S1 = 2 056, S2 = 351 576, S3 = 67 634 176.

34	30	28	26	146	83	85	115	142	172	174	111	231	229	227	223
52	40	124	64	234	110	207	219	38	50	147	23	193	133	217	205
178	168	226	212	169	245	151	42	215	106	12	88	45	31	89	79
125	201	5	249	112	91	49	103	154	208	166	145	8	252	56	132
196	180	176	232	199	59	96	241	16	161	198	58	25	81	77	61
62	78	82	118	247	214	114	15	242	143	43	10	139	175	179	195
203	253	107	127	97	44	13	102	155	244	213	160	130	150	4	54
119	55	71	189	210	236	20	164	93	237	21	47	68	186	202	138
255	99	185	67	66	76	238	94	163	19	181	191	190	72	158	2
137	157	251	129	24	182	171	18	239	86	75	233	128	6	100	120
131	135	183	187	9	173	36	240	17	221	84	248	70	74	122	126
53	3	149	69	192	148	243	156	101	14	109	65	188	108	254	204
224	228	230	140	159	197	144	37	220	113	60	98	117	27	29	33
1	121	73	7	48	165	162	153	104	95	92	209	250	184	136	256
80	90	32	46	87	11	105	216	41	152	246	170	211	225	167	177
206	218	134	194	57	22	222	141	116	35	235	200	63	123	39	51

• Download the 16th-order trimagic square of Chen Mutian and Chen Qinwu, Excel file 44Kb.
• Download the story of this 16th-order trimagic square, written by Chen Qinwu, Word file 132Kb.
• Download the 16th-order trimagic square of Li Wen, Excel file 44Kb.

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