Premier carré bimagique d

Dear friends,

Thanks for the 20th-order non-normal pandiagonal bimagic square, very nice.

On this subject, two interesting questions:

-is it possible to construct a non-normal pandiagonal bimagic square of an order smaller than 18?

-is it possible to construct a normal pandiagonal bimagic square of an order smaller than 32?

About your other square, I had problems to understand what you meant by “Dual magic square”.

After the analysis of your file, I call such a square an “additive-multiplicative magic square”.

Did you see my best 8^th and 9^th-order additive-multiplicative magic squares at www.multimagie.com/English/Multiplicative8_9.htm?

Is it possible to construct better 8^th and 9^th-order additive-multiplicative magic squares? (better = with smaller sums, smaller products, or smaller max numbers)

And is it possible to construct 5^th, 6^th or 7^th-order additive-multiplicative magic squares?

In my www.multimagie.com/English/Multiplicative6_7.htm page, I have 6^th and 7^th-order multiplicative square with some additive-multiplicative properties: all their rows are additive-multiplicative.

Attached, you will find the 18^th-order bimagic square by Jacques Guéron.

Best regards.

Christian Boyer.

De : gzy [mailto:gy1397@163.com]
Envoyé : jeudi 18 mai 2006 06:15
À : Christian Boyer
Cc : '李文'; 'QinWuChen'; panfch@126.com
Objet : 20th-order Dual magic square and 20th-order pandiagonal bimagic square

Dear friends,

Concerning Who was the first, we think consistently that Assurance it, should with announce of the day.Perhaps thare are his ories on Very early, but real magic square result is distinguishing with theories.

To Your latest news: I will add the first 18 th-order bimagic square constructed by Jacques Gu é ron, received today!

We are very happy, because it is an important result.We say to Jacques Gu é ron heartfelt congratulation!But we haven't see his 18^th bimagic square, my website also wants to announce this importance achievement, hoping you send his the square, letting the joy that we share him as early as possible.

I send 20th-order pandiagonal bimagic square and 20th-order Dual magic square by Su Maoting, China 2006 /2 /8.

I agree the first authors mentioned in the table of your site will be:

48 Pan Fengchu, 2004

50 Gao Zhiyuan, 2006

55 Li Wen, 2006

56 Gao Zhiyuan, 2006

63 Pan Fengchu, 2006

Best regards. Gao Zhiyuan 2006/5/18

0	308	37	71	73	107	109	143	145	177	266	233	202	188	290	240	283	35
34	151	282	267	235	220	215	182	311	142	1	110	106	36	70	74	292	179
278	33	38	69	72	105	111	165	2	214	141	307	257	249	232	146	197	291
68	289	274	173	246	231	108	310	75	104	150	3	39	210	269	186	32	140
4	31	40	112	250	149	139	313	201	191	229	67	103	76	288	268	171	275
102	295	178	66	309	251	213	77	137	144	30	113	230	181	5	263	41	272
204	42	176	29	136	78	6	153	265	227	180	114	276	101	312	65	247	296
134	203	100	252	228	316	43	79	279	64	196	237	28	168	156	302	115	7
293	148	172	248	63	8	261	44	212	80	194	27	284	116	99	127	306	225
122	163	270	160	193	200	81	62	26	300	244	262	45	320	223	129	9	98
158	205	303	183	222	259	318	135	97	25	46	281	167	121	61	10	234	82
271	159	297	322	11	216	211	264	130	60	184	170	24	47	83	124	92	242
217	258	131	96	23	117	174	323	58	88	304	198	238	273	154	48	12	195
253	49	95	20	120	294	317	13	162	286	219	87	192	245	56	133	157	209
155	123	93	298	175	14	52	126	277	189	239	86	254	55	224	321	207	19
152	206	243	190	51	118	256	169	84	314	22	305	90	15	132	280	221	59
161	18	128	260	185	16	57	94	299	285	241	218	85	319	199	53	125	164
301	226	50	91	315	208	236	255	147	17	21	89	287	187	54	138	166	119

0	308	37	71	73	107	109	143	145	177	266	233	202	188	290	240	283	35
34	151	282	267	235	220	215	182	311	142	1	110	106	36	70	74	292	179
278	33	38	69	72	105	111	165	2	214	141	307	257	249	232	146	197	291
68	289	274	173	246	231	108	310	75	104	150	3	39	210	269	186	32	140
4	31	40	112	250	149	139	313	201	191	229	67	103	76	288	268	171	275
102	295	178	66	309	251	213	77	137	144	30	113	230	181	5	263	41	272
204	42	176	29	136	78	6	153	265	227	180	114	276	101	312	65	247	296
134	203	100	252	228	316	43	79	279	64	196	237	28	168	156	302	115	7
293	148	172	248	63	8	261	44	212	80	194	27	284	116	99	127	306	225
122	163	270	160	193	200	81	62	26	300	244	262	45	320	223	129	9	98
158	205	303	183	222	259	318	135	97	25	46	281	167	121	61	10	234	82
271	159	297	322	11	216	211	264	130	60	184	170	24	47	83	124	92	242
217	258	131	96	23	117	174	323	58	88	304	198	238	273	154	48	12	195
253	49	95	20	120	294	317	13	162	286	219	87	192	245	56	133	157	209
155	123	93	298	175	14	52	126	277	189	239	86	254	55	224	321	207	19
152	206	243	190	51	118	256	169	84	314	22	305	90	15	132	280	221	59
161	18	128	260	185	16	57	94	299	285	241	218	85	319	199	53	125	164
301	226	50	91	315	208	236	255	147	17	21	89	287	187	54	138	166	119

0	308	37	71	73	107	109	143	145	177	266	233	202	188	290	240	283	35
34	151	282	267	235	220	215	182	311	142	1	110	106	36	70	74	292	179
278	33	38	69	72	105	111	165	2	214	141	307	257	249	232	146	197	291
68	289	274	173	246	231	108	310	75	104	150	3	39	210	269	186	32	140
4	31	40	112	250	149	139	313	201	191	229	67	103	76	288	268	171	275
102	295	178	66	309	251	213	77	137	144	30	113	230	181	5	263	41	272
204	42	176	29	136	78	6	153	265	227	180	114	276	101	312	65	247	296
134	203	100	252	228	316	43	79	279	64	196	237	28	168	156	302	115	7
293	148	172	248	63	8	261	44	212	80	194	27	284	116	99	127	306	225
122	163	270	160	193	200	81	62	26	300	244	262	45	320	223	129	9	98
158	205	303	183	222	259	318	135	97	25	46	281	167	121	61	10	234	82
271	159	297	322	11	216	211	264	130	60	184	170	24	47	83	124	92	242
217	258	131	96	23	117	174	323	58	88	304	198	238	273	154	48	12	195
253	49	95	20	120	294	317	13	162	286	219	87	192	245	56	133	157	209
155	123	93	298	175	14	52	126	277	189	239	86	254	55	224	321	207	19
152	206	243	190	51	118	256	169	84	314	22	305	90	15	132	280	221	59
161	18	128	260	185	16	57	94	299	285	241	218	85	319	199	53	125	164
301	226	50	91	315	208	236	255	147	17	21	89	287	187	54	138	166	119

0	308	37	71	73	107	109	143	145	177	266	233	202	188	290	240	283	35
34	151	282	267	235	220	215	182	311	142	1	110	106	36	70	74	292	179
278	33	38	69	72	105	111	165	2	214	141	307	257	249	232	146	197	291
68	289	274	173	246	231	108	310	75	104	150	3	39	210	269	186	32	140
4	31	40	112	250	149	139	313	201	191	229	67	103	76	288	268	171	275
102	295	178	66	309	251	213	77	137	144	30	113	230	181	5	263	41	272
204	42	176	29	136	78	6	153	265	227	180	114	276	101	312	65	247	296
134	203	100	252	228	316	43	79	279	64	196	237	28	168	156	302	115	7
293	148	172	248	63	8	261	44	212	80	194	27	284	116	99	127	306	225
122	163	270	160	193	200	81	62	26	300	244	262	45	320	223	129	9	98
158	205	303	183	222	259	318	135	97	25	46	281	167	121	61	10	234	82
271	159	297	322	11	216	211	264	130	60	184	170	24	47	83	124	92	242
217	258	131	96	23	117	174	323	58	88	304	198	238	273	154	48	12	195
253	49	95	20	120	294	317	13	162	286	219	87	192	245	56	133	157	209
155	123	93	298	175	14	52	126	277	189	239	86	254	55	224	321	207	19
152	206	243	190	51	118	256	169	84	314	22	305	90	15	132	280	221	59
161	18	128	260	185	16	57	94	299	285	241	218	85	319	199	53	125	164
301	226	50	91	315	208	236	255	147	17	21	89	287	187	54	138	166	119