MULTIMAGIE.COM - Multiplicative magic squares, 10th-order and +

Smallest multiplicative magic squares, 10th-order and above

After multiplicative magic squares of orders 3-4-5, 6-7, 8-9, here are my results on orders from 10 to 17.

My best 10x10 examples, "best" meaning with smallest product and/or smallest maximum number, are:

2005: 10x10 multiplicative magic square,
**P = 277 563 225 139 200 000**, Max nb = 375
1	345	80	56	38	110	78	204	75	36
160	14	285	45	150	92	12	11	104	51
35	152	27	300	64	34	230	6	165	13
200	17	130	44	3	108	95	105	32	138
66	4	184	170	65	21	240	50	9	228
156	55	2	69	255	96	25	72	76	70
57	54	100	16	84	195	136	115	10	22
68	39	132	15	23	125	18	190	42	128
135	250	102	26	88	19	28	48	276	5
46	192	7	114	90	8	33	52	85	375

2005: 10x10 multiplicative magic square,
P = 939 088 911 720 960 000, **Max nb = 290**
192	133	26	100	153	5	33	84	230	29
65	225	119	58	69	19	144	8	44	140
50	68	290	138	13	126	95	72	7	88
92	112	55	1	240	174	34	175	117	57
3	24	76	70	22	250	91	207	232	102
66	2	216	190	98	39	184	116	17	125
170	87	23	104	150	77	56	38	120	9
14	110	6	168	152	46	261	85	75	52
203	115	42	99	4	136	25	130	114	48
171	78	200	51	145	96	10	11	28	161

It should be possible to construct better 10x10 squares. Luke Pebody, England, also studied 10x10 multiplicative magic squares: he thinks that the smallest possible P is 43,716,207,959,424,000 (more than 6 times smaller than my best P above), but has not yet succeeded to construct such a square.

Pandiagonal additive magic squares (using consecutive integers) of order 10 are impossible, and more generally all 4k+2 orders. But pandiagonal multiplicative magic squares of order 10 are possible. This example is also a most-perfect square, because all its 2x2 subsquares (as for example the blue one) have the same product P'. And all its 5x5 subsquares (as for example the green one) have the same product P''.

2006: 10x10 pandiagonal multiplicative magic square,
P = 1 667 988 097 820 100 000 000 000 000 000 000, Max nb = 441 000
441	80	11025	2000	39375	784	45	19600	1125	70000
1250	441000	50	17640	14	45000	12250	1800	490	504
1764	20	44100	500	157500	196	180	4900	4500	17500
5000	110250	200	4410	56	11250	49000	450	1960	126
2352	15	58800	375	210000	147	240	3675	6000	13125
5625	98000	225	3920	63	10000	55125	400	2205	112
98	360	2450	9000	8750	3528	10	88200	250	315000
22500	24500	900	980	252	2500	220500	100	8820	28
392	90	9800	2250	35000	882	40	22050	1000	78750
30000	18375	1200	735	336	1875	294000	75	11760	21

With Jaroslaw Wroblewski, Poland, I constructed in May-June 2007 two better pandiagonal squares of order 10 having smaller P and Max nb than the previous one. However, they are no longer most-perfect squares. Here they are:

2007: 10x10 pandiagonal multiplicative magic square,
**P = 26 480 706 717 104 640 000**, Max nb = 1 617,
by J. Wroblewski and C. Boyer.
1	196	51	80	156	9	1225	187	528	195
66	510	26	56	21	10	408	234	350	77
325	11	1617	255	16	52	3	245	204	144
84	18	850	286	462	105	2	136	78	70
48	65	12	441	425	176	429	15	49	68
14	28	6	170	312	126	175	22	1122	390
561	240	13	4	147	85	192	117	25	539
650	154	231	30	34	104	42	35	24	306
588	153	400	143	33	735	17	64	39	5
102	130	168	63	50	374	858	210	7	8

2007: 10x10 pandiagonal multiplicative magic square,
P = 393 968 065 240 189 440 000,**Max nb = 850**,
by J. Wroblewski and C. Boyer.
1	102	165	182	270	8	714	297	624	450
96	675	22	51	65	28	405	176	357	117
378	9	816	825	26	54	5	238	495	208
195	32	567	198	408	325	4	81	110	119
130	126	15	272	693	234	432	25	34	99
17	39	20	189	330	136	273	36	648	550
792	650	18	3	170	231	390	144	21	306
462	153	312	100	27	66	85	91	60	216
510	264	546	162	24	850	33	78	90	7
135	154	255	104	84	243	528	425	13	12

I also constructed 11x11 pandiagonal multiplicative magic squares. I think that it will not be possible to construct better 11x11 pandiagonal squares.

2005: 11x11 pandiagonal multiplicative magic square,
**P = 160 986 670 580 736 000 000**, Max nb = 580
260	160	85	38	345	200	116	1	84	54	33
14	135	88	52	16	204	114	69	500	290	5
100	29	12	42	27	220	130	80	34	285	184
102	57	460	250	145	2	105	72	44	13	192
110	65	32	255	152	92	25	348	6	21	180
15	56	36	11	156	96	51	380	230	125	58
23	300	174	3	140	90	55	26	240	136	76
48	340	190	115	50	435	8	28	9	132	78
45	22	195	128	68	19	276	150	87	20	70
232	4	7	108	66	39	320	170	95	46	375
228	138	75	580	10	35	18	165	104	64	17

2005: 11x11 pandiagonal multiplicative magic square,
P = 1 441 031 935 035 813 120 000, **Max nb = 341**
104	85	38	220	161	100	27	252	174	93	11
290	217	4	13	153	114	66	253	200	135	56
25	243	168	87	341	8	65	34	190	154	92
57	242	184	125	54	280	203	124	1	117	102
5	26	170	133	88	23	225	162	84	319	248
196	116	31	9	78	51	209	176	115	50	270
207	150	81	308	232	155	2	130	119	76	22
187	152	110	46	250	189	112	29	279	6	39
62	10	91	68	19	198	138	75	297	224	145
108	28	261	186	3	143	136	95	44	230	175
132	69	275	216	140	58	310	7	52	17	171

I also constructed examples from 12x12 to 17x17. It would be boring to directly display all of them in this page. But you can get them, and all the other squares of this page:

Download my multiplicative magic squares from order 10 to order 17 (Excel file of 57Kb),
note that the add-mult magic squares of orders 11, 12, 13, 14, 15, 17 constructed by Toshihiro Shirakawa in 2010 and 2013 have smaller P and Max Nb than my squares
Download the best known examples of pandiagonal multiplicative magic squares from order 3 to order 17 (zipped Excel file of 108Kb)

The pandiagonal results are summarized in this table. One of the two biggest squares of these Excel files:

2005: 17x17 pandiagonal multiplicative magic square,
**P = 136 260 600 278 658 342 262 589 318 529 024 000 000**, Max nb = 1272
375	235	306	222	220	189	1029	152	984	117	29	530	46	516	48	434	4
686	76	615	65	522	318	460	301	336	248	24	225	47	170	74	132	81
276	129	224	124	15	125	846	102	740	77	567	392	456	369	13	290	106
470	34	444	33	378	196	285	205	234	174	1060	161	903	128	744	9	25
171	41	130	58	636	69	602	64	465	5	450	282	340	259	231	216	1176
344	384	279	1	250	94	204	111	154	108	735	95	738	78	580	371	483
119	777	88	648	441	19	410	26	348	159	322	172	240	155	18	150	940
246	260	203	1113	184	1032	144	31	10	50	564	51	518	44	405	245	342
80	558	6	500	329	357	296	264	243	49	190	82	156	87	742	92	645
148	165	135	882	114	820	91	609	424	552	387	16	310	2	300	141	238
39	406	212	345	215	288	186	20	175	987	136	888	99	27	490	38	492
62	12	75	658	68	555	55	486	294	380	287	273	232	1272	207	43	160
11	270	98	228	123	182	116	795	115	774	96	620	7	525	376	408	333
696	477	23	430	32	372	3	350	188	255	185	198	162	980	133	861	104
21	200	1128	153	37	110	54	588	57	574	52	435	265	414	258	320	217
540	343	399	328	312	261	53	230	86	192	93	14	100	705	85	666	66
954	138	860	112	651	8	600	423	17	370	22	324	147	266	164	195	145

In the files, you will find another 17x17 pandiagonal multiplicative magic square, with a bigger P, but with a smaller max nb = 1003.

Many thanks to Edwin Clark, Don Reble, and Günter Stertenbrink for checking these big multiplicative squares, confirming that they have all the announced properties... and that all their integers are really distinct!

Return to the home page http://www.multimagie.com

1	345	80	56	38	110	78	204	75	36
160	14	285	45	150	92	12	11	104	51
35	152	27	300	64	34	230	6	165	13
200	17	130	44	3	108	95	105	32	138
66	4	184	170	65	21	240	50	9	228
156	55	2	69	255	96	25	72	76	70
57	54	100	16	84	195	136	115	10	22
68	39	132	15	23	125	18	190	42	128
135	250	102	26	88	19	28	48	276	5
46	192	7	114	90	8	33	52	85	375

192	133	26	100	153	5	33	84	230	29
65	225	119	58	69	19	144	8	44	140
50	68	290	138	13	126	95	72	7	88
92	112	55	1	240	174	34	175	117	57
3	24	76	70	22	250	91	207	232	102
66	2	216	190	98	39	184	116	17	125
170	87	23	104	150	77	56	38	120	9
14	110	6	168	152	46	261	85	75	52
203	115	42	99	4	136	25	130	114	48
171	78	200	51	145	96	10	11	28	161

1	102	165	182	270	8	714	297	624	450
96	675	22	51	65	28	405	176	357	117
378	9	816	825	26	54	5	238	495	208
195	32	567	198	408	325	4	81	110	119
130	126	15	272	693	234	432	25	34	99
17	39	20	189	330	136	273	36	648	550
792	650	18	3	170	231	390	144	21	306
462	153	312	100	27	66	85	91	60	216
510	264	546	162	24	850	33	78	90	7
135	154	255	104	84	243	528	425	13	12

260	160	85	38	345	200	116	1	84	54	33
14	135	88	52	16	204	114	69	500	290	5
100	29	12	42	27	220	130	80	34	285	184
102	57	460	250	145	2	105	72	44	13	192
110	65	32	255	152	92	25	348	6	21	180
15	56	36	11	156	96	51	380	230	125	58
23	300	174	3	140	90	55	26	240	136	76
48	340	190	115	50	435	8	28	9	132	78
45	22	195	128	68	19	276	150	87	20	70
232	4	7	108	66	39	320	170	95	46	375
228	138	75	580	10	35	18	165	104	64	17

104	85	38	220	161	100	27	252	174	93	11
290	217	4	13	153	114	66	253	200	135	56
25	243	168	87	341	8	65	34	190	154	92
57	242	184	125	54	280	203	124	1	117	102
5	26	170	133	88	23	225	162	84	319	248
196	116	31	9	78	51	209	176	115	50	270
207	150	81	308	232	155	2	130	119	76	22
187	152	110	46	250	189	112	29	279	6	39
62	10	91	68	19	198	138	75	297	224	145
108	28	261	186	3	143	136	95	44	230	175
132	69	275	216	140	58	310	7	52	17	171

1	345	80	56	38	110	78	204	75	36
160	14	285	45	150	92	12	11	104	51
35	152	27	300	64	34	230	6	165	13
200	17	130	44	3	108	95	105	32	138
66	4	184	170	65	21	240	50	9	228
156	55	2	69	255	96	25	72	76	70
57	54	100	16	84	195	136	115	10	22
68	39	132	15	23	125	18	190	42	128
135	250	102	26	88	19	28	48	276	5
46	192	7	114	90	8	33	52	85	375

192	133	26	100	153	5	33	84	230	29
65	225	119	58	69	19	144	8	44	140
50	68	290	138	13	126	95	72	7	88
92	112	55	1	240	174	34	175	117	57
3	24	76	70	22	250	91	207	232	102
66	2	216	190	98	39	184	116	17	125
170	87	23	104	150	77	56	38	120	9
14	110	6	168	152	46	261	85	75	52
203	115	42	99	4	136	25	130	114	48
171	78	200	51	145	96	10	11	28	161

1	102	165	182	270	8	714	297	624	450
96	675	22	51	65	28	405	176	357	117
378	9	816	825	26	54	5	238	495	208
195	32	567	198	408	325	4	81	110	119
130	126	15	272	693	234	432	25	34	99
17	39	20	189	330	136	273	36	648	550
792	650	18	3	170	231	390	144	21	306
462	153	312	100	27	66	85	91	60	216
510	264	546	162	24	850	33	78	90	7
135	154	255	104	84	243	528	425	13	12

260	160	85	38	345	200	116	1	84	54	33
14	135	88	52	16	204	114	69	500	290	5
100	29	12	42	27	220	130	80	34	285	184
102	57	460	250	145	2	105	72	44	13	192
110	65	32	255	152	92	25	348	6	21	180
15	56	36	11	156	96	51	380	230	125	58
23	300	174	3	140	90	55	26	240	136	76
48	340	190	115	50	435	8	28	9	132	78
45	22	195	128	68	19	276	150	87	20	70
232	4	7	108	66	39	320	170	95	46	375
228	138	75	580	10	35	18	165	104	64	17

104	85	38	220	161	100	27	252	174	93	11
290	217	4	13	153	114	66	253	200	135	56
25	243	168	87	341	8	65	34	190	154	92
57	242	184	125	54	280	203	124	1	117	102
5	26	170	133	88	23	225	162	84	319	248
196	116	31	9	78	51	209	176	115	50	270
207	150	81	308	232	155	2	130	119	76	22
187	152	110	46	250	189	112	29	279	6	39
62	10	91	68	19	198	138	75	297	224	145
108	28	261	186	3	143	136	95	44	230	175
132	69	275	216	140	58	310	7	52	17	171

1	345	80	56	38	110	78	204	75	36
160	14	285	45	150	92	12	11	104	51
35	152	27	300	64	34	230	6	165	13
200	17	130	44	3	108	95	105	32	138
66	4	184	170	65	21	240	50	9	228
156	55	2	69	255	96	25	72	76	70
57	54	100	16	84	195	136	115	10	22
68	39	132	15	23	125	18	190	42	128
135	250	102	26	88	19	28	48	276	5
46	192	7	114	90	8	33	52	85	375

192	133	26	100	153	5	33	84	230	29
65	225	119	58	69	19	144	8	44	140
50	68	290	138	13	126	95	72	7	88
92	112	55	1	240	174	34	175	117	57
3	24	76	70	22	250	91	207	232	102
66	2	216	190	98	39	184	116	17	125
170	87	23	104	150	77	56	38	120	9
14	110	6	168	152	46	261	85	75	52
203	115	42	99	4	136	25	130	114	48
171	78	200	51	145	96	10	11	28	161

1	102	165	182	270	8	714	297	624	450
96	675	22	51	65	28	405	176	357	117
378	9	816	825	26	54	5	238	495	208
195	32	567	198	408	325	4	81	110	119
130	126	15	272	693	234	432	25	34	99
17	39	20	189	330	136	273	36	648	550
792	650	18	3	170	231	390	144	21	306
462	153	312	100	27	66	85	91	60	216
510	264	546	162	24	850	33	78	90	7
135	154	255	104	84	243	528	425	13	12

260	160	85	38	345	200	116	1	84	54	33
14	135	88	52	16	204	114	69	500	290	5
100	29	12	42	27	220	130	80	34	285	184
102	57	460	250	145	2	105	72	44	13	192
110	65	32	255	152	92	25	348	6	21	180
15	56	36	11	156	96	51	380	230	125	58
23	300	174	3	140	90	55	26	240	136	76
48	340	190	115	50	435	8	28	9	132	78
45	22	195	128	68	19	276	150	87	20	70
232	4	7	108	66	39	320	170	95	46	375
228	138	75	580	10	35	18	165	104	64	17

104	85	38	220	161	100	27	252	174	93	11
290	217	4	13	153	114	66	253	200	135	56
25	243	168	87	341	8	65	34	190	154	92
57	242	184	125	54	280	203	124	1	117	102
5	26	170	133	88	23	225	162	84	319	248
196	116	31	9	78	51	209	176	115	50	270
207	150	81	308	232	155	2	130	119	76	22
187	152	110	46	250	189	112	29	279	6	39
62	10	91	68	19	198	138	75	297	224	145
108	28	261	186	3	143	136	95	44	230	175
132	69	275	216	140	58	310	7	52	17	171