Additive-Multiplicative Magic Square

This page describes the magic square which is simultaneously a magic square and multiplicative magic square.

Multiplicative magic square

Multiplicative magic square is a square which is magic using multiplication instead of addition. It is not so difficult, using powering to make this type of square. Following is order 3 example.

P = 4096 (= 2¹²)
2³ 2⁸ 2¹

2² 2⁴ 2⁶

2⁷ 2⁰ 2⁵

P = 4096 (= 2¹²)
2³	2⁸	2¹
2²	2⁴	2⁶
2⁷	2⁰	2⁵

Some square that satisfy simultaneously magic sum and magic multiply. Today (May 2006), I know order 8 and order 9 are constructed, but other order squares are unknown. Belows are current status.

Order 3 : Impossible

Folloing is order-3 basic magic square.

a b c

d e f

g h i

The conditions of magic square provide the following equations.

    b + c = d + g … (1)
    b * c = d * g … (2)

b=d or b=g is solution which is satisfied both (1) and (2), those conditions don't satisfy the condition of magic square. Then, non normal order-3 bimagic square cannot exist.

Order 4 : Impossible

Each element of order-4 magic square describe a₁₁～a₄₄.

a₁₁ a₁₂ a₁₃ a₁₄

a₂₁ a₂₂ a₂₃ a₂₄

a₃₁ a₃₂ a₃₃ a₃₄

a₄₁ a₄₂ a₄₃ a₄₄

a₁₁	a₁₂	a₁₃	a₁₄
a₂₁	a₂₂	a₂₃	a₂₄
a₃₁	a₃₂	a₃₃	a₃₄
a₄₁	a₄₂	a₄₃	a₄₄

The conditions of magic square provide the following equations.

    a₁₁ + a₁₂ + a₁₃ + a₁₄ = S ... (1)

On order 4 square, sum of corner is the constant (proof ellipsis),

    a₁₁ + a₁₄ + a₄₁ + a₄₄ = S ... (2)

(1)-(2)

    a₁₂ + a₁₃ = a₄₁ + a₄₄ ... (3)

Same conditions can apply to squared number,

    a₁₂ * a₁₃ = a₄₁ * a₄₄ ... (4)

When we solve (3) and (4), we have following equation.
a₁₂ = a₄₁ or a₄₄
Those condtions don't satisfy the condition of magic square. Then, non normal order 4 bimagic square cannot exist.

Order 5 : Unknown

Order 5 additive-multiplicative magic square exeistance is unknown.

Order 6 : Unknown

Order 6 additive-multiplicative magic square exeistance is unknown. Following is possible to have all multiplicative and some additive properties.

2005: Order 6 multiplicative square with 6 additive-multiplicative rows< by Christian Boyer
S (row only) = 173
P = 39,916,800
(=2^8*3^4*5^2*7*11)
Max number = 88

1 33 36 48 35 20

54 55 2 10 24 28

50 14 72 22 12 3

88 16 30 7 5 27

8 6 70 60 18 11

21 15 4 9 44 80

Order 7 : Unknown

Order 7 additive-multiplicative magic square exeistance is unknown. Following is possible to have all multiplicative and some additive properties.

2005: Order 7 multiplicative square with 7 additive-multiplicative rows by Christian Boyer
S (row only) = 238
P = 3,632,428,800
(=2^8*3^4*5^2*7^2*11*13)
Max number = 91

35 48 1 39 40 33 42

77 65 20 16 36 3 21

4 56 54 7 26 25 66

9 14 44 24 6 91 50

52 5 70 63 22 18 8

12 11 84 10 15 28 78

60 27 13 55 49 32 2

Order 8 : Possible

First order 8 square is constructed by Walter H. Horner (1955).

1955: Order 8 additive-multiplicative magic square, by Walter H. Horner,
S = 840
P = 2,058,068,231,856,000
(=2^7*3^8*5^3*7*13*17*19*23*29)
Max number = 261

46 81 117 102 15 76 200 203

19 60 232 175 54 69 153 78

216 161 17 52 171 90 58 75

135 114 50 87 184 189 13 68

150 261 45 38 91 136 92 27

119 104 108 23 174 225 57 30

116 25 133 120 51 26 162 207

39 34 138 243 100 29 105 152

In November 2005, Christian Boyer constructed better order 8 additive-multiplicative squares with smaller constants.

2005: Order 8 additive-multiplicative magic square, by Christian Boyer
S = 760
P = 51,407,948,592,000
(=2^7*3^6*5^3*7^2*11*13*17*37)
Max number = 333

222 66 225 63 5 7 68 104

1 35 52 136 198 74 189 75

132 296 21 175 9 15 78 34

45 3 102 26 148 264 25 147

51 117 10 6 200 84 259 33

168 100 231 37 39 153 2 30

91 17 8 20 42 150 99 333

50 126 111 297 119 13 40 4

2005: Order 8 additive-multiplicative magic square, by Christian Boyer
S = 600
P = 67,463,283,888,000
(=2^7*3^4*5^3*7^2*11*13*17*19*23)
Max number = 225

75 38 207 102 11 20 91 56

5 44 49 104 57 50 153 138

133 200 17 92 45 66 21 26

99 30 39 14 175 152 23 68

78 63 22 15 184 119 100 19

136 161 76 25 42 117 10 33

28 13 40 77 34 69 114 225

46 51 150 171 52 7 88 35

Order 9 : Possible

First order 9 square is constructed by Horner and Madachy (1975).

1952: Order 9 additive-multiplicative magic square, by Walter H. Horner,
S = 848
P = 5,804,807,833,440,000
(=2^8*3^7*5^4*7*11*17*19*23*29)
Max number = 290

200 87 95 42 99 1 46 108 170

14 44 10 184 81 85 150 261 19

138 243 17 50 116 190 56 33 5

57 125 232 9 7 66 68 230 54

4 70 22 51 115 216 171 25 174

153 23 162 76 250 58 3 35 88

145 152 75 11 6 63 270 34 92

110 2 28 135 136 69 29 114 225

27 102 207 290 38 100 55 8 21

In November 2005, Christian Boyer constructed better order 9 additive-multiplicative squares with smaller constants.

2005: Order 9 additive-multiplicative magic square, by Christian Boyer
S = 784
P = 2,987,659,715,040,000
(=2^8*3^4*5^4*7*11*13*17*19*23*31)
Max number = 310

38 150 248 10 7 65 44 153 69

4 63 39 22 102 184 190 25 155

110 17 115 76 225 93 2 42 104

186 152 50 13 5 70 207 33 68

117 3 28 138 88 34 31 95 250

23 55 170 279 57 100 78 8 14

200 62 114 35 130 1 51 92 99

21 52 9 136 46 66 125 310 19

85 230 11 75 124 171 56 26 6

2005: Order 9 additive-multiplicative magic square, by Christian Boyer
S = 840
P = 7,349,391,483,033,600
(=2^11*3^7*5^2*7^2*11*13*17*19*29)
Max number = 261

84 145 133 80 11 6 104 243 34

40 99 2 78 135 119 224 29 114

208 27 102 112 261 38 30 55 7

95 196 87 1 60 88 153 52 108

9 20 44 85 182 81 19 168 232

17 156 216 171 56 116 5 70 33

203 57 140 66 8 10 54 68 234

22 4 90 189 51 130 174 152 28

162 136 26 58 76 252 77 3 50

Order 16 : Possible

In December 2008, first order 16 square is constructed by Mohamad Moosavi.

2008: Order 16 additive-multiplicative magic square, by Mohamad Moosavi
S = 8432
P = 5,520,657,501,700,315,221,083,892,340,123,238,400,000
(= 2^24*3^13*5^5*7^3*11*13*17*19*23*37*41*43*53*67*71*103*107)
Max number = 1968

1 63 492 206 962 1005 800 798 460 749 256 102 729 583 516 710

246 412 3 21 700 912 1110 871 192 136 644 535 430 852 891 477

309 123 42 4 855 650 938 1184 119 160 642 736 781 387 530 972

84 2 103 369 1072 1036 741 750 856 552 85 224 636 810 639 473

405 371 344 426 828 1177 384 170 74 201 200 114 13 315 1968 1442

258 568 567 265 320 204 1012 963 100 228 222 67 1722 1648 15 273

497 215 318 648 187 288 1070 1104 171 50 134 296 1545 1599 294 16

424 486 355 301 1284 920 153 352 268 148 57 150 336 14 1339 1845

666 737 600 570 5 147 984 618 1053 795 688 994 92 321 128 34

500 684 814 603 738 824 7 105 602 1136 1215 689 64 68 276 107

627 450 670 888 721 615 126 8 1065 559 742 1296 51 32 214 368

804 740 513 550 168 6 515 861 848 1134 923 645 428 184 17 96

1196 1605 512 238 81 159 172 142 9 231 1476 1030 370 469 400 342

448 272 1380 1391 86 284 243 53 1230 1236 11 189 300 456 518 335

255 416 1498 1472 213 43 106 324 1133 1107 210 12 399 250 402 592

1712 1288 221 480 212 162 71 129 252 10 927 1353 536 444 285 350

Other order 10 and upper : Unknown

I think 10 and upper squares may exist.

Home > Additive-Multiplicative Magic Square

1	33	36	48	35	20
54	55	2	10	24	28
50	14	72	22	12	3
88	16	30	7	5	27
8	6	70	60	18	11
21	15	4	9	44	80

35	48	1	39	40	33	42
77	65	20	16	36	3	21
4	56	54	7	26	25	66
9	14	44	24	6	91	50
52	5	70	63	22	18	8
12	11	84	10	15	28	78
60	27	13	55	49	32	2

46	81	117	102	15	76	200	203
19	60	232	175	54	69	153	78
216	161	17	52	171	90	58	75
135	114	50	87	184	189	13	68
150	261	45	38	91	136	92	27
119	104	108	23	174	225	57	30
116	25	133	120	51	26	162	207
39	34	138	243	100	29	105	152

222	66	225	63	5	7	68	104
1	35	52	136	198	74	189	75
132	296	21	175	9	15	78	34
45	3	102	26	148	264	25	147
51	117	10	6	200	84	259	33
168	100	231	37	39	153	2	30
91	17	8	20	42	150	99	333
50	126	111	297	119	13	40	4

75	38	207	102	11	20	91	56
5	44	49	104	57	50	153	138
133	200	17	92	45	66	21	26
99	30	39	14	175	152	23	68
78	63	22	15	184	119	100	19
136	161	76	25	42	117	10	33
28	13	40	77	34	69	114	225
46	51	150	171	52	7	88	35

200	87	95	42	99	1	46	108	170
14	44	10	184	81	85	150	261	19
138	243	17	50	116	190	56	33	5
57	125	232	9	7	66	68	230	54
4	70	22	51	115	216	171	25	174
153	23	162	76	250	58	3	35	88
145	152	75	11	6	63	270	34	92
110	2	28	135	136	69	29	114	225
27	102	207	290	38	100	55	8	21

38	150	248	10	7	65	44	153	69
4	63	39	22	102	184	190	25	155
110	17	115	76	225	93	2	42	104
186	152	50	13	5	70	207	33	68
117	3	28	138	88	34	31	95	250
23	55	170	279	57	100	78	8	14
200	62	114	35	130	1	51	92	99
21	52	9	136	46	66	125	310	19
85	230	11	75	124	171	56	26	6

84	145	133	80	11	6	104	243	34
40	99	2	78	135	119	224	29	114
208	27	102	112	261	38	30	55	7
95	196	87	1	60	88	153	52	108
9	20	44	85	182	81	19	168	232
17	156	216	171	56	116	5	70	33
203	57	140	66	8	10	54	68	234
22	4	90	189	51	130	174	152	28
162	136	26	58	76	252	77	3	50

1	63	492	206	962	1005	800	798	460	749	256	102	729	583	516	710
246	412	3	21	700	912	1110	871	192	136	644	535	430	852	891	477
309	123	42	4	855	650	938	1184	119	160	642	736	781	387	530	972
84	2	103	369	1072	1036	741	750	856	552	85	224	636	810	639	473
405	371	344	426	828	1177	384	170	74	201	200	114	13	315	1968	1442
258	568	567	265	320	204	1012	963	100	228	222	67	1722	1648	15	273
497	215	318	648	187	288	1070	1104	171	50	134	296	1545	1599	294	16
424	486	355	301	1284	920	153	352	268	148	57	150	336	14	1339	1845
666	737	600	570	5	147	984	618	1053	795	688	994	92	321	128	34
500	684	814	603	738	824	7	105	602	1136	1215	689	64	68	276	107
627	450	670	888	721	615	126	8	1065	559	742	1296	51	32	214	368
804	740	513	550	168	6	515	861	848	1134	923	645	428	184	17	96
1196	1605	512	238	81	159	172	142	9	231	1476	1030	370	469	400	342
448	272	1380	1391	86	284	243	53	1230	1236	11	189	300	456	518	335
255	416	1498	1472	213	43	106	324	1133	1107	210	12	399	250	402	592
1712	1288	221	480	212	162	71	129	252	10	927	1353	536	444	285	350

1	33	36	48	35	20
54	55	2	10	24	28
50	14	72	22	12	3
88	16	30	7	5	27
8	6	70	60	18	11
21	15	4	9	44	80

35	48	1	39	40	33	42
77	65	20	16	36	3	21
4	56	54	7	26	25	66
9	14	44	24	6	91	50
52	5	70	63	22	18	8
12	11	84	10	15	28	78
60	27	13	55	49	32	2

46	81	117	102	15	76	200	203
19	60	232	175	54	69	153	78
216	161	17	52	171	90	58	75
135	114	50	87	184	189	13	68
150	261	45	38	91	136	92	27
119	104	108	23	174	225	57	30
116	25	133	120	51	26	162	207
39	34	138	243	100	29	105	152

222	66	225	63	5	7	68	104
1	35	52	136	198	74	189	75
132	296	21	175	9	15	78	34
45	3	102	26	148	264	25	147
51	117	10	6	200	84	259	33
168	100	231	37	39	153	2	30
91	17	8	20	42	150	99	333
50	126	111	297	119	13	40	4

75	38	207	102	11	20	91	56
5	44	49	104	57	50	153	138
133	200	17	92	45	66	21	26
99	30	39	14	175	152	23	68
78	63	22	15	184	119	100	19
136	161	76	25	42	117	10	33
28	13	40	77	34	69	114	225
46	51	150	171	52	7	88	35

200	87	95	42	99	1	46	108	170
14	44	10	184	81	85	150	261	19
138	243	17	50	116	190	56	33	5
57	125	232	9	7	66	68	230	54
4	70	22	51	115	216	171	25	174
153	23	162	76	250	58	3	35	88
145	152	75	11	6	63	270	34	92
110	2	28	135	136	69	29	114	225
27	102	207	290	38	100	55	8	21

38	150	248	10	7	65	44	153	69
4	63	39	22	102	184	190	25	155
110	17	115	76	225	93	2	42	104
186	152	50	13	5	70	207	33	68
117	3	28	138	88	34	31	95	250
23	55	170	279	57	100	78	8	14
200	62	114	35	130	1	51	92	99
21	52	9	136	46	66	125	310	19
85	230	11	75	124	171	56	26	6

84	145	133	80	11	6	104	243	34
40	99	2	78	135	119	224	29	114
208	27	102	112	261	38	30	55	7
95	196	87	1	60	88	153	52	108
9	20	44	85	182	81	19	168	232
17	156	216	171	56	116	5	70	33
203	57	140	66	8	10	54	68	234
22	4	90	189	51	130	174	152	28
162	136	26	58	76	252	77	3	50

1	33	36	48	35	20
54	55	2	10	24	28
50	14	72	22	12	3
88	16	30	7	5	27
8	6	70	60	18	11
21	15	4	9	44	80

35	48	1	39	40	33	42
77	65	20	16	36	3	21
4	56	54	7	26	25	66
9	14	44	24	6	91	50
52	5	70	63	22	18	8
12	11	84	10	15	28	78
60	27	13	55	49	32	2

46	81	117	102	15	76	200	203
19	60	232	175	54	69	153	78
216	161	17	52	171	90	58	75
135	114	50	87	184	189	13	68
150	261	45	38	91	136	92	27
119	104	108	23	174	225	57	30
116	25	133	120	51	26	162	207
39	34	138	243	100	29	105	152

222	66	225	63	5	7	68	104
1	35	52	136	198	74	189	75
132	296	21	175	9	15	78	34
45	3	102	26	148	264	25	147
51	117	10	6	200	84	259	33
168	100	231	37	39	153	2	30
91	17	8	20	42	150	99	333
50	126	111	297	119	13	40	4

75	38	207	102	11	20	91	56
5	44	49	104	57	50	153	138
133	200	17	92	45	66	21	26
99	30	39	14	175	152	23	68
78	63	22	15	184	119	100	19
136	161	76	25	42	117	10	33
28	13	40	77	34	69	114	225
46	51	150	171	52	7	88	35

200	87	95	42	99	1	46	108	170
14	44	10	184	81	85	150	261	19
138	243	17	50	116	190	56	33	5
57	125	232	9	7	66	68	230	54
4	70	22	51	115	216	171	25	174
153	23	162	76	250	58	3	35	88
145	152	75	11	6	63	270	34	92
110	2	28	135	136	69	29	114	225
27	102	207	290	38	100	55	8	21

38	150	248	10	7	65	44	153	69
4	63	39	22	102	184	190	25	155
110	17	115	76	225	93	2	42	104
186	152	50	13	5	70	207	33	68
117	3	28	138	88	34	31	95	250
23	55	170	279	57	100	78	8	14
200	62	114	35	130	1	51	92	99
21	52	9	136	46	66	125	310	19
85	230	11	75	124	171	56	26	6

84	145	133	80	11	6	104	243	34
40	99	2	78	135	119	224	29	114
208	27	102	112	261	38	30	55	7
95	196	87	1	60	88	153	52	108
9	20	44	85	182	81	19	168	232
17	156	216	171	56	116	5	70	33
203	57	140	66	8	10	54	68	234
22	4	90	189	51	130	174	152	28
162	136	26	58	76	252	77	3	50