MULTIMAGIE.COM - Smallest additive-multiplicative magic square

What is the smallest possible additive-multiplicative (or addition-multiplication) magic square? 5x5, 6x6, 7x7, or 8x8: today nobody knows!

Reminder: an additive-multiplicative magic square has to be magic when we add the cells of any line (same sum S), and has to be also magic when we multiply the cells of any line (same product P). All the used integers have to be distinct. The smallest known additive-multiplicative magic squares are 8x8 and 9x9 squares, first constructed by Walter Horner in the 50's. In 2005, I constructed other 8x8 and 9x9 squares with smaller magic products and smaller magic sums than the Horner's squares: they are still the best known add-mult magic squares.

Here is the status on the subject, most of the results ≤ 7 coming from Lee Morgenstern, USA.

Semi-magic and magic squares are impossible. Here is the demonstration given by Lee Morgenstern.

*First known additive-mutiplicative squares*
Order	Semi-magic	Magic
3	Impossible. L. Morgenstern (2007)
4	L. Morgenstern (2007)	Impossible. L. Morgenstern (2007)
5	Unknown!
6
7	Unknown!
8	first known by W. Horner (1955), best known by C. Boyer (2005)
9	first known by W. Horner (1952), best known by C. Boyer (2005)
≥ 10	See this page and its table

Duplication Lemma
Given
     a + b = c + d
and
     ab = cd,
then
     c = a or c = b.

Proof
Use the 2nd equation and set d = ab/c. Substitute this into the 1st equation to get
     a + b = c + ab/c
or
     c(a + b) = c^2 + ab
or
     (c - a)(c - b) = 0.

Thus c = a or c = b.

Theorem
A 3x3 add-mult semi-magic square of distinct entries is impossible.

Proof
Suppose the following is a 3x3 add-mult semi-magic square.
     a b M
     N P c
     Q R d

Because the square is additive semi-magic, we have
a + b + M = M + c + d
or
(1) a + b = c + d.

Because the square is multiplicative semi-magic, we have
abM = Mcd
or
(2) ab = cd.

From (1) and (2) and the Duplication Lemma, all 3x3 add-mult semi-magic squares must have duplicate entries.

Corollary
A 3x3 add-mult magic square of distinct entries is impossible.

4x4 add-mult semi-magic squares are possible. Lee Morgenstern found 54 semi-magic examples with Max nb < 256, the best possible being:

2007: 4x4 add-mult semi-magic square,
**Smallest Max nb = 110**,
S = 325, P = 39 916 800,
by L. Morgenstern
110	72	63	80
64	105	66	90
81	88	100	56
70	60	96	99

2007: 4x4 add-mult semi-magic square, Max nb = 156,
Smallest S = 247,
Smallest P = 3 369 600,
by L. Morgenstern
156	18	48	25
30	144	60	13
16	20	130	81
45	65	9	128

Theorem
A 4x4 add-mult magic square of distinct entries is impossible.

Proof
Suppose the following is a 4x4 add-mult magic square.
     a M N b
     P Q R S
     T U V W
     X c d Y

Because the square is additive magic, we have
     (1) a + M + N + b = X + c + d + Y,
     (2) a + Q + V + Y = M + Q + U + c,
     (3) b + R + U + X = N + R + V + d.

Add (1), (2), and (3), cancel common terms, and divide by 2,
(4) a + b = c + d.

Because the square is multiplicative magic, we have
     (5) aMNb = XcdY
     (6) aQVY = MQUc
     (7) bRUX = NRVd

Multiply (5), (6), and (7),
MNQRUVXY(ab)^2 = MNQRUVXY(cd)^2
or
(8) ab = cd.

From (4) and (8) and the Duplication Lemma, all 4x4 add-mult magic squares must have duplicate entries.

We don't know if 5x5 add-mult magic squares are possible. But 5x5 add-mult semi-magic squares are possible. Lee Morgenstern found 15 semi-magic examples with Max nb < 256. Here is the square having the smallest possible Max nb. It has a very interesting supplemental property: it is also an additive panmagic square, the sums of all its broken diagonals giving again the same S.

2007: 5x5 add-mult semi-magic square,
and additive panmagic square,
**Smallest Max nb = 190**,
S = 520, P = 8 216 208 000,
by L. Morgenstern
95	176	108	91	50
144	57	65	100	154
70	78	190	110	72
156	125	77	48	114
55	84	80	171	130

2007: 5x5 add-mult semi-magic square,
Max nb = 208,Smallest S = 444,
Smallest P = 926 640 000,
by L. Morgenstern
208	20	110	25	81
11	195	80	108	50
75	30	180	16	143
60	144	65	165	10
90	55	9	130	160

But his very best square is this other one. Better than an add-mult semi-magic square, because it has one add-mult magic diagonal (and always 5 add-mult magic rows, and 5 add-mult magic columns). Very close to a 5x5 add-mult magic square!

2007: 5x5 add-mult **nearly** magic square,
Max nb = 220, S = 570,
P = 6 810 804 000,
only one non add-mult magic diagonal
by L. Morgenstern
105	182	40	198	45
78	216	66	175	35
220	42	65	63	180
140	55	189	30	156
27	75	210	104	154

Results of the search done by Lee Morgenstern in 2007: there is no magic example with Max nb < 259.
And results of my own search of February 2009: there is no magic example with Max nb < 1000 and P < 10^9 * center cell.

Who will be the first to construct a 5x5 add-mult magic square? Or prove that it is impossible?

In 2005, I constructed a 6x6 multiplicative magic square with partial add-mult properties: its 6 rows are additive-multiplicative. In 2007, Lee Morgenstern constructed the first known 6x6 add-mult semi-magic squares. Here are the best possible semi-magic examples. The first square has one additive magic diagonal. And the second square has one additive magic diagonal and one multiplicative magic diagonal.

Interesting observation on the previous square: it uses the first consecutive integers 1, 2, 3, 4, 5, 6, 7. Not fully sure that this square has the smallest possible P, but it has the smallest known P.

2007: 6x6 add-mult semi-magic square,
**Smallest Max nb = 105,**
S = 316, P = 7 264 857 600,
by L. Morgenstern
105	32	42	78	44	15
36	49	60	12	104	55
16	99	84	35	30	52
65	48	11	40	54	98
28	75	39	88	14	72
66	13	80	63	70	24

2007: 6x6 add-mult semi-magic square,
Max nb = 135, **Smallest S = 289**,
P = 1 596 672 000,
by L. Morgenstern
24	88	18	120	25	14
135	28	64	22	30	10
8	75	45	16	112	33
35	4	80	90	44	36
32	54	77	20	6	100
55	40	5	21	72	96

2007: 6x6 add-mult semi-magic square,
Max nb = 200, S = 327,
**Smallest (known) P = 508 032 000**,
by L. Morgenstern
200	42	24	30	28	3
16	160	6	63	75	7
70	10	144	2	56	45
9	21	5	140	32	120
14	4	50	12	135	112
18	90	98	80	1	40

We don't know if 6x6 add-mult magic squares are possible. The second square above (Smallest S = 289) has 13 of 14 correct sums, and 13 of 14 correct products: total 26 out of 28, not an additive magic square, and not a multiplicative magic square. Here is a square with 14 of 14 correct sums, and 12 of 14 correct products: still 26 out of 28, still not a multiplicative magic square, but this is now an additive magic square!

Results of the search done by Lee Morgenstern in 2007: there is no magic example with Max nb < 136.

2008: 6x6 add-mult semi-magic square
and additive magic square,
Max nb = 156, S = 360, P = 8 491 392 000,
by L. Morgenstern
27	25	156	48	84	20
75	144	18	56	52	15
24	12	45	117	50	112
16	65	21	30	108	120
140	72	40	9	60	39
78	42	80	100	6	54

Who will be the first to construct a 6x6 add-mult magic square? Or prove that it is impossible?

In 2005, I constructed a 7x7 multiplicative magic square with partial add-mult properties: its 7 rows are additive-multiplicative. We don't know any 7x7 add-mult semi-magic square. Results of the search done by Lee Morgenstern in 2007: there is no semi-magic example with Max nb < 91.

Who will be the first to construct a 7x7 add-mult magic square? Or prove that it is impossible?

95	176	108	91	50
144	57	65	100	154
70	78	190	110	72
156	125	77	48	114
55	84	80	171	130

105	182	40	198	45
78	216	66	175	35
220	42	65	63	180
140	55	189	30	156
27	75	210	104	154

105	32	42	78	44	15
36	49	60	12	104	55
16	99	84	35	30	52
65	48	11	40	54	98
28	75	39	88	14	72
66	13	80	63	70	24

24	88	18	120	25	14
135	28	64	22	30	10
8	75	45	16	112	33
35	4	80	90	44	36
32	54	77	20	6	100
55	40	5	21	72	96

200	42	24	30	28	3
16	160	6	63	75	7
70	10	144	2	56	45
9	21	5	140	32	120
14	4	50	12	135	112
18	90	98	80	1	40

27	25	156	48	84	20
75	144	18	56	52	15
24	12	45	117	50	112
16	65	21	30	108	120
140	72	40	9	60	39
78	42	80	100	6	54

95	176	108	91	50
144	57	65	100	154
70	78	190	110	72
156	125	77	48	114
55	84	80	171	130

105	182	40	198	45
78	216	66	175	35
220	42	65	63	180
140	55	189	30	156
27	75	210	104	154

105	32	42	78	44	15
36	49	60	12	104	55
16	99	84	35	30	52
65	48	11	40	54	98
28	75	39	88	14	72
66	13	80	63	70	24

24	88	18	120	25	14
135	28	64	22	30	10
8	75	45	16	112	33
35	4	80	90	44	36
32	54	77	20	6	100
55	40	5	21	72	96

200	42	24	30	28	3
16	160	6	63	75	7
70	10	144	2	56	45
9	21	5	140	32	120
14	4	50	12	135	112
18	90	98	80	1	40

27	25	156	48	84	20
75	144	18	56	52	15
24	12	45	117	50	112
16	65	21	30	108	120
140	72	40	9	60	39
78	42	80	100	6	54

95	176	108	91	50
144	57	65	100	154
70	78	190	110	72
156	125	77	48	114
55	84	80	171	130

105	182	40	198	45
78	216	66	175	35
220	42	65	63	180
140	55	189	30	156
27	75	210	104	154

105	32	42	78	44	15
36	49	60	12	104	55
16	99	84	35	30	52
65	48	11	40	54	98
28	75	39	88	14	72
66	13	80	63	70	24

24	88	18	120	25	14
135	28	64	22	30	10
8	75	45	16	112	33
35	4	80	90	44	36
32	54	77	20	6	100
55	40	5	21	72	96

200	42	24	30	28	3
16	160	6	63	75	7
70	10	144	2	56	45
9	21	5	140	32	120
14	4	50	12	135	112
18	90	98	80	1	40

27	25	156	48	84	20
75	144	18	56	52	15
24	12	45	117	50	112
16	65	21	30	108	120
140	72	40	9	60	39
78	42	80	100	6	54