The smallest possible additivemultiplicative magic square
What is the smallest possible additivemultiplicative (or additionmultiplication) magic square? 5x5, 6x6, or 7x7: today nobody knows!
Reminder: an additivemultiplicative magic square must be magic when we add the cells of any line (same sum S), and also must be magic when we multiply the cells of any line (same product P). All the integers used must be distinct. The smallest known additivemultiplicative magic square is a 7x7 square constructed by Sébastien Miquel in 2016. Before, the smallest known squares were 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 Horner's squares.
Order 
Semimagic 
Magic 
Impossible. L. Morgenstern (2007) 

L. Morgenstern (2007) 
Impossible. L. Morgenstern (2007) 

Unknown! 

T. Shirakawa (2010) 
S. Miquel (2016) 

first known by W. Horner (1955), best known by C. Boyer (2005) 

first known by W. Horner (1952), best known by C. Boyer (2005) 

≥ 10 
Semimagic and magic squares are impossible. Here is the demonstration given by Lee Morgenstern.
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 addmult semimagic square of
distinct entries is impossible.
Proof
Suppose the following is a 3x3 addmult
semimagic square.
a b M
N P c
Q R d
Because the square is additive semimagic, we
have
a + b + M = M + c + d
or
(1) a + b = c +
d.
Because the square is multiplicative semimagic, we
have
abM = Mcd
or
(2) ab = cd.
From (1) and (2) and the Duplication Lemma, all 3x3 addmult semimagic squares must have duplicate entries.
Corollary
A 3x3 addmult magic square of
distinct entries is impossible.
4x4 addmult semimagic squares are possible. Lee Morgenstern found 54 semimagic examples with Max nb < 256, the best possible being:
110 
72 
63 
80 
64 
105 
66 
90 
81 
88 
100 
56 
70 
60 
96 
99 
156 
18 
48 
25 
30 
144 
60 
13 
16 
20 
130 
81 
45 
65 
9 
128 
But 4x4 addmult magic squares are impossible, as proved below by Lee.
Proof
Suppose the following is a 4x4 addmult
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 addmult magic squares must have duplicate entries.
We don't know if 5x5 addmult magic squares are possible. But 5x5 addmult semimagic squares are possible. Lee Morgenstern found 20 semimagic examples with Max nb < 276. Here is the square having the smallest possible Max nb:
60 
66 
132 
182 
36 
33 
112 
52 
99 
180 
72 
110 
176 
27 
91 
168 
26 
81 
80 
121 
143 
162 
35 
88 
48 
This other square 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.
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 
Here is the square having the smallest S:
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 addmult semimagic square, because it has one addmult magic diagonal (and always 5 addmult magic rows, and 5 addmult magic columns). Very close to a 5x5 addmult magic square!
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 < 276.
And results of my own search of February
2009: there is no magic example with Max nb < 1000 and P < 10^9 * center
cell.
In April 2010, Jérôme Crétaux, famous for cracking in 2005 the chip of the French ehealth insurance card (the "Vitale" card, see for example http://ec.europa.eu/idabc/servlets/Doc?id=22448 in English or www.internetactu.net/2005/09/09/yauratilunscandalesesamvitale in French), nominee for the Big Brother Awards France 2005 (http://bigbrotherawards.eu.org/article550.html), found a 5x5 addmult semimagic square with (Max nb, S, P) = (253, 674, 26854027200). This same square was previously found by Lee Morgenstern, among his 20 examples with Max nb < 276 not fully given here. Will Jérôme also crack the enigma #6?
In January 2010, Javier Cilleruelo, Universidad Autonoma de Madrid, and Florian Luca, Universidad Nacional Autonoma de Mexico, published a paper titled "On multiplicative magic squares" in The Electronics Journal of Combinatorics (www.combinatorics.org) in which they "give a lower bound for the distance between the maximal and minimal element in a multiplicative magic square of dimension r whose entries are distinct positive integers." After their proof that a 4x4 addmult magic square cannot exist, they propose: "Problem 2. Are there additivemultiplicative magic squares of order r = 5 with distinct entries?". This is the same problem as ours here! Look at their interesting paper : www.combinatorics.org/Volume_17/PDF/v17i1n8.pdf
In December 2016, Keith F. Lynch, USA, found more than a thousand 5x5 addmult semimagic squares with P < 2x10^12 (and some other constraints), but none of them has two addmult magic diagonals. Disappointing, he didn't find any new example with one addmult magic diagonal. Before Lynch's work, Morgenstern's square given above with P = 926640000 was considered to be the square having the smallest possible product. But because Lee searched only squares with Max nb < 276, he didn't find Keith's square below. Here is the semimagic square having the smallest P:
4 
57 
72 
220 
112 
64 
308 
20 
54 
19 
70 
16 
11 
152 
216 
99 
60 
266 
32 
8 
228 
24 
96 
7 
110 
Among his thousand semimagic squares, Keith found only one square using 1, and only one square using only odd numbers. Here are these rare squares:
1 
84 
288 
330 
459 

29 
245 
351 
585 
715 
99 
918 
9 
56 
80 
273 
195 
45 
957 
455 

204 
18 
770 
162 
8 
375 
377 
1001 
91 
81 

378 
120 
68 
2 
594 
429 
1053 
203 
175 
65 

480 
22 
27 
612 
21 
819 
55 
325 
117 
609 
Analyzing the two addmult semimagic squares (given above, from Lee and Keith) which are also additive panmagic squares, I observe that they are GraecoLatin squares. We can obtain them with:
A+a 
B+b 
C+c 
D+d 
E+e 

0+95 
76+100 
30+78 
23+114 
30+80 

0+4 
3+54 
56+16 
212+8 
16+96 
C+d 
D+e 
E+a 
A+b 
B+c 
30+114 
23+80 
30+95 
0+100 
76+78 
56+8 
212+96 
16+4 
0+54 
3+16 

E+b 
A+c 
B+d 
C+e 
D+a 
30+100 
0+78 
76+114 
30+80 
23+95 
16+54 
0+16 
3+8 
56+96 
212+4 

B+e 
C+a 
D+b 
E+c 
A+d 
76+80 
30+95 
23+100 
30+78 
0+114 
3+96 
56+4 
212+54 
16+16 
0+8 

D+c 
E+d 
A+e 
B+a 
C+b 
23+78 
30+114 
0+80 
76+95 
30+100 
212+16 
16+8 
0+96 
3+4 
56+54 
Among the thousand of Keith's semimagic addmult squares, I succeeded in moving the cells of 63 of them in order to add the panmagic additive property. ALL of these 63 squares have a GraecoLatin additive structure. But none has mult magic diagonals.
Because such GraecoLatin squares have ten (and not just two) additive magic diagonals, we have ten possibilities (and not just two) that some of their diagonals are also multiplicatively magic. Of course, not easy that rows and columns are also multiplicatively magic... But it's perhaps a method allowing a 5x5 addmult magic square with some good (A, B, C, D, E) and (a, b, c, d, e)?
Who will be the first to construct a 5x5 addmult magic square? Or prove that it is impossible?
In 2005, I constructed a 6x6 multiplicative magic square with partial addmult properties: its 6 rows are additivemultiplicative. In 2007, Lee Morgenstern constructed the first known 6x6 addmult semimagic squares. Here are the best possible semimagic examples. The first square has one additive magic diagonal. And the second square has one additive magic diagonal and one multiplicative magic diagonal.
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 
Interesting observation on the previous square: it uses the first consecutive integers 1, 2, 3, 4, 5, 6, 7. It is not for sure that this square has the smallest possible P, but it has the smallest known P.
We don't know if 6x6 addmult 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!
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 
Results of the search done by Lee Morgenstern in 2007: there is no magic example with Max nb < 136.
Who will be the first to construct a 6x6 addmult magic square? Or prove that it is impossible?
In 2005, I constructed a 7x7 multiplicative magic square with partial addmult properties: its 7 rows are additivemultiplicative.
Results of the search done by Lee Morgenstern in 2007: there is no semimagic example with Max nb < 91.
Who will be the first to construct a 7x7 addmult semimagic square?
In AprilMay 2010, Toshihiro Shirakawa constructed the first known 7x7 addmult semimagic squares, his two best squares being:
36 
70 
91 
3 
22 
24 
60 

7 
9 
32 
110 
14 
78 
60 
105 
12 
14 
44 
45 
8 
78 
96 
65 
35 
63 
2 
16 
33 

5 
21 
16 
54 
112 
65 
33 
52 
98 
15 
3 
88 
30 
24 

26 
110 
9 
40 
7 
42 
72 
22 
20 
168 
26 
45 
21 
8 

88 
39 
20 
98 
27 
30 
4 
10 
4 
36 
40 
39 
132 
49 

28 
48 
66 
15 
13 
126 
10 
105 
66 
13 
12 
80 
28 
6 

18 
6 
90 
52 
80 
11 
49 
18 
48 
11 
56 
42 
5 
130 
The diagonals are add magic, but not mult magic. Some months later, he found this better square, this time one diagonal is addmult magic, the other one is mult magic (however the other diagonal can also be add magic, reordering the cells with 108 + 78 + 39 + 20 + 54 + 63 + 18, but then this diag is no more mult magic):
30 
21 
4 
104 
110 
63 
48 
98 
6 
50 
132 
24 
52 
18 
3 
128 
39 
45 
60 
28 
77 
32 
90 
154 
14 
54 
10 
26 
65 
22 
27 
20 
112 
8 
126 
108 
35 
64 
56 
13 
99 
5 
44 
78 
42 
9 
7 
120 
80 
Who will be the first to construct a 7x7 addmult magic square? Or prove that it is impossible?
August 15th, 2016: Sébastien Miquel, France, is the first to solve my enigma #6b. He previously solved the enigma #4c also on 7x7 squares. Sébastien is now a doctoral student at Université ParisSud, working in mathematics on Lie groups. He used his own application written in Rust, running on his personal computer with i7920 processor. He choose some heuristically "good" magic products (here 150885504000 = 2^10 * 3^7 * 5^3 * 7^2 * 11), before searching squares with selected magic sums. For this product, the search of a magic square with sum 465 took ~600h of computing. Other solutions may exist with smaller products and/or smaller sums.
126 
66 
50 
90 
48 
1 
84 
20 
70 
16 
54 
189 
110 
6 
100 
2 
22 
98 
36 
72 
135 
96 
60 
81 
4 
10 
49 
165 
3 
63 
30 
176 
120 
45 
28 
99 
180 
14 
25 
7 
108 
32 
21 
24 
252 
18 
55 
80 
15 
This 7x7 is currently the SMALLEST known addmult magic square: 5x5 and 6x6 still unknown! (enigmas #6, #6a)
8x8 and 9x9 addmult magic squares?
Examples are known. Look at this other page.
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