The smallest possible additive-multiplicative magic square

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

Reminder: an additive-multiplicative 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 additive-multiplicative 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 Semi-magic Magic 3 Impossible. L. Morgenstern (2007) 4 L. Morgenstern (2007) Impossible. L. Morgenstern (2007) 5 Unknown! 6 7 T. Shirakawa (2010) S. Miquel (2016) 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

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

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:

 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 add-mult magic squares are impossible, as proved below by Lee.

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 20 semi-magic 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 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!

 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 e-health 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/yaura-t-il-un-scandale-sesam-vitale in French), nominee for the Big Brother Awards France 2005 (http://bigbrotherawards.eu.org/article550.html), found a 5x5 add-mult semi-magic 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 add-mult magic square cannot exist, they propose: "Problem 2. Are there additive-multiplicative 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 add-mult semi-magic squares with P < 2x10^12 (and some other constraints), but none of them has two add-mult magic diagonals. Disappointing, he didn't find any new example with one add-mult 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 semi-magic 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 semi-magic 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 add-mult semi-magic squares (given above, from Lee and Keith) which are also additive panmagic squares, I observe that they are Graeco-Latin squares. We can obtain them with:

• (A, B, C, D, E) = (k, 76+k, 30+k, -23+k, -30+k) and (a, b, c, d, e) = (95-k, 100-k, 78-k, 114-k, 80-k) for Morgenstern's square
• (A, B, C, D, E) = (k, 3+k, 56+k, 212+k, 16+k) and (a, b, c, d, e) = (4-k, 54-k, 16-k, 8-k, 96-k) for Lynch's square

•  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 semi-magic add-mult 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 Graeco-Latin additive structure. But none has mult magic diagonals.

• Download the 63 add-mult semi-magic squares of order 5 which are also Graeco-Latin additive squares, with their (A, B, C, D, E) and (a, b, c, d, e) parameters,
(Excel file, 107Kb), by Keith Lynch and Christian Boyer, square #1 is Lynch's square given above, square #8 is Lee's square given above.

Because such Graeco-Latin 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 add-mult 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 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.

 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 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!

 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 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.

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 semi-magic square?

In April-May 2010, Toshihiro Shirakawa constructed the first known 7x7 add-mult semi-magic 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 add-mult 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 add-mult 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 Paris-Sud, working in mathematics on Lie groups. He used his own application written in Rust, running on his personal computer with i7-920 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 add-mult magic square: 5x5 and 6x6 still unknown! (enigmas #6, #6a)

8x8 and 9x9 add-mult magic squares?

Examples are known. Look at this other page.