The smallest possible multiplicative magic squares

• In this page:
• What are the smallest possible multiplicative squares?
• What is a multiplicative magic square?
• Smallest 3rd-order multiplicative magic squares
• Smallest 4th-order multiplicative magic squares
• Smallest 5th-order multiplicative magic squares
• In other pages:
• Smallest 6th and 7th-order multiplicative magic squares
• Smallest 8th and 9th-order multiplicative magic squares, and additive-multiplicative magic squares
• Smallest multiplicative magic squares of orders >= 10
• Smallest additive-multiplicative magic squares
• Additive-multiplicative magic squares of orders >= 10
• Pandiagonal multiplicative magic squares

What are the smallest possible multiplicative squares?

In September 2005, Ed Pegg Jr., author of the well-known MathPuzzle web site, asked the question: "Is there a list of smallest multiplicative constants for various NxN multiplicative magic squares somewhere?"

Strangely, it seems that no list has never been published. After the question asked by Ed, I immediately studied the question and you will find in this website the answer to the Ed's question: the 3x3, 4x4, 5x5, 6x6, 7x7 lists, and much more information... A lot of new results: for example, the smallest P=302400 for 5x5 squares is new, found in September 2005. As far as I know, all the examples of 5x5 multiplicative squares published before were bigger.

Thanks to Ed for his very interesting question! After the first answers to his question, Ed wrote an interesting paper in his Math Games column of MAA Online:

http://www.maa.org/editorial/mathgames/mathgames_11_07_05.html

Thanks to Dan Asimov, Michael Kleber, Richard Schroeppel, and David Wilson for their ideas. And thanks to Edwin Clark, Don Reble, and Günter Stertenbrink for their checkings of my big multiplicative squares, confirming that they have all the announced properties.

Smallest possible magic products P of multiplicative magic squares
(look also at the tables of best known pandiagonal multiplicative magic squares
and best known additive-multiplicative magic squares)

Order	Semi-magic	Magic	Smallest max nb (*)
3	120	(a) 216	(a) 36
4	(c) 4 320	(b) 5 040	(b) 28
5	277 200	302 400	45
6	25 945 920		66 (!)
7	3 632 428 800		91
8	(d) 670 442 572 800		≤ 160 (!)
9	(d) 140 792 940 288 000		≤ 225 (!)
10	≤ 277 563 225 139 200 000		≤ 290 (!)
11	≤ 160 986 670 580 736 000 000		≤ 341 (!)
12	≤ 119 774 082 912 067 584 000 000		≤ 444 (!)
13	≤ 89 740 731 621 866 637 312 000 000		≤ 546 (!)
14	≤ 86 138 139 806 868 813 305 241 600 000		≤ 645 (!)
15	≤ 80 969 851 418 456 684 506 927 104 000 000		≤ 735 (!)
16	≤ 102 993 651 004 276 902 692 811 276 288 000 000		≤ 848 (!)
17	≤ 136 260 600 278 658 342 262 589 318 529 024 000 000		≤ 1003 (!)

• Results of this table were first found in 2005 by Christian Boyer, except (a) first found in 1893 by G. Pfeffermann, (b) first found in 1913 by Harry Sayles, (c) first found in 2005 by Ed Pegg Jr, (d) found in 2005 by Luke Pebody.
• A less or equal sign "≤" means that this product (or this maximum number) is the smallest known: the true smallest possible product (or the true smallest max nb) is less or equal this number. If you succeed in getting a smaller product (or a smaller max nb), send me a message! Your results will be added in this website.
• (*) Smallest max nb. Each square use distinct integers, most of the integers being non-consecutive. It is interesting to know what is the smallest possible set of integers, from 1 to x, for each order of multiplicative magic squares. The goal is to find the smallest possible x.
• (!) means that this max nb is currently known with a product different and bigger than the smallest possible or smallest known product. For example, for the order 6, the smallest max nb 66 is impossible in a multiplicative magic square having P = 25 945 920, but possible with known squares having a bigger product.
• The examples of big multiplicative squares, orders from 10 to 17, are in an Excel file downloadable from this page

What is a multiplicative magic square?

It is a square which is magic using multiplication instead of addition.

A "multiplicative" magic square is very easy to construct from a standard "additive" magic square, using the numbers of the additive magic square as powers of a fixed integer. For example:

Construction of a multiplicative magic square P=4096,
from a standard additive magic square S=12.

Additive			=12	>>	(powering step)			>>	Multiplicative			=4096
3	8	1	=12		23	28	21		8	256	2	=4096
2	4	6	=12		22	24	26		4	16	64	=4096
7	0	5	=12		27	20	25		128	1	32	=4096
=12	=12	=12	=12						=4096	=4096	=4096	=4096

When we add numbers of any line in the left square, we get always the same number (here 12). When we multiply numbers of any line in the right square, we get always the same number (here 4096). This 3rd-order multiplicative square on the right was published by Antoine Arnauld in Nouveaux Eléments de Géométrie, Paris, in... 1667... a long time ago!

Some properties on multiplicative magic squares:

• It is impossible to construct a multiplicative magic square using consecutive integers
• A multiplicative square remains magic when all its numbers are multiplied by the same constant.
• A multiplicative square remains magic when all its numbers are squared. Idem when they are cubed, or raised at the 4th power, and so on... Very interesting for our www.multimagie.com website: multiplicative squares are infinitely multimagic!!!

I recommend the book Magic squares and cubes by W.S. Andrews republished by Dover in 1960: it includes, pages 283-294, an excellent paper on multiplicative squares written by Harry A. Sayles, a paper which was initially published in The Monist in 1913. Various examples of multiplicative magic squares are given. Some of them are used in this page, referenced by Sayles[page in the Andrews book, figure number]. For example, the above square P=4096 can also be found in this book: Sayles[284, 510]. And I recommend the French old magazine Les Tablettes du Chercheur, year 1893, in which G. Pfeffermann published various multiplicative squares as games.

Smallest 3rd-order multiplicative magic squares

Because the construction method seen above uses powers, the generated numbers are big. It is possible to construct 3x3 squares with smaller magic products than 4096.

Sayles published in 1913 two examples:

P=216, by Sayles[287, 532]
P=1000, by Sayles[286, 524]

18	1	12		50	1	20
4	6	9		4	10	25
3	36	2		5	100	2

I discovered in 2006 that G. Pfeffermann published a lot of 3x3 multiplicative squares 20 years before Sayles: he published 17 squares in 1893! These squares were games, as for his first bimagic squares 8x8 and 9x9. His nine first 3x3 multiplicative squares were the following, including the smallest P=216. Will you succeed in filling them?

Les Tablettes du Chercheur, 1893, Oct. 15.
Game by G. Pfeffermann: complete the
missing numbers (distinct integers in each square)
in order to get 3x3 multiplicative magic squares.

In 1917, in his Amusements in Mathematics, Henry E. Dudeney published also the square P=216. We can remark that these squares can be generated by the same method:

P=a³b³

ab²	1	a²b
a²	ab	b²
b	a²b²	a

With a=2 and b=3, we get the square P=216. With a=2 and b=5, we get the square P=1000.

As first proved in a paper published in 1983 in Discrete Mathematics by Debra K. Borkovitz (currently at Wheeklock College, Boston, USA) and Frank K.-M. Hwang (currently at National Chiao Tung University, Taiwan), the minimum magic product for 3x3 multiplicative squares is 216. In 2005, just after the above question asked by Ed, here is another -and very short- proof given by Rich Schroeppel:

"There's a standard proof that the center of a 3x3 addition magic square is K/3, where K is the row sum (Add up the four lines through the center, subtract the whole square.)  Of course this works for multiplication too, so the magic product is always the cube of  the center.
Minimality of K = 216 for 3x3 mulgic square, Proof sketch:
K must be a cube, with at least 9 divisors.
1, 8, 27, 64, 125 have 1, 4, 4, 7, 4 divisors. Done!"

If we do not consider the diagonals, here is the smallest possible semi-magic square. It was not required, but one diagonal (not the other diag) does have the correct product P:

Semi-magic with P=120

1	20	6
12	2	5
10	3	4

And now, here is the list of the smallest possible magic products asked by Ed:

The ten smallest magic products P
of 3x3 multiplicative magic squares

#	P	= 2^	· 3^	· 5^	· 7^	· 11^	· 13^
1	216	3	3	0	0	0	0
2	1000	3	0	3	0	0	0
3	1728	6	3	0	0	0	0
4	2744	3	0	0	3	0	0
5	3375	0	3	3	0	0	0
6	4096	12	0	0	0	0	0
7	5832	3	6	0	0	0	0
8	8000	6	0	3	0	0	0
9	9261	0	3	0	3	0	0
10	10648	3	0	0	0	3	0

For more terms: see the 3x3 list referenced in Oct. 2005 under the number A111029 in the Neil Sloane's Encyclopedia of Integer Sequences, AT&T Research.

Within the 17 squares (3x3) published by G. Pfeffermann in 1893, the above 10 first smallest P were already all present. Excellent analysis done more than one century ago!

My formulation above with P = a³b³ is the complete formulation in normalized rational numbers. Here is the nice complete all-integer formulation constructed by Lee Morgenstern in December 2007, where ab = cd. All solutions may be obtained from this by scaling the 9 entries by the same constant. In an unscaled solution, the 4 middle-side entries are always squares.

P = (ab)³
because ab = cd

bc	a²	bd
d²	ab	c²
ac	b²	ad

Smallest 4th-order multiplicative magic squares

In 1893, G. Pfeffermann published a 4th-order pandiagonal square as a game, as usual for him. Complete the next square P=28224 with the numbers 2, 3, 4, 6, 8, 12, 14, 21, 28, 42, 56, 84. The numbers in the four quarters and in the central 2x2 square are also asked to have the same P. Will you succeed? His problem has 3 solutions.

4x4 pandiag. magic.
P=28224.
Game: fill the missing numbers,
by Pfeffermann

1	24
		168	7

In 1913, Sayles published various samples of 4th-order squares: P = 5040, 7560, 11760, 14112, 14400, 21000, 2985984. In their Discrete Mathematics paper of 1983, Borkovitz and Hwang proved that the minimum magic product for 4x4 multiplicative squares is 5040, and produced a different example than the Sayles's one.

P = 5040, on the left by Sayles[291, 555],
and on the right by Borkovitz-Hwang

1	15	24	14		1	14	12	30
12	28	3	5		15	24	7	2
21	6	10	4		42	3	10	4
20	2	7	18		8	5	6	21

These 2 squares can be constructed with a multiplication of cells of two latin squares (the two squares producing an Eulerian square), using two set of numbers (A, B, C, D) and (a, b, c, d). For example, the Sayles's square can be built using:

Construction method for 4th order squares.
(A, B, C, D) = (1, 2, 3, 4) and (a, b, c, d) = (1, 5, 6, 7)
produces the Sayles's square: P = ABCDabcd = 7! = 5040

A

C

D

B

X

a

b

c

d

=

Aa

Cb

Dc

Bd

B

D

C

A

c

d

a

b

Bc

Dd

Ca

Ab

C

A

B

D

d

c

b

a

Cd

Ac

Bb

Da

D

B

A

C

b

a

d

c

Db

Ba

Ad

Cc

and the Borkovitz & Hwang example can be produced by the same method, but using other latin squares and other sets: (1, 2, 3, 6) and (1, 4, 5, 7).

Sayles published also a pandiagonal multiplicative square, pandiagonal meaning that all the broken diagonals give also the same magic product. As it is for the above 4x4 Pfeffermann's square, it is also a most-perfect magic square: all 2x2 subsquares have the same magic product.

Smallest 4x4 pandiagonal magic, P=14400,
by Sayles[288, 540]

1	24	10	60
30	20	3	8
12	2	120	5
40	15	4	6

In 1957, Ronald B. Edwards published in Scripta Mathematica (Vol XXII, p.202) an astonishing 4x4 multiplicative square. When we write all its numbers backward, we get again a 4x4 multiplicative square!!! So surprising square! What was the method used to build it, 50 years ago, without any computer? Its magic product 4558554 is a palindromic number. And we can border the square to get a 6x6 additive square.

4x4 multiplicative square giving another 4x4 multiplicative square when its numbers are reversed, and a 6x6 additive square whenbordered, by Ronald B. Edwards, 1957.
P(1) = 4,558,554.     P(2) = 8,642,304.     S(3) = 438.

										98	102	51	64	79	44
46	39	231	11		64	93	132	11		58	46	39	231	11	53
121	21	26	69		121	12	62	96		95	121	21	26	69	106
13	253	33	42		31	352	33	24		50	13	253	33	42	47
63	22	23	143		36	22	32	341		96	63	22	23	143	91
										41	93	52	61	94	97

The main results of my research are:

• the smallest possible product of 4x4 multiplicative pandiagonal squares is P=14400, meaning that the Sayles's square cannot be beaten
• the smallest possible product of 4x4 multiplicative magic squares is P=5040, as already proved by Borkovitz & Hwang
• the second smallest magic product of 4x4 multiplicative magic squares, which was unknown, is P=6480. For example with:
   

Second smallest 4x4, P=6480,
produced by (a, b, c) = (2, 3, 5),
P = a⁴b⁴c = 2⁴3⁴5 = 6480

1	bc	ab3	a3	>>	1	15	54	8
a3b	ab2	c	b		24	18	5	3
ac	ab	a2b	b2		10	6	12	9
b3	a2	a	abc		27	4	2	30

• the smallest possible product of 4x4 multiplicative semi-magic squares is P=4320 (and at least one magic diagonal is impossible for that P)
   

Smallest 4x4 semi-magic, P=4320
(nearest possible diag  4608)

16	1	10	27
5	24	18	2
6	12	3	20
9	15	8	4

the list of the smallest possible magic products is:
 

The ten smallest magic products P
of 4x4 multiplicative magic squares

#	P	= 2^	· 3^	· 5^	· 7^	· 11^	· 13^
1	5040	4	2	1	1	0	0
2	6480	4	4	1	0	0	0
3	6720	6	1	1	1	0	0
4	7200	5	2	2	0	0	0
5	7560	3	3	1	1	0	0
6	7920	4	2	1	0	1	0
7	8400	4	1	2	1	0	0
8	8640	6	3	1	0	0	0
9	9072	4	4	0	1	0	0
10	9240	3	1	1	1	1	0

For more terms: see the 4x4 list referenced in Oct. 2005 under the number A111030 in the Neil Sloane's Encyclopedia of Integer Sequences, AT&T Research.

In December 2007, Lee Morgenstern gives this complete formulation in normalized rational numbers of 4x4 multiplicative magic squares. It requires 7 terms. For example (a, b, c, d, e, f, g) = (1, 2, 3, 4, 5, 6, 7) gives the smallest solution P=5040 of Sayles (see his square above).

2007: Formulation of 4x4 multiplicative magic squares,
P = a²bcdefg,
by Lee Morgenstern.

1	ce	adf	abg
bf	adg	c	ae
acg	f	abe	d
ade	ab	g	cf

From this first formulation, here is his reasonment producing the next formulation in normalized rational numbers of 4x4 pandiagonal multiplicative magic squares:

2007: Formulation of 4x4 pandiagonal magic squares,
P = (cdfg)²,
by Lee Morgenstern.

1	cfg	df	cdg
cdf	dg	c	fg
cg	f	cdfg	d
dfg	cd	g	cf

Smallest 5th-order multiplicative magic squares

In 1893, G. Pfeffermann published two samples of 5th-order pandiagonal squares: P=665280 and 1182720. A supplemental info given by Pfeffermann about the square P=1182720: the missing numbers are 1, 2, 3, 4, 5, 6, 7, 8, 10, 11. Will you succeed?

Two 5x5 pandiagonal magic squares.
P=665280 (left) and 1182720 (right).
Game: fill the missing numbers, by Pfeffermann

		5	2				28	48		88
10	4			7			40			112
	8	14	20	9		44	16	14	24
		11	3	16		56		20	176
1	6					80	22			12

In 1913, Sayles published two other pandiagonal samples: P = 362880 and 720720. And Dudeney a bigger example P=60466176. Here is the Sayles's smallest example which can be built with two latin squares (producing an Eulerian square), and which is pandiagonal:

Smallest 5x5 pandiagonal magic,
P=362880 by Sayles[291, 556], Max nb = 54,
produced by (A, B, C, D, E) = (1, 2, 3, 4, 6)
and (a, b, c, d, e) = (1, 5, 7, 8, 9)
P = ABCDEabcde = 9! = 362880

Aa	Bb	Cc	Dd	Ee	>>	1	10	21	32	54
Dc	Ed	Ae	Ba	Cb		28	48	9	2	15
Be	Ca	Db	Ec	Ad		18	3	20	42	8
Eb	Ac	Bd	Ce	Da		30	7	16	27	4
Cd	De	Ea	Ab	Bc		24	36	6	5	14

The main results of my research are:

• the smallest possible product of 5x5 multiplicative pandiagonal squares is P=362880, meaning that the Sayles's square cannot be beaten
• the smallest possible product of 5x5 multiplicative magic squares is P=302400, never published before. Here is one of the possible examples:
   

Smallest 5x5 magic, P=302400, Max nb = 45,
produced by (a, b, c, d) = (2, 3, 5, 7),
P = a⁶b³c²d = 2⁶3³5²7 = 302400

a2b	cd	1	a3c	ab2	>>	12	35	1	40	18
a2b2	a	a3b	d	c2		36	2	24	7	25
ad	b2c	bc	a2	a3		14	45	15	4	8
c	a4	abd	abc	b		5	16	42	30	3
ac	ab	a2c	b2	a2d		10	6	20	9	28

• the smallest possible product of 5x5 multiplicative semi-magic squares is P=277200, and allows one magic diagonal
   

Smallest 5x5 semi-magic, P=277200 (and one magic diagonal)

9	1	14	44	50
2	35	55	18	4
25	33	8	7	6
22	20	3	10	21
28	12	15	5	11

the list of the smallest possible magic products is:
 

The ten smallest magic products P
of 5x5 multiplicative magic squares

#	P	= 2^	· 3^	· 5^	· 7^	· 11^	· 13^
1	302400	6	3	2	1	0	0
2	332640	5	3	1	1	1	0
3	362880	7	4	1	1	0	0
4	393120	5	3	1	1	0	1
5	403200	8	2	2	1	0	0
6	415800	3	3	2	1	1	0
7	423360	6	3	1	2	0	0
8	443520	7	2	1	1	1	0
9	453600	5	4	2	1	0	0
10	475200	6	3	2	0	1	0

For more terms: see the 5x5 list referenced in Oct. 2005 under the number A111031 in the Neil Sloane's Encyclopedia of Integer Sequences, AT&T Research.

In December 2007, Lee Morgenstern gives this complete formulation in normalized rational numbers of 5x5 multiplicative magic squares. It requires 14 terms. For example, setting s = t = u = v = w = x = y = z = 1, then setting e = f = a, we get my previous formulation above where P = a⁶b³c²d.

2007: Formulation of 5x5 multiplicative magic squares,
P = (abvy)³(cfstu)²(dewxz), by Lee Morgenstern.

abfstuvyz	cd	1	a2cftxy	b2esuv2wy
ab2fv2w	a	abefs2tuy	d	c2tuvxy2z
de	b2cs2tuwy3	bcvx	afuv2z	a2ft
cx	a2f2tz	abduv2y	bcestuvwy2	bs
acstuy2	beuv3x	acftwyz	b2s	adf

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