Pandiagonal multiplicative magic squares

Pandiagonal multiplicative magic squares are multiplicative magic squares with the "pandiagonal" supplemental feature: all their broken diagonals are also magic, same magic product than the other rows, columns and 2 standard diagonals.

A 3x3 pandiagonal multiplicative magic square is impossible. The smallest possible size is 4x4:

1913: Smallest 4x4 pandiagonal magic, P=14 400,
MaxNb = 120
by Harry Sayles

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

You can check that for example 10*8*12*15 = 8*120*15*1 = 14 400.

As it is for all multiplicative squares, pandiagonal or not, the main interest (and game!) is to construct squares using the SMALLEST possible numbers and/or producing the smallest possible products. The research on these pandiagonal squares is detailed in other multiplicative pages of this site: 4x4, 5x5, 6x6, 7x7, 8x8, 9x9, 10x10, 11x11,...

Best known PANDIAGONAL multiplicative magic squares
(look also at the table of the smallest possible magic products of multiplicative magic squares)

Order	Smallest known product	Smallest known MaxNb	Additional properties (**)	Author (***)	Year
3	Impossible
4	(*) 14 400	120	most-perfect	(a)	1913
5	(*) 362 880	54
6	46 656 000 000	1 800	3x3	(d)	2012-14
7	8 821 612 800	136		(b)	2005
	17 643 225 600	119
8	42 074 422 790 400	546	4x4	(c)	2007
	398 337 730 560 000	528
9	294 451 250 429 952 000	1 760	3x3	(f)	2007-17
	10 107 959 331 165 696 000	1 365		(g)	2017
10	26 480 706 717 104 640 000	1 617	5x5	(c)	2007
	393 968 065 240 189 440 000	850
11	160 986 670 580 736 000 000	580		(b)	2005
	1 441 031 935 035 813 120 000	341
12	1 858 314 876 538 115 309 174 784 000 000	110 880	most-perfect		2006
13	89 740 731 621 866 637 312 000 000	740			2005
	1 259 162 176 578 813 217 751 040 000	546
14	61 242 221 119 253 958 615 896 064 000 000	3 480	7x7	(c)	2007
15	12 374 909 661 880 770 361 264 383 140 560 896 000	9 108	5x5	(e)	2014
16	1 262 776 099 266 522 647 454 419 588 772 695 900 160 000	10 374	4x4	(g)	2017
	272 667 072 835 526 978 538 805 734 677 268 513 423 360 000	8 372
17	136 260 600 278 658 342 262 589 318 529 024 000 000	1 272		(b)	2005
	5 170 296 729 462 351 156 714 529 991 349 043 200 000	1 003

• (*) These 4x4 and 5x5 squares are proved to be the smallest possible pandiagonal 4x4 and 5x5. But for all other squares (6x6 or higher), the question is open. 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.
• (**) Additional properties:
• most-perfect = all 2x2 subsquares with same product
• 3x3, 4x4, 5x5, or 7x7 = all 3x3, 4x4, 5x5, or 7x7 subsquares with same product
• (***) Author:
• (a) Harry A. Sayles in 1913
• (b) Christian Boyer in 2005-2006
• (c) Jaroslaw Wroblewski and Christian Boyer in 2007
• (d) Radko Nachev in 2012, improved by Oscar Lanzi in 2014
• (e) Oscar Lanzi in 2014
• (f) Jaroslaw Wroblewski and Christian Boyer in 2007, improved by Elbert Krison in 2017
• (g) Elbert Krison in 2017

• Download the best known examples of pandiagonal multiplicative magic squares from order 3 to order 17 (zipped Excel file of 159Kb)

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