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,... For example, the best known 6x6 example with the smallest known product is:

2006: 6x6 pandiagonal multiplicative square,
P = 2 985 984 000 000,
Max nb = 4 800

5	720	160	45	80	1440
4800	12	150	192	300	6
9	400	288	25	144	800
320	180	10	2880	20	90
75	48	2400	3	1200	96
576	100	18	1600	36	50

Smallest known product... but not sure that it is THE smallest possible for 6x6. It has also additional properties of subsquares, "most-perfect square, and 3x3", as summarized in the table below.
 

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	2 985 984 000 000	4 800	most-perfect, and 3x3	(b)	2006
	85 766 121 000 000	4 410	most-perfect, and 3x3
7	8 821 612 800	136			2005
	17 643 225 600	119
8	42 074 422 790 400	546	4x4	(c)	2007
	398 337 730 560 000	528	4x4
9	294 451 250 429 952 000	1 760
10	26 480 706 717 104 640 000	1 617	5x5
	393 968 065 240 189 440 000	850	5x5
11	160 986 670 580 736 000 000	580	also best multiplicative	(b)	2005
	1 441 031 935 035 813 120 000	341	also best multiplicative
12	29 036 169 945 908 051 705 856 000 000	55 440	most-perfect	(c)	2007
13	89 740 731 621 866 637 312 000 000	740	also best multiplicative	(b)	2005
	1 259 162 176 578 813 217 751 040 000	546	also best multiplicative
14	61 242 221 119 253 958 615 896 064 000 000	3 480	7x7	(c)	2007
15	32 030 091 312 206 494 919 248 937 189 376 000 000 000 000 000	1 587 600	3x3 and 5x5	(b)	2006
16	1 865 789 581 791 785 139 922 532 277 874 750 134 681 600 000 000	1 081 080	most-perfect
17	136 260 600 278 658 342 262 589 318 529 024 000 000	1 272	also best multiplicative		2005
	5 170 296 729 462 351 156 714 529 991 349 043 200 000	1 003	also best multiplicative

• (*) 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
• also best multiplicative = these 6 pandiagonal squares are also the best known squares of all multiplicative squares: nobody knowns non-pandiagonal magic squares of orders 11, 13, 17 with smaller products or smaller max nbs
• (***) Author:
• (a) Harry A. Sayles, in 1913
• (b) Christian Boyer, in 2005-2006
• (c) Jaroslaw Wroblewski and Christian Boyer, in 2007

• Download the best known examples of pandiagonal multiplicative magic squares from order 3 to order 17 (zipped Excel file of 102Kb)

Return to the home page http://www.multimagie.com