Pandiagonal perfect multiplicative magic cubes

What are pandiagonal perfect multiplicative magic cubes?

As explained in the multiplicative magic cubes page, a pandiagonal perfect magic cube has all its lines (rows, columns, pillars, triagonals, plane diagonals, broken triagonals and broken diagonals) magic. It is a perfect magic cube that remains perfect magic when any face of the cube is moved parallel to itself, from its side to the opposite side of the cube. They are the BEST possible cubes: any line, entire or broken, in any direction gives always the same product!

Because a cell in a cube has 3^3 - 1 = 26 adjacent cells, a pandiagonal perfect multiplicative magic cube of order n has all its 13n² lines which are magic.

My 4 best pandiagonal perfect multiplicative magic cubes, and Shirakawa's cube, extracted from the general table, are:

 Order Magic product P Max nb Comments 8 89518183823250314294722560000 (~ 8.95 E+28) 17 297 280 The best known pandiag. perfect magic cube (of any order) using the smallest P 9 265237261271449982022984892416000 (~ 2.65 E+32) 591 192 The best known 9th-order pandiag. perfect magic cube 10 (*) 352180956945527724652234727828937579440100000000000000000000 (~  3.52 E+59) 9 018 009 000 The best known 10th-order pandiag. perfect magic cube 11 9009441144967875033124980845568000000 (~ 9.01 E+36) 46 620 The best known 11th-order pandiag. perfect magic cube using the smallest P 174930251129029312377859321968844800000 (~ 1.75 E+38) 24 992 The best known pandiag. perfect magic cube (of any order) using the smallest Max nb
• Cubes of this table were first found in 2006 by Christian Boyer, except (*) first found in 2013 by Toshihiro Shirakawa
• If you succeed to get smaller products (or smaller max numbers), send me a message! Your results will be added in this website.

• Download the multiplicative cubes, orders 8 to 11 (Excel file of 201Kb)

Thanks again to Edwin Clark, Mathematics Department of the University of South Florida, USA, for checking in 2006 all my pandiagonal perfect multiplicative magic cubes, confirming that they have all the announced properties.

What is the smallest possible magic product P?
(of pandiagonal perfect multiplicative magic cubes)

This problem is also a factorization problem in 3 dimensions:

What is the smallest composite integer P from which we can pick and organize distinct factors in a multiplicative cube? The result of the multiplication of these numbers in any line of any direction (including any broken diagonal and broken triagonal) has always to be equal to this integer P. And what is the size (=order) of this cube?

>> In 2 dimensions (squares), the answer is known. The smallest possible integer is 14400, organizing distinct factors in a 4th-order square that way:

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

When we multiply the integers of any row, column, or diagonal (including all broken diagonals), we get always 14400. There are 32 different ways to get 14400.

>> In 3 dimensions (cubes), the question is an open problem. I do not have the answer, but my smallest composite integer is:

89 518 183 823 250 314 294 722 560 000

organizing 512 of its factors in a 8th-order magic cube. Because it is a 8th-order pandiagonal perfect cube, there are 13·8² = 832 different ways to get the same product P.

The best known pandiagonal perfect magic cube
using smallest possible magic product P:
multiplying integers of ANY possible line
(entire or broken) give always P.
This 8th-order cube contains 512 distinct factors of
P = 89518183823250314294722560000.
The maximum used factor is 17 297 280, called Max nb.
(click on the image to enlarge it)

 576576 38610 87360 26 576 1890 10560 154 24 3024 3960 6160 24024 61776 32760 1040 2002 82368 24570 12480 2 4032 2970 73920 80 168 4752 27720 80080 3432 39312 4680 960960 286 52416 3510 960 14 6336 20790 360 560 264 33264 360360 11440 2184 5616 270270 137280 182 7488 270 6720 22 44352 432 2520 880 1848 432432 51480 7280 312 21120 77 1153152 19305 174720 13 1152 945 16380 2080 12 6048 1980 12320 12012 123552 1485 147840 1001 164736 12285 24960 1 8064 78624 2340 160 84 9504 13860 160160 1716 12672 10395 1921920 143 104832 1755 1920 7 1092 11232 180 1120 132 66528 180180 22880 11 88704 135135 274560 91 14976 135 13440 14560 156 864 1260 1760 924 864864 25740

What is the smallest possible maximum number?
(of pandiagonal perfect multiplicative magic cubes)

My best result is 24 992, used in a 11th-order cube.

It means that the numbers used in this cube are smaller than the numbers used in smaller cubes! The smallest known Max nbs for 8th-order and 9th-order are bigger: 17 297 280 and 591 192.

Because it is a 11th-order pandiagonal perfect cube, there are 13·11² = 1573 different ways to get the same product P.

 2006 1 3219 12000 3116 7137 6880 20770 7614 17963 5194 16740 5535 14030 3612 12529 329 4118 1325 3363 104 4736 7525 2546 611 6816 13144 6372 207 2590 10200 246 19459 3976 8109 295 290 5550 12540 3731 3904 1333 5427 8648 352 8029 3240 943 2562 5848 268 12267 8875 10070 4602 9512 6100 7353 4355 15040 13206 15741 9499 28 629 180 9246 7238 8449 106 1711 75 5624 3120 11808 9455 11610 2067 15104 124 8991 6900 5740 6222 473 13601 2350 1349 370 10440 11275 8113 1118 2144 4371 15336 4876 7434 85 1647 2967 7504 3196 639 7685 14750 114 5291 13440 2542 4465 9230 10176 20119 189 1702 840 2091 488 4988 15075 11224 2408 10251 235 20590 7950 12331 91 2368 1860 3321 3666 24992 11501 3186 23 1554 8160 164 15921 5375 12730 177 232 3700 10260 2665 19520 7998 19899 7567 1988 901 16200 5658 9394 5117 134 1363 5325 8056 3068 288 5735 817 2613 12032 8804 12879 6785 140 3774 660 8323 3050 6035 530 10266 275 4921 1560 1312 5673 9288 6164 5922 62 999 4140 4592 4148 387 9715 11750 8094 7579 13216 9225 5795 5590 12864 16027 13419 2438 826 51 296 6960 6566 1598 71 4611 11800 76 4329 9600 12710 9882 10879 6784 16461 135 8510 5040 7667 427 2494 1675 2679 7384 11803 10500 1558 793 4128 16616 5076 14697 3710 10030 6