Multiplicative magic cubes

• In another page:

What are multiplicative magic cubes?

Multiplicative magic cubes are cubes which are magic using multiplication instead of addition. The numbers used can't be consecutive, but they must be distinct. A reminder of some definitions, similar to additive cubes:

• a semi-magic cube has all rows, columns and pillars magic
• a magic cube has all rows, columns, pillars and triagonals (=the 4 space diagonals) magic
• a perfect magic cube has all rows, columns, pillars, triagonals and plane diagonals (=diagonals of squares parallel to faces) magic
• a pandiagonal perfect magic cube has all rows, columns, pillars, triagonals, plane diagonals, broken triagonals and broken diagonals magic

It seems that the first multiplicative magic cubes were published in 1913 by Harry A. Sayles in The Monist in 1913. This paper was republished in the book Magic squares and cubes by W.S. Andrews, where we can find (page 293) two cubes:

• a 3rd-order cube, with magic product = P = 27000, and with maximum number used = Max nb = 900
• a 4th-order cube, P = 57153600, Max nb = 7560
 1 90 300 150 4 45 180 75 2 60 25 18 9 30 100 50 36 15 450 12 5 20 225 6 3 10 900

(click on the image to enlarge the cube)

In 2002, Marián Trenkler, Slovakia, published a paper on multiplicative magic squares and cubes in Obzory Matematiky, Fyziky a Informatiky 1/2002 (31), pages 9-16, with:

• a 3rd-order cube, P = 27000, Max nb = 900 (same set of 27 numbers as Sayles's cube, but differently placed)
• a 4th-order cube, P = 57153600, Max nb = 7560 (same set of 64 numbers as Sayles's cube, but differently placed)
• a 5th-order cube, P = 35286451200, Max nb = 2448

Is it possible to do better? Cube of the same orders, but using smaller products P, or smaller Max nb? Yes, both for the orders 4 and 5!

After multiplicative magic squares, here are the best known cubes, 3rd-order to 11th-order. The goal is always to minimize the magic product and the max nb.

 Order Magic product P Max nb Multiplicative cube Comments 3 (c) 6 720 160 semi-magic (c) 15 120 105 (a) 27 000 900 magic The best known magic cube (of any order) using the smallest P 216 000 400 4 (b) 4 324 320 351 magic The best known magic cubes (of any order) using the smallest Max nb 5 (b) 13 967 553 600 855 magic + magic broken triagonals (b) 101 625 502 003 200 000 2 976 750 perfect magic The best known perfect magic cube (of any order)  using the smallest P (b) 104 064 514 051 276 800 000 250 880 6 115 651 343 808 000 3 300 semi-magic 223 592 598 028 800 2 262 (b) 117 327 450 240 000 5 225 magic (b) 23 959 607 303 503 872 000 849 420 perfect magic (b) 80 863 674 649 325 568 000 000 66 924 7 (b) 897 612 484 786 617 600 3 367 magic + magic broken triagonals (b) 19 407 837 508 899 840 000 2 912 + magic broken triagonals 1 411 407 979 783 492 239 360 000 000 1 259 712 perfect magic 5 750 476 043 814 094 602 240 000 000 862 400 8 (c) 13 248 760 275 450 475 776 000 4 797 magic 89 518 183 823 250 314 294 722 560 000 17 297 280 pandiag. perfect magic (*) The best known pandiag. perfect magic cube (of any order) using the smallest P 9 (c) 321 308 934 200 224 938 519 552 000 7 191 magic 265 237 261 271 449 982 022 984 892 416 000 591 192 pandiag. perfect magic (*) 10 117 218 854 345 145 394 654 241 228 800 000 22 125 semi-magic 602 839 822 346 462 029 650 383 462 400 000 17 400 (c) 2 108 555 793 636 163 874 695 070 438 860 800 000 568 875 magic (c) 23 827 892 281 763 155 326 511 017 263 976 960 000 511 270 (b) ~  2.62 E+51 1 920 996 000 perfect magic (b) ~  3.52 E+59 9 018 009 000 pandiag. perfect magic (*) 11 (c) 386 505 025 119 121 838 921 061 678 274 867 200 000 20 801 magic 9 009 441 144 967 875 033 124 980 845 568 000 000 46 620 pandiag. perfect magic (*) 174 930 251 129 029 312 377 859 321 968 844 800 000 24 992 The best known perfect magic cube (of any order), andthe 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 (a) first found in 1913 by Harry Sayles, (b) in 2010-2012-2013 by Toshihiro Shirakawa, (c) in 2017 by Elbert Krison
• (*) See the page on pandiagonal perfect multiplicative magic cubes
• If you succeed in getting smaller products (or smaller max numbers), send me a message! Your results will be added to this website.
• Download the best multiplicative cubes, orders 3 to 7 (Excel file of 146Kb)
• Download the best multiplicative cubes, orders 8 to 11 (Excel file of 276Kb)
 Many thanks to Edwin Clark, Mathematics Department of the University of South Florida, USA, for checking in 2006 all my multiplicative cubes, confirming that they have all the announced properties.

3rd-order multiplicative magic cubes

It is easy to prove that any 3rd-order multiplicative magic cube must have P = (center)^3. With two different possible constructions using the integer 1

 1 abc² a²b²c a a²bc² b²c ab²c a² bc² a²b²c 1 abc² a²bc² b²c a bc² ab²c a² a²bc b² ac² a²bc b² ac² c² abc a²b² c² abc a²b² ab² a²c² bc ab² a²c² bc ab²c² a²c b b²c² ac a²b a²b b²c² ac ab a²b²c² c c ab a²b²c² a²c b ab²c²

we can produce exactly 4 different 3rd-order multiplicative magic cubes with the same P = (2·3·5)^3 = 27000: one cube from the first construction, and three from the second construction. They use the same set of integers, but they are "different" because from any of these 4 cubes, it is impossible to produce any other ones, using symmetries and rotations.

 1 150 180 2 300 45 3 450 20 5 450 12 90 4 75 180 1 150 180 1 150 300 1 90 300 45 2 75 90 4 50 60 9 18 60 25 60 9 50 60 9 50 90 4 75 150 4 45 25 30 36 25 30 36 25 30 36 9 30 100 18 100 15 18 100 15 12 225 10 20 225 6 450 20 3 225 10 12 100 15 18 36 15 50 12 225 10 6 900 5 6 900 5 10 900 3 5 6 900 20 3 450 45 2 300 75 2 180

It is impossible to construct better 3rd-order cubes with a smaller P. But, using Min nb > 1, it is possible to construct 3rd-order cubes with a smaller Max nb (and bigger P), the smallest possible Max nb being 400. Here are some examples with Max nb < 900:

 90 80 30 320 150 36 200 192 45 240 9 100 180 32 300 288 25 240 10 300 72 30 360 160 30 360 160 48 225 20 60 288 100 96 225 80 25 60 144 200 120 72 100 120 144 180 16 75 144 50 240 180 64 150 50 12 360 90 40 480 90 40 480 36 400 15 48 450 80 60 576 50 120 45 40 400 96 45 320 75 72

And it is possible to construct 3rd-order semi-magic cubes using smaller constants. In the three examples below, all the rows, columns and pillars have the same magic product P = 7560, but some of their 4 triagonals do not have the same product: the example on the right is a very nearly magic cube, only one triagonal is incorrect!!!

 1 56 135 1 42 180 1 30 252 72 15 7 54 20 7 90 28 3 105 9 8 140 9 6 84 9 10 84 45 2 84 45 2 60 63 2 5 14 108 5 14 108 7 6 180 18 12 35 18 12 35 18 20 21 90 3 28 90 4 21 126 4 15 21 36 10 28 27 10 12 45 14 4 70 27 3 70 36 5 42 36

In February 2013, André LFS Bacci, Brasilia, Brazil, constructed this semi-magic cube with a smaller P:

 21 20 16 32 42 5 10 8 84 2 56 60 15 4 112 224 30 1 160 6 7 14 40 12 3 28 80

In June 2017, I was surprised to receive still better 3rd-order semi-magic cubes! From Elbert Krison, a 15-year-old schoolboy of Jakarta, Indonesia. Elbert sent also new best known magic cubes of orders 8 to 11 listed in the table.

 3 28 80 8 105 18 160 6 7 21 48 15 14 40 12 90 3 56 16 5 84 30 6 84 21 32 10 12 35 36 20 42 8 42 72 5 140 48 1 63 24 10 2 35 96 60 9 28 24 4 70 4 70 54

And what about adding magic plane diagonals? Both for additive and multiplicative magic cubes, it is impossible to construct 3rd-order perfect magic cubes.

4th-order multiplicative magic cubes

Sayles's cube and Trenkler's cube have the same characteristics: P = 57153600, Max nb = 7560. Is it possible to construct 4th-order cubes with smaller constants? Yes! My best cubes have a P more than 8 times smaller, and a Max nb more than 20 times smaller:

4th-order multiplicative magic cubes, by Christian Boyer
P = 6,486,480 and Max nb = 546 (left cube, January 2006),
P = 17,297,280 and
Max nb = 364 (right cube, June 2007)
(click on the image to enlarge a cube)

Some plane diagonals of my two cubes are magic, but not all of them: both for additive and multiplicative magic cubes, it is impossible to construct 4th-order perfect magic cubes.

These were my two best cubes, but not sure I found the best possible cubes. That's why I asked these next questions. Who will be able to construct better 4th-order cubes (with smaller P or smaller Max nb)?  Is it possible to construct a multiplicative magic cube (of any order!) using integers < 364? Of any order, because we can notice that the Max nb = 364 of the 4th-order cube on the right is smaller than the Max nb = 400 used for the best possible 3rd-order cube.

In July 2008, Michael Quist worked on this problem, and found that any 4th-order cube must have Max nb ≥ 221 and that any 5th-order (or higher) cube must have Max nb ≥ 442. If we suppose that his work is correct (read it here), and if we suppose that any 3rd-order cube must have Max nb ≥ 400, then the enigma is limited to the 4th-order, and is equivalent to: is it possible to construct a multiplicative magic cube, of 4th order, and having 221 ≤ Max nb < 364?

In January 2010, Max Alekseyev, Dept of Computer Science & Engineering, University of South Carolina, found another 4th-order magic cube having the same Max nb = 364 as my above cube, but with a smaller P. None of its 24 small diagonals are magic (8 small diagonals are magic in my cube), but this is not a problem because not needed in a magic cube: only rows + columns + pillars + 4 triagonals have to be magic. An excellent cube constructed by Max!

 1 110 224 351 130 8 297 28 308 27 13 80 216 364 10 11 231 12 78 40 96 273 5 66 6 55 168 156 65 48 132 21 144 91 15 44 77 18 52 120 195 32 198 7 4 165 56 234 260 72 33 14 9 220 112 39 24 182 20 99 154 3 117 160

Toshiro Shirakawa in 2010 (Kuwana, Japan, 1983 - )

And finally in April 2010, Toshihiro Shirakawa solved my enigma #5 with this excellent cube using smaller integers: Max nb = 351 < 364. The magic product is also smaller than the above cubes. Congratulations!!! Toshihiro lives in Ebina Kanagawa, Japan. He is a programmer, and does mathematics as a hobby.

April 2010. Best known 4th-order multiplicative magic cube, by Toshihiro Shirakawa
and best known cube -of any order- using the smallest possible integers
P = 4,324,320 and Max nb = 351
(click on the image to enlarge it)

His method is easy and ingenious, without any computing. He used, directly from this website, my own 4x4 multiplicative semi-magic square having the smallest possible product that I constructed in 2005, with magic product 4320 = 25 * 33 * 5 and Max nb 27:

 16 1 10 27 = 1A 1B 1C 1D 5 24 18 2 2A 2B 2C 2D 6 12 3 20 3A 3B 3C 3D 9 15 8 4 4A 4B 4C 4D

Noticing that this square used only factors 2, 3 and 5, he cleverly reused it, combined with the four factors (E, F, G, H) = (1, 11, 7, 13). That's why his magic product is 4320(from the square) *7*11*13(his factors) = 4324320, and his Max nb is 27(from the square) * 13(his max factor) = 351. And 351 is less than 364, as asked in the enigma. That's it! Er... shame on me... to have been unable to think to this ingenious method... using my own square...

 1A 1B 1C 1D * E F G H 1B 1A 1D 1C H G F E 1C 1D 1A 1B F E H G 1D 1C 1B 1A G H E F 2A 2B 2C 2D F E H G 2B 2A 2D 2C G H E F 2C 2D 2A 2B E F G H 2D 2C 2B 2A H G F E 3A 3B 3C 3D G H E F 3B 3A 3D 3C F E H G 3C 3D 3A 3B H G F E 3D 3C 3B 3A E F G H 4A 4B 4C 4D H G F E 4B 4A 4D 4C E F G H 4C 4D 4A 4B G H E F 4D 4C 4B 4A F E H G

I can now give the method used for my cube (P, MaxNb) = (17297280, 364) which was challenged with enigma #5. This cube was constructed from the Eulerian (or Graeco-Latin) cube below, using the best possible set (A, B, C, D) (I, J, K, L) (a, b, c, d) generating 64 distinct integers and generating the smallest MaxNb:

• (A, B, C, D) = (1, 2, 3, 4)
• (I, J, K, L) = (1, 5, 6, 7)
• (a, b, c, d) = (3, 8, 11, 13)

 D C A B * I L J K * d b a c B A C D K J L I a c d b C D B A L I K J c a b d A B D C J K I L b d c a B A C D L I K J a c d b D C A B J K I L d b a c A B D C I L J K b d c a C D B A K J L I c a b d C D B A J K I L c a b d A B D C L I K J b d c a D C A B K J L I d b a c B A C D I L J K a c d b A B D C K J L I b d c a C D B A I L J K c a b d B A C D J K I L a c d b D C A B L I K J d b a c

My other cube (P, MaxNb) = (6486480, 546) used a similar Eulerian cube, but with the best possible set generating 64 distinct integers and generating the smallest product: (1, 2, 3, 6), (1, 4, 5, 7), (1, 9, 11, 13).

5th-order multiplicative magic cubes

Trenkler's cube published in 2002 has P = 35286451200, Max nb = 2448. Is it possible to construct 5th-order cubes with smaller constants? Yes! My best cube has a P more than 2 times smaller, and a Max nb more than 2 times smaller:

January 2006: 5th-order multiplicative magic cube, by Christian Boyer
P = 16,761,064,320 and Max nb = 1026
(click on the image to enlarge it)

All its rows, columns, pillars and 4 triagonals are magic. And this cube has a very nice additional property: all its broken triagonals are magic, as the example in blue.

In May 2010, Toshihiro Shirakawa constructed a 5th-order multiplicative magic cube with smaller characteristics than my above cube: P = 13,967,553,600 and MaxNb = 855. Also with magic broken triagonals. This is today the best known 5th-order magic cube! You can see it in the downloadable Excel file, below the table. Similarly to his 4th-order method, he used my 5th-order multiplicative magic square (reproduced below), combined with (1, 11, 13, 17, 19). My above cube was an Eulerian cube (1, 2, 3, 4, 6) (1, 5, 7, 8, 9) (1, 11, 13, 17, 19).

 12 35 1 40 18 36 2 24 7 25 14 45 15 4 8 5 16 42 30 3 10 6 20 9 28

Unfortunately the plane diagonals of our two cubes are not magic: they are not perfect magic cubes. However, it is possible to construct 5th-order perfect magic cubes: use the first known 5th-order (additive) perfect magic cube constructed by Walter Trump and me in 2003, and replace each number n by 2^(n-1). Then you will get a multiplicative perfect magic cube... but tedious... using very big numbers: Max nb = 2^(125-1) = 2.13 · 10^37, and P = 2^310 = 2.09 · 10^93. In December 2012, Toshihiro Shirakawa constructed two far better 5th-order perfect magic cubes: one with Max nb = 250 880, one with P = 2520^5 = 1.01 · 10^17. Astonishingly, but with a larger order, perfect cubes using smaller integers are known: see my 11th-order pandiagonal perfect cube with Max nb = 24 992 < 250 880.

6th-order multiplicative magic cubes

In January 2006, I constructed two cubes:

• P = 115 651 343 808 000, Max nb = 3 300
• P = 223 592 598 028 800, Max nb = 2 262

These cubes are only semi-magic: their 4 triagonals are not magic.

Later, in May 2006, I constructed a magic cube, this time with 4 magic triagonals. This added feature has a cost, bigger P and Max nb than the previous semi-magic cubes:

• P = 170 400 029 184 000 000, Max nb = 554 400

Then in June 2010, Toshihiro Shirakawa constructed a magic cube with much smaller characteristics than my poor above cube:

• P = 117 327 450 240 000, Max nb = 5 225

This is today the best known 6th-order magic cube!

It is possible to construct 6th-order perfect magic cubes: use the first known 6th-order (additive) perfect magic cube constructed by Walter Trump in 2003, and replace each number n by 2^(n-1). Then you will get a multiplicative perfect magic cube... but tedious... using very big numbers: Max nb = 2^(216-1) = 5.27 · 10^64, and P = 2^645 = 1.46 · 10^194. In December 2012 and February 2013, Toshihiro Shirakawa constructed two far better 6th-order perfect magic cubes: one with Max nb = 66 924, one with P = 2 882 880^3 = 2.39 · 10^19. Astonishingly, but with a larger order, perfect cubes using smaller integers are known: see my 11th-order pandiagonal perfect cube with Max nb = 24 992 < 66 924.

7th-order multiplicative magic cubes

In January 2006, my two best magic cubes were:

• P = 1 663 528 929 334 272 000, Max nb = 5 952
• P = 3 881 567 501 779 968 000, Max nb = 4 352

But in May 2010, Toshihiro Shirakawa constructed better cubes:

• P = 897 612 484 786 617 600, Max nb = 3 367
• P = 19 407 837 508 899 840 000, Max nb = 2 912

As the above 5th-order magic cubes, all their broken triagonals and 4 entire triagonals are magic, but their plane diagonals are not magic: they are not perfect magic cubes. With the same P and Max nbs, I have successfully constructed cubes with all their magic plane diagonals... but unfortunately loosing 2 magic triagonals on the 4 triagonals.

Keeping magic plane diagonals, I have constructed -still in January 2006- two cubes with 3 magic triagonals (now, only one triagonal is bad!) with the cost of using bigger characteristics:

• P = 16 579 776 648 314 880 000, Max nb = 24 000
• P = 101 059 382 457 057 024 000, Max nb = 9 486

and finally succeeded in constructing perfect magic cubes with 4 magic triagonals, but with again bigger characteristics. To get the 4th triagonal is very expensive!

• P = 1 411 407 979 783 492 239 360 000 000, Max nb = 1 259 712
• P = 5 750 476 043 814 094 602 240 000 000, Max nb = 862 400

8th to 11th-order multiplicative magic cubes

See the summary in the table at the beginning of this page. And see the page on the pandiagonal perfect multiplicative magic cubes.

To celebrate January 2006 when the cubes were created, the two first numbers used in my 10th and 11th-order cubes are: "2006" and "1"!