# Magic Cubes - Multiply         Multiply magic cubes have the same basic requirements as the more familiar additive magic cubes, except that the magic constant is a product instead of a sum. So all rows, columns, pillars, and the 4 triagonals must produce the same magic product. Note this difference however. While it is desirable to use a series of consecutive numbers (and all normal magic squares and cubes do), this is an impossible condition for multiply magic squares, cubes, etc.

This page starts with two order 3 multiply magic cubes constructed by myself using the principles of exponential and ratio multiply magic squares. Then I show an order 3 cube constructed by Marian Trenkler .

These order 3 cubes are all associated, as are all order 3 magic hypercubes. Therefore the 3 central planar squares are also multiply associated magic squares. Notice the difference in magic product size between the 3 types of cubes.

Finally I show an order 4 associative and an order 5 not associative. Both of these were constructed by Marian Trenkler. 

Addendum: After writing and posting this page, I obtained a copy of a paper by Harry A. Sayles on this subject that was published 90 years ago! 

Rather then re-write the entire page, I have simply added material extracted from his paper to the end. His paper covers multiply magic squares quite extensively but he shows only two magic cubes, both constructed using the ratio method.

Sayles uses the term “geometric” when referring to this type of magic object, but does not claim to be the originator of either the name or the type of square or cube.
He demonstrates three methods of construction; exponential, ratio, and factorial.
Trenkler, by contrast, uses two completely different methods; formulae (modular equations) and binary number patterns.

Addendum-2: I posted a new cube-update page in February 2010 in which is included news on new order-4 multiply cubes with low maximum number and low product.

 Marián Trenkler, Additive and Multiplicative Magic Cubes., 6th Summer school on applications of modern math. methods, TU Košice 2002, 23-25
 H. A. Sayles, Geometric Magic Squares and Cubes, The Monist, 23, 1913, pp 631-640

 Heinz Trenkler Sayles Add/Multiply Heinz Special Heinz Order-3 - Exponential

The Horizontal planes

 I II III 2 131072 16777216 6388608 8 65536 262144 4194304 4 32768 524288 256 128 16384 2097152 1048576 512 8192 67108864 64 1024 4096 33554432 32 16 2048 134217728

The generator for both Heinz multiply cubes.

 I II III 1 17 24 23 3 16 18 22 2 15 19 8 7 14 21 20 9 13 26 6 10 12 25 5 4 11 27

The numbers in this additive magic cube (left) were used as powers of 2, to form the multiply cube above.
Actually, this generating cube is a non-normalized (rotated version) of index # 1 of the 4 basic order 3 cubes.

Constant product of this multiply cube is 4,398,046,511,104.
All these order 3 multiply magic cubes are multiply associated so the center horizontal plane, as well as the center vertical planes parallel to the front, and parallel to the sides, are magic squares with the same magic product.
All order 3 magic squares, cubes, tesseracts, etc. are associated. Heinz Order-3 - Ratio

 I II III 1 486 8748 4374 4 243 972 2187 2 324 729 18 9 162 2916 1458 36 81 13122 12 27 108 6561 6 3 54 26244
The constant product of this ratio type multiply magic cube is 4,251,528. Trenkler's product is only 27,000.
This cube is multiply associated and semi-pantriagonal.

This cube uses the index # 1 magic cube shown above (for the exponential generator) for a pattern and these 9 ratio series: 1, 2, 4; 3, 6, 12; 9, 18, 36; 27, 54, 108;…, 6561, 13122, 26244.

Or, to compare directly with the Trenkler order 3 below, these series.

 Horizontal: Plane 1 1 27 729 12 324 8748 18 486 13122 Plane 2 4 108 2916 6 162 4374 9 243 6561 Plane 3 2 54 1458 3 81 2187 36 972 26244  Trenkler - Order-3

A rotated version of this cube is in Trenkler's paper 
 I II III 18 20 75 60 225 2 25 6 180 300 9 10 1 30 900 90 100 3 5 150 36 450 4 15 12 45 50 This cube is multiply associated (but not additive associated). Constant product is 27,000. Compare with the Heinz exponential and Heinz ratio cubes with constants of 4,398,046,511,104 and 4,251,528.

As mentioned previously, because the cube is associated, the 3 central planar squares are also multiply associated magic squares.

The series used for this cube are:

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

 I II 1 840 1080 63 2520 27 21 40 1512 45 35 24 15 56 72 945 1890 36 28 30 12 70 90 756 20 42 54 1260 126 540 420 2 III IV 3780 18 14 60 6 140 180 378 10 84 108 630 252 270 210 4 8 105 135 504 315 216 168 5 189 360 280 3 120 7 9 7560

This is a multiply magic cube so the magic constant is obtained by multiplying the four numbers in each line together. The magic product is 57,153,600 and appears in the required 48 orthogonal rows and the 4 main diagonals.

In addition, all rows of 2 of the 6 oblique squares are correct with all columns correct on the other 4 oblique squares. On each of the oblique squares, 2 of the broken diagonals are correct and in the 3-dimensional space, 11 of the broken triagonals are correct.

Finally, any 2 x 2 array of cells that start on an odd row and an odd column also produces the correct product. So 16 such arrays out of the 64 possible (including wrap-around) in each orthogonal direction have the correct product. There are no correct 2 x 2 arrays in the oblique squares.

This cube is multiply associated (but not additive associated). There are no multiply (and no additive) magic squares in the cube. Trenkler - Order-5

 I II III 99 182 300 408 16 26 540 952 80 33 180 136 144 77 130 1456 75 102 4 792 135 238 20 264 208 34 36 616 1040 45 600 816 1 198 364 1904 5 66 52 1080 9 154 260 360 272 204 8 1584 91 150 40 528 13 270 476 1232 65 90 68 72 2 396 728 1200 51 132 104 2160 119 10 520 720 17 18 308 IV V 680 48 11 234 420 112 55 78 60 1224 12 88 1872 105 170 440 624 15 306 28 22 468 840 1360 3 156 120 2448 7 110 117 210 340 24 176 30 612 56 880 39 1680 85 6 44 936 153 14 220 312 240

This cube is multiply magic because all rows, columns, pillars, and the 4 main triagonals produce the correct product. It is pantriagonal because all broken triagonal pairs produce the magic product. It is not associative. The magic product is 35,286,451,200. There are no magic squares or other special feature in this cube.

Each horizontal plane of the cube consists of 5 series of 5 numbers starting with an odd number then successively doubling it 4 times. The series used in each horizontal plane start with:

 The series used in each horiz. plane start with: For example, plane 1 has the 5 series Plane 1 1 51 75 91 99 1 2 4 8 16 Plane 2 5 13 33 119 135 51 102 204 408 816 Plane 3 9 17 45 65 77 75 150 300 600 1200 Plane 4 3 11 85 105 117 91 182 364 728 1456 Plane 5 7 15 39 55 153 99 198 396 792 1584  Sayles Geometric magic cubes

The following 2 cubes were published by Harry A. Sayles in 1913! 
Both use the ratio method of construction.
The order 3 cube contains 3 order 3 multiply magic squares. This is expected, because it is an odd order associated magic cube.

### Order 3 multiply magic cube

 Following is the cubic series used to construct the multiply cube to the right. The blue numbers are the ratios between the rows, columns and planes. The illustration below is from Sayles paper. P = 27,000 By coincidence (?), Trenkler's order 3 cube uses the same series. However, he used modular equations to design his cube, and made no mention of ratios. Some of the ratios are disguised in my illustration of his series. Sayles order 4

 This series, formed from ratios shown by the blue numbers, was used by Sayles to construct the following order 4 multiply magic cube. The cube was simply constructed by exchanging each of the violet numbers (above) with it's complement. The complement of each number in this series is the difference between the number and 7561 (which is 1 + 7560, the first and last numbers in the series).

Order 4 multiply (geometric) cube

 I II III IV 7560 2 5 756 7 540 216 70 9 420 168 90 120 126 315 12 3 1260 504 30 360 42 105 36 280 54 135 28 189 20 8 1890 4 945 378 40 270 56 140 27 210 72 180 21 252 15 6 2520 630 24 60 63 84 45 18 840 108 35 14 1080 10 1512 3780 1

This cube is associated. It contains no multiply magic squares. P = 57,153,600
As with the order 3 cube, Trenkler's order 4 uses the same series as used here by Sayles. However, again no mention is made of ratios. He does show this equation:
q4 (i, j, k) = 2b13b24b35b47b59b6.

In both the Sayles and Trenkler order 4 cubes, the product of the 4 cells in each quadrant of each plane is also magic i.e. equals 57,153,600.

 H. A. Sayles, Geometric Magic Squares and Cubes, The Monist, 23, 1913, pp 631-640  17 171 126 54 230 100 93 264 145 124 66 290 85 57 168 162 23 225 216 115 75 279 198 29 170 76 42 261 186 33 210 68 38 200 135 69 50 270 92 87 248 165 21 153 114 105 51 152 150 27 207 116 62 330 138 25 243 132 58 310 95 63 136 190 84 34 184 125 81 297 174 31 99 232 155 19 189 102 46 250 108

This is a regular magic square that produces a constant sum, but with the added property that all lines have a constant product when the numbers are multiplied instead of added.
The magic sum is of this square is 1,200, and the magic product is 1,619,541,385,529,760,000. It was constructed in 1962 by Gakuho Abe in Japan.

It should be possible to construct a magic cube with these characteristics, also. However, I am unaware of any such cube being constructed. Anyone wish to try?  Heinz special Multiply cube 2002

This cube is not add or multiply magic in the normal sense. Neither is it associated
Rather, it is the basis for an amusing parlour trick

Pick 3 numbers that are not in mutual rows, columns, pillars or planes. When you multiply them together you always get 510,510. For example: the numbers 154, 13, and 255 are all in different rows, columns, pillars and planes. Multiplied together they give the product 510,510.

 I II III 154 22 110 77 11 55 231 33 165 238 34 170 119 17 85 357 51 255 182 26 130 91 13 62 273 39 195

This is an extension of Martin Gardner’s magic square puzzle. I constructed it by putting the numbers 11, 17 and 13 in the center column of the center plane. Then multiply this column by 5 and 7 to get the two outside columns of this plane. Then multiply this plane by 2 and 3 to get the third and first planes. By using primes for these key numbers, there is no risk of duplicate numbers appearing when forming the cube.      This page was originally posted February 2003 It was last updated March 02, 2010 Harvey Heinz   harveyheinz@shaw.ca Copyright © 1998-2009 by Harvey D. Heinz