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 [1].

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. [1]

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! [2]

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.

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

Heinz Order 3 - Exponential

The Horizontal planes

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

The constant product of this ratio type multiply magic cube is 4,251,528. Trenkler's product is only 27,000.

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.

This cube is multiply associated and semi-pantriagonal.

Trenkler Order 3

The horizontal planes 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 (Actually, Marián Trenkler illustrates this cube rotated 90 degrees forward in his paper.) [1]
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:

Trenkler Order 4

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 additional 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 magic squares in the cube.  

Trenkler Order 5

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:

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. This illustration is from Sayles paper. 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.

P = 27,000

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     

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) = 2^b13^b24^b35^b47^b59^b6.
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

Add/Multiply magic cubes

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.

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.

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Harvey Heinz   harveyheinz@shaw.ca
This page last updated October 15, 2009
Copyright © 2002, 2003 by Harvey D. Heinz