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 rewrite 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. Addendum2: I posted a new cubeupdate page in February 2010 in which is included news on new order4 multiply cubes with low maximum number and low product. [1] Marián Trenkler, Additive and
Multiplicative Magic Cubes., 6th Summer school on applications of modern math.
methods, TU Košice 2002, 2325
Heinz Order3  Exponential The Horizontal planes
Constant product of this multiply cube is
4,398,046,511,104. Heinz Order3  Ratio
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.
Trenkler  Order3
Trenkler  Order4
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 3dimensional 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 wraparound) 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  Order5
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! [3] Order 3 multiply magic cube
Sayles order 4
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 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. [3] H. A. Sayles, Geometric Magic Squares and Cubes, The Monist, 23, 1913, pp 631640 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 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.
