Magic Cubes - Multiply

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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. 

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.

[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 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 [1]

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! [3]
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.

[3] H. A. Sayles, Geometric Magic Squares and Cubes, The Monist, 23, 1913, pp 631-640

 Add/Multiply magic cubes?

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