Magic Cubes - Order 3

The Four Basic Cubes	What is considered the standard position
Magic Cube Variations (Aspects)	There are 48 variations due to rotations and reflections
Associated Magic Cubes	The central planes of an order 3 hypercube are always associated
An Early Cube	Emile Fourrey constructed this magic cube in 1899.

The Four Basic Cubes

It was proven in 1972 [1] that there are four basic magic cubes of order 3. Each one may be shown in 47 other variations due to rotations or reflections. These are called basic cubes because no one of them may be transformed to another one by rotations and reflects. Also they are in the standard position with the lowest corner in the bottom left position, and the 3 numbers adjacent to that corner are in increasing order in the x, y and z directions.

Notice that a vertical plane parallel to sides, of each of the last 3 cubes appear as center planes in the first cube!
Notice that the center horizontal planes of the first three cubes all appear as center planes in the fourth cube!
Notice that each face of the cube has one odd numbered corner and three even numbered corners.

The set of 4 numbers below each illustration uniquely identifies that cube. Also, the cube can be reconstructed from these numbers, with the aid of the center number of the cube, which is always 14.

To obtain a rotated version of one of these cubes, just assign a different corner of the cube to the front bottom left position. Then place the 3 adjacent numbers in adjacent x, y and z positions, and put the 14 in the center of the cube. Finally subtract the total of the two numbers in a row from 42, to obtain the third number until the new cube is completed.

# 1: 1, 15, 17, 23

# 2: 2, 15, 18, 24

# 3: 4, 17, 18, 26

# 4: 6, 16, 17, 26

[1] John R. Hendricks, The Third-Order Magic Cube Complete, Journal of Recreational Mathematics 5:1:1972, pp 43-50

Magic Cube Variations (Aspects)

There are 48 variations, or aspects, of a magic cube of any order. These variations are due to rotations and/or reflections. Usually, these are not considered to be different cubes. In comparison, there are 8 aspect for each magic square, and 384 aspect for each magic tesseract, regardless of the order.

In 1908 W. S. Andrews published the first edition of his Magic Squares and Cubes [1]. In this book he presented four order 3 magic cubes. It turns out that they are different aspects of the four basic cubes presented above. These cubes appear on pages 66 and 69 . However, he seemed unaware that these were the only basic cubes of order 3. In fact, he shows two aspects for each of numbers 2, 3, and 4, seemingly unaware that they were variations on the basic cubes.
However, later [2, page 364] Dr. Planck established that there are 192 variations of order 3 magic cubes (4 [basic] x 48 [aspects]).

Here I show the above basic cubes in text format. Then I will show the 4 Andrews cubes in the same format. Compare Andrews cubes with the basic cubes to get a feeling for aspect (a disguised version).

Standard position
The above four cubes represented in text format. Horizontal planes: Top, middle, bottom

#1             #2             #3             #4
 2 13 27       03 13 26       07 11 24       08 12 22  Top
22 09 11       23 09 10       23 09 10       24 07 11
18 20 04       16 20 06       12 22 08       10 23 09

16 21 05       17 21 04       15 25 02       15 25 02  middle
03 14 25       01 14 27       01 14 27       01 14 27
23 07 12       24 07 11       26 03 13       26 03 13

24 08 10       22 08 12       20 06 16       19 05 18  bottom     
17 19 06       18 19 05       18 19 05       17 21 04
01 15 26       02 15 25       04 17 21       06 16 20

Non-standard positions as shown in Andrews book [1][2]

#1             #2             #3             #4
01 17 24       02 24 16       04 26 12       10 24 08
15 19 08       18 01 23       18 01 23       26 01 15
26 06 10       22 17 03       20 15 07       06 17 19

23 03 16       15 07 20       17 03 22       23 07 12
07 14 21       19 14 09       19 14 09       03 14 25
12 25 05       08 21 13       06 25 11       16 21 05

18 22 02       25 11 06       21 13 08       09 11 22
20 09 13       05 27 10       05 27 10       13 27 02
04 11 27       12 04 26       16 02 24       20 04 18

[1] W. S. Andrews, Magic Squares & Cubes, Open Court, 1908, p.69.
[2] W. S. Andrews, Magic Squares & Cubes, 2^nd edition, Dover Publ. 1960, p.69.
This is an unaltered reprint of the 1917 Open Court Publication of the second (1917) edition.

Associated Magic Cubes

A magic cube is called normal if it consists of the numbers 1 to m³ (or 0 to m³ – 1).

A magic cube is called associated if all pairs of two numbers diametrically equidistant from the center of the cube equal the sum of the first and last number in the series. If the associated cube (or other dimension of hypercube) is an odd order, then the center of the cube is a cell containing one half the sum of the first and last number in the series.

To illustrate both these points, I present the middle cube from an order 3 magic tesseract (dimension 4 hypercube).

Horizontal plane 1 – top     Vertical plane 1 – back    Vertical plane 1 – left
                  69                         33                         39
   36   67   20 123          36   67   20 123          36   65   22 123
   65   27   31 123          77    3   43 123          77    7   39 123
   22   29   72 123          10   53   60 123          10   51   62 123
123 123 123 135         123 123 123   99         123 123 123 105

Horizontal plane 2           Vertical plane 2           Vertical plane 2
                 123                        123                        123
   77    3   43 123          65   27   31 123          67   27   29 123
    7   41   75 123           7   41   75 123           3   41   79 123
   39   79    5 123          51   55   17 123          53   55   15 123
123 123 123 123         123 123 123 123         123 123 123 123

Horizontal plane 3 –bottom   Vertical plane 3- front    Vertical plane 3 – right
                 177                        213                        207
   10   53   60 123          22   29   72 123          20   31   72 123
   51   55   17 123          39   79    5 123          43   75    5 123
   62   15   46 123          62   15   46 123          60   17   46 123
123 123 123 111         123 123 123 147         123 123 123 141

Note the following regarding the above magic cube.

Not normal magic
First, note that the above cube contains scattered numbers from 5 to 77, so it is not a normal magic cube. However it is magic because all orthogonal rows and the 4 triagonals sum the same.

Associated
The 3 central orthogonal planes are magic squares because the diagonals also sum correctly. If you enlarge the tesseract to the right and examine it, you will find the other 3 central hyperplanes are also magic cubes. Both of these cases are because any order 3 hypercube and the central planes within it are associated. (If you have a problem seeing the fourth central cube in the tesseract, the corners are 8, 37, 6, 73, 76, 9, 74 and 45.)

Now check that the above magic cube is associated by confirming that pairs of numbers diametrically equidistant either side of center, sum to the 1st plus last number of the series ( 77 + 5). Check that the center number of the cube is 77 + 5/2. And finally check that the magic squares in the cube are also associated.

The above is the middle horizontal cube of this tesseract. [1]
(Click to enlarge)

[1] H.D. Heinz and J.R. Hendricks, Magic Square Lexicon: Illustrated, Self-published, 2000, 0-9687985-0-0
[2] John R. Hendricks, All Third-Order Magic Tesseracts, self-published, 1999, 0-9684700-2-5, 36+ pages.

An Early Cube

Emile Fourrey [1] constructed this magic cube in 1899, using a method proposed by Joseph Sauveur [2] in 1710. It is not magic by today's standards because rows and columns do not sum to the constant 42. The 4 main triagonals are correct. All pantriagonals are correct in one of the four directions.
Diagonals are correct on all planar squares. All broken diagonals in one direction are correct on the horizontal planar squares and the 6 oblique squares. They are correct in both directions in the 6 vertical planar squares.

The total of all 9 cells in each of the 9 orthogonal planes and the 6 oblique planes sum to 126 (or 3 times 42).

I have illustrated the cube as Fourrey did in his book.[1]

[1] E. Fourrey, Recreations arithmetiques, (Arithmetical Recreations) 8^th edition, Vuibert, 2001. Originally published in 1899. (French).
[2] More on Joseph Sauveur on my Early Cubes page.