Magic Cubes  Order 3

The Four Basic CubesIt 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 reflections. 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! 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.
[1] John R. Hendricks, The ThirdOrder Magic Cube Complete, Journal of Recreational Mathematics 5:1:1972, pp 4350 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. 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 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 #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 Nonstandard 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. Associated Magic CubesA magic cube is called normal if it consists of the numbers 1 to m^{3} (or 0 to m^{3} – 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).
[2]
Horizontal plane
1 – top Vertical plane 1 – back Vertical plane 1 – left
Horizontal plane
2 Vertical plane 2 Vertical plane 2
Horizontal plane
3 –bottom Vertical plane 3 front Vertical plane 3 – right Note the following regarding the above magic cube.
[1]
H.D. Heinz and
J.R. Hendricks, Magic Square Lexicon: Illustrated, Selfpublished,
2000, 0968798500 An Early Cube
