Magic Cubes - Order 4

Order 3 has only one type of magic cube. It is simple but has the extra feature of being associated.

The next larger order is 4 and has cubes with a variety of features. Of most importance, probably, is the introduction of a major class of magic cubes, the pantriagonal cube.

Simple, associated or not	Schubert's simple associated order 4 cube was published in 1898. Weidemann's simple cube, published in 1922, is not associated.
Pantriagonal	John Hendricks introduced the name pan-3-agonal in several JRM articles in 1972, and it's mathematical relationship to pandiagonal magic squares in 1980.
More Order 4 Cubes	Royal Heath published a pantriagonal cube in 1930. Kanji Setsuda has produced many pantriagonal cubes with different features.
Magic Squares to Cubes	Adler and Shuo-yen demonstrate conversion between magic squares and cubes.

Simple and associated

[1]Hermann Schubert, Mathematical Essays and Recreations. Translated from German to English by Thomas J. McCormack, Open Court, 1899.
[2] W. S. Andrews, Magic Squares & Cubes, Open Court, 1908.
[3] W. S. Andrews, Magic Squares & Cubes, 2^nd edition, Dover Publ. 1960.

This cube was published in Germany in 1898 [1] along with an order 5 magic cube. These are the first published cubes I have been able to locate that are magic by the now accepted rules (except fot A.H. Frost and F. A. P. Barnard). These rules, first published by W.S. Andrews [2][3] page 69 state

A magic cube consists of a series of numbers so arranged in cubical form that each row of numbers running parallel with any of its edges, and also the four great diagonals shall sum the same amount.

This is an associated magic cube because number pairs on opposite sides of the center of the cube sum to 65, which is the sum of the first and last numbers in the series used. Examples are : 1 + 64, 51 + 14, 42 + 23, etc.
It contains no magic squares. However, it is unique because both diagonals sum to the same value for each of the 12 planar arrays.

Different aspects of this cube were also published by Adler and Shuo-yen (1977) and Benson & Jacoby (1981).

Alternate horizontal planes are shown in these illustrations in alternate colors as an aid to clarity.

This cube by Ingenieur Weidemann was first published in 1922. [4]

A semi-pantriagonal, bent triagonal (see next cube) magic cube. Not associated, but diametrically opposed cells sum to either 64 or 66.
The 4 horizontal squares only, are pandiagonal magic. Also, all 2x2 squares in these 4 planes sum to the magic constant.

[4] Weidemann, Ingenieur, Zauberquadrate und andere magische Zahlenfiguren der Ebene und des Raumes, Oscar Leiner, 1922 (German) Translated title is Magic squares and other plane and solid magic figures.

Hendricks order 4 magic cube with bent triagonals

This semi-pantriagonal magic cube by Hendricks appears in his 1999 book [5].

It is magic because all rows, columns and the 4 main triagonals sum correctly. It also is not associated.

It has the unique feature of containing bent triagonals. Examples of these are 56 + 9 + 24 + 41, 56 + 9 + 3 + 62, 56 + 9 + 54 + 11, 56 + 9 + 64 + 1, etc.

It is semi-pantriagonal because opposite short triagonals such as 13 + 52 + 39 + 26 = 130.
All associated (center symmetric) cubes are also semi-pantriagonal.

[5] John R. Hendricks, Inlaid Magic Squares and Cubes, self-published, 1999, 0-9684700-1-7

Pantriagonal

Pantriagonal magic is the third order of complexity after simple and associated. John R. Hendricks published this cube in 1972 when he introduced the concept [1][2].

Of course, the pantriagonal magic cube was not a new invention. In fact, a rotated version of this same cube was published in 1939 [3]. However, the characteristics were not recognized (and a name for this class not assigned).

Required for the cube to be pantriagonal magic:

Rows, columns pillars and 4 main triagonals must sum correctly (the basic requirement for a magic cube).
All broken triagonals must also sum correctly. These will each consist of 2 or 3 segments. There are a total of 4m² triagonals in a cube.

Hendricks order 4 pantriagonal first published in 1972

This cube is not associated. Associated pantriagonal magic cubes start at order 5. One triagonal is shown in red for clarity.
Because 3 segment triagonals are rather difficult to envision, here is one that parallels the main triagonal shown in red: 13, 38, 52, 27.

This cube has two other features only found in order 4 cubes (or 4x). All the pantriagonals cubes I have looked at, have either one or both of these. Is this true for all pantriagonals?
The two additional features are:

Compact: every 2 x 2 square, horizontal or vertical, in the cube sums to 130, the magic constant. Abe [4] defined the term in reference to magic squares. Kanji Setsuda extended it to cubes [5] but used the term ‘composite’. I have reverted back to ‘compact’ because composite has a different meaning when used with magic squares (and by extension, with magic cubes).
Complete: Every pantriagonal contains m/2 complement pairs spaced m/2 apart. This is a characteristic of most-perfect magic squares.

The pandiagonal magic square can be transformed to a different magic square by moving a row or column from 1 side of the square to the other. In exactly the same way, a magic cube may be transformed into another magic cube. However, this time we move an orthogonal plane from one side to the other. Here the bottom plane of the above cube has been moved to the top.

John Hendricks published another paper on Pantriagonal magic cubes in 1980 [6]. In it, he explored the mathematics, and various transformations.

a modification of the above cube - 1 plane moved to top

[1] John R. Hendricks, The Pan-3-Agonal Magic Cube, JRM 5:1:1972, pp 51-54
[2] John R. Hendricks, The Pan-3-Agonal Magic Cube of Order 5, JRM 5:3:1972, pp 205-206
[3] RouseBall & Coxeter, Mathematical Recreations & Essays, 11th edition, 1939.
[4] Gakuho Abe, Fifty Problems of Magic Squares, Self published 1950. Later republished in Discrete Math, 127, 1994, pp 3-13.
[5] . http://homepage2.nifty.com/KanjiSetsuda/pages/EnglishP1.html
[6] John R. Hendricks, the Pan-3-agonal Magic Cube of Order 4, JRM 13:4, 1980-81, pp274-281

More Order 4 Cubes

Below I show some more order 4 cubes. Click on the image to get an enlarged view.

Encyclopedia Britanica - 1911 edition
This is simple associated magic. No special features.

Heath 1938 pantriagonal [1]
Because the four horizontal planes of this cube are simple magic, they may form the quadrants of an order 8 magic square. This square has the unique feature that alternating numbers in each row, column and the two main diagonals sum to 130. Two oblique squares are pandiagonal magic. It is ‘complete’ because all triagonals consist of 2 complement pairs.

Kanji Setsuda lists a great many order 4 magic cubes on his site [2]. Here I show 4 pantriagonal cubes. None of Setsuda's cubes shown here contain magic squares.

Encyclopedia Britanica - 1911	Heath 1938 - Pantriagonal	Setsuda -Compact & Complete
Setsuda -Compact The 2 complement pair differences in each triagonal is 64.	Setsuda -Compact The 2 complement pair differences in each triagonal is 32.	Setsuda - Complete The 2 complement pair differences in each triagonal is 0 (each pair = 65).

[1] RouseBall & Coxeter, Mathematical Recreations & Essays, 11 edition, 1939. Chapter VII. Also
Editions 12, University of Toronto Press, 1974, 0-8020-6189-9 and
editions 13, Dover Publ., 1989, 0-486-25357-0
[2] Kanji Setsuda’s http://homepage2.nifty.com/KanjiSetsuda/pages/EnglishP1.html

Magic Squares to Cubes

A. Adler, & R. Shuo-yen, in a 1977 paper, demonstrate a method for transforming an order m magic cube into an order 2m magic square. Diagram (a) shows the originating cube. Diagram (b) the resulting magic square when every 2 rows of the cube are put into 1 row of the square. (c) shows transposing 2 columns into 1 column, and (d) shows transposing 2 pillars into 1 row.

Each of the resulting magic squares can then be transformed to another magic cube by changing rows or columns into 2 rows, columns or pillars of a new magic cube.

I tried this method on six different types of magic cubes, including a cube with prime numbers and another cube using only every sixth number. It worked for all of them except for the Setsuda complete (the last cube shown above). so it seems this method works for most, but not all, order 4 magic cubes. The method can probably be extended to other orders.
By the way. The Adler cube shown here is semi-pantriagonal, associated. It is the same as the cube published in 1911 in the Encyclopedia Britanica. The other cubes I tried were simple, not associated; semi-pantriagonal, associated; pantriagonal, not associated (normal and not normal); and pantriagonal associated (not normal, prime).

Top                2                  3                  Bottom
01 63 62 04     48 18 19 45    32 34 35 29     49 15 14 52
60 06 07 57     21 43 42 24     37 27 26 40     12 54 55 09
56 10 11 53     25 39 38 28     41 23 22 44     08 58 59 05
13 51 50 16     36 30 31 33     20 46 47 17     61 03 02 64
(a)        originating magic cube

01 63 62 04 60 06 07 57    01 62 48 19 32 35 49 14    01 48 32 49 63 18 34 15
56 10 11 53 13 51 50 16    60 07 21 42 37 26 12 55    62 19 35 14 04 45 29 52
48 18 19 45 21 43 42 24    56 11 25 38 41 22 08 59    60 21 37 12 06 48 27 54
25 39 38 28 36 30 31 33    13 30 36 31 20 47 61 02    07 42 26 55 57 24 40 09
32 34 35 29 37 27 26 40    63 04 18 45 34 29 15 52    56 25 41 08 10 39 23 58
41 23 22 44 20 46 47 17    06 57 43 24 27 40 54 09    11 38 22 59 53 28 44 05
49 15 14 52 12 54 55 09    10 53 39 28 23 44 58 05    13 36 20 61 51 30 46 03
08 58 59 05 61 03 02 64    51 16 30 33 46 17 03 64    50 31 47 02 16 33 17 64
(b) row-by-row              (c) column-by-column       (d) pillar-by-pillar

[1] A. Adler, & R. Shuo-yen, Magic Cubes and Prouhet Sequences, American Mathematical Monthly, vol. 84 (8), 1977, p. 618-627.

Magic Cubes - Order 4