On this page we show features or characteristics
common to all magic cubes. Many special magic cubes contain features that
are unique to that type of cube. These features will be presented as we
discuss the appropriate cubes.
Presentation

This is the traditional method.
It was used by W.S. Andrews and others circa 1900.
[1] [2]
In recent years, the grid is often not used, so just the cell
numbers are presented.The horizontal
planes of the cube are shown, in reverse order, from top to bottom
(plane 1 on the bottom). For smaller order cubes, the m planes are
usually printed side by side to conserve vertical space on the page.


This graphic illustration gives a
clearer picture of the relationship of the numbers in the cube.
This method of presentation is very good for
cubes of low orders. However, for orders higher then 5 or 6, it is
time consuming to prepare, and it becomes increasingly harder to
follow the pillars and triagonals in the diagram. 

This method was used by early
investigators and is currently favored by
M. Trenkler. [3] (However, he also uses the first
method, show above.)Notice that this
displays vertical, instead of horizontal planes.
Two pages on representation for all dimensions
of hypercubes is here. 
[1] W. S. Andrews, Magic
Squares & Cubes, Open Court, 1908, 193+ pages.
[2]
Hermann Schubert,
Mathematical
Essays and Recreations.
Translated from German to English by Thomas J. McCormack, Open Court,
1899. 143+ pages.
[3] Marián Trenkler, A construction of
magic cubes, The Mathematical Gazette, 84(2000), 3641,.
Basic parts

There are 3m^{2} rows, columns
and pillars in a magic cube. All are required to sum to the magic
constant.
There are 4 triagonals. All 4 must sum to the correct constant.
These are the minimum requirements for a simple
magic cube.
There may be some diagonals that sum
correctly, but that is not a requirement for a simple magic cube.
This cube in this illustration
is the basis for other diagrams on this page. 
Orthogonal
(planar) squares
The rows, columns, pillars, and triagonals may be
considered the primary building blocks of a magic cube. They are the
elements we are usually concerned with when constructing a magic cube.
However, when we wish to construct a cube with more
advanced features, there is another element we must consider. That is the
square arrays within the cube.
There are 2 types of square arrays. The first type are parallel to the
sides of the cube and are often called orthogonal or planar squares. There
are 3m squares of this type in a magic cube.
If each of the 2 diagonals of one of these squares sum correctly, that
square is magic.
The 2nd type of square array goes on the diagonal
from the junction of 2 faces, through the center of the cube, to the
junction of the opposite 2 faces. These will be discussed a little further
down the page.

I have chosen to illustrate the
planar squares by showing the central square in each direction.
Obviously, there are central squares only if the cube is of odd
order. Each cell in the cube is common
to 3 squares. Here the 14 is common to all 3 central squares. The
20, for instance, is common to the top horizontal square, the front
vertical square, and the central square parallel to the sides.
In this order 3 cube, the only cells not
included in the 3 central planar squares are the 8 corners of the
cube. As the order of the cube increases the number of cells not
included in the central planes becomes an ever greater percentage of
the total cells.
The horizontal planes are often used to
present a magic cube on paper in text only format. For larger
orders, it is much more practical then using the grid diagram. 
Associated
magic cubes
Reproduced here is the central
horizontal plane of the above magic cube. 
Notice that the rows and columns sum
correctly to 42.
Check the pair of numbers on opposite sides of the central 14. In
all cases the pair sums to 28 which is equal to m3 + 1 and 28 + 14 =
42, the magic constant of this cube. Because the pairs we checked
include the 2 diagonals, this plane is an associated magic square.
This feature is common to the 3 central planes of
all odd order associated magic hypercubes, regardless of order or
dimension. So, if an odd order magic cube is associated, it contains
at least 3 magic squares. 
BTW. Order 3 hypercubes of all dimensions are
associated. i.e squares, tesseracts, etc
Associated magic squares, cubes, etc., are centersymmetric. Magic
hypercubes of higher dimensions have other types of symmetry as well.
There is much more information on these and other types of symmetrical
cubes on my Selfsimilar cubes page.
Pandiagonal
and semipandiagonal magic squares and cubes
Both main diagonals must sum correctly in order for
a square to be considered magic.
The four main triagonals of a cube must sum correctly in order for the
cube to be considered magic.
Pandiagonal and semipandiagonal are extra features a magic square or cube
may have.
Pandiagonal magic squares
01
08 13
12
15 10
03 06
04
05 16 09
14 11
02 07
Order 4 pandiagonal
01
07 25 19
13
20 14
03 06 22
08
21 17 15 04
12 05
09 23
16
24 18 11
02 10
Order 5 pandiagonal

If all the broken diagonals in a
magic square also sum correctly, the square is classed as
pandiagonal.
The equivalent in a magic cube is when all the broken triagonals sum
correctly. The cube is then called pantriagonal.
In a magic square, the broken diagonals consist
of 2 segments. In a magic cube, the broken triagonals may consist of
2 or 3 segments.
In each of these pandiagonal magic square
examples, I highlight a main diagonal in red, and 2 broken diagonal
pairs in blue and violet.
A pandiagonal magic square has the feature
that a row or column may be moved from one edge to the opposite edge
and the square remains magic.
A pantriagonal magic cube has the feature that an entire plane may
be moved from one edge to the opposite edge and the cube remains
magic. 
Semipandiagonal m. s.
03 06 13 12
10 15 08 01
16 09 02 07
05 04 11 14
Order 4 semipandiagonal
12 01 20
09 23
21 15 04 18
07
10 24
13 02
16
19 08
22 11 05
03 17
06 25 14
Order 5 semipandiagonal

This is another broad
classification for magic hypercubes. In
this case, we consider a broken diagonal pair where each of two
parallel segments have an equal number of cells. For an even order
square or cube, each segment contains m/2 cells that together sum to
S.
An odd order square or cube there are several
combinations that sum to S:
The two segments that each contain m1 cells sum to S when the
center cell is added.
The two segments that each contain m+1 cells sum to S when the
center cell is subtracted.
If both segments of each broken diagonal pair
of an even order square sum to S/2, then that is a bent diagonal
square. It is not associated. Some even order and all odd order
semipandiagonal magic squares are associated.
The same characteristics apply to semipantriagonal magic cubes. 
Pandiagonal and semipandiagonal magic
squares and cubes are covered in much more detail and examples on my
c_semipan.htm page.
My c_groups.htm page shows the relationships
between simple, associated, pandiagonal/pantriagonal and semipan squares
and cubes of order 4.
Oblique
squares
Besides the 3m orthogonal squares, a second
type of square array goes on the diagonal from the junction of 2 faces,
through the center of the cube, to the junction of the opposite 2 faces.
There are 6 squares of this type in any cube, regardless of the order.
Reproduced from top of page 

Above are the 3 pairs of oblique squares, with the
complete grid shown for 1 square of each pair.
The most significant lines in these square arrays
are the red lines. They are diagonals in these squares but are the
triagonals of the cube. Actually, each triagonal appears as a diagonal in
3 oblique squares (6 squares times 2 = 12 and 12/3 = 4 triagonals).
The green lines are rows, columns or pillars in the orthogonal square
arrays.
The oblique squares (planes) are not nearly as
significant as the orthogonal ones are. However, they are another feature
of a magic cube, and contribute to the total count of magic squares within
the cube.
Broken oblique planes
Each of the six oblique planes has m1 parallel planes that consist
of two segments. these are analogous to the broken diagonals of a magic
square.
And just as these broken diagonals sum correctly in a pandiagonal magic
square, so are the planes correct (i.e. magic squares) in a perfect magic
cube. In fact a perfect magic cube contains 9m pandiagonal magic
squares.
To review characteristics of a magic cube.
 There are 3m orthogonal square arrays. Rows and
columns and pillars must sum correctly, but diagonals need not.
However in an odd order associated magic cube, the 3 central planes will
have correct diagonals, meaning that these cubes contain at least 3
magic squares.
 There are 6 oblique square arrays in any type,
and any order, of magic cube.
Diagonals of these squares must sum correctly (they are the
triagonals of the cube). Rows and columns of these arrays need not
sum correctly.
 Higher types of magic cubes (not yet discussed)
will contain varying numbers and types of magic squares.
Basic
magic squares
There is 1 basic magic square of order 3. However,
it may be shown in 7 other 'disguised' versions that are obtained by
rotations and reflections. These 8 variations of the magic square are
called aspects and are all considered equivalent when enumerating or
comparing magic squares. Before 1675, Frénicle de Bessy listed all 880
basic solutions for the order 4 magic squares. To do so, he devised a
method to determine which of the 8 aspects of each square should be called
the basic square. There are only two simple rules to determine the basic
square:
 Rotate the magic square so that the cell in the
top left corner of the square (the first cell of row 1) is the lowest
value of all the corner cells.
 If necessary, reflect the square around the
leading diagonal so that the cell to the right of the top left corner is
a lower value then the first cell of row 2.

Because there is just 1 order 3 square,
indexing is rather irrelevant. Therefore, which is the basic magic square
of this order is not important. However, for higher orders, this is an
important concept! BTW Note that the order 3
magic square is associated, as per the statement made previously, that all
order 3 hypercubes are associated. 
Basic cubes and aspects
The order 3 magic square has 1 basic solution and 8
variations (aspects).
The order 3 magic cube has 4 basic solutions and 48 variations (aspects).
There are 8 possible corners of a magic cube, that
may be placed lower, front, left. There are 3 rays extending from that
corner which may be labeled in 6 different ways. That gives 8 times 6 = 48
variations due to rotations and reflections.
[1]
The index for an order 3 cube consists of 4 numbers.
To normalize a magic cube to the standard position, there are 2 rules
(actually 4 steps).
 Rotate the cube so the smallest corner is in the
bottom left front position.
 Rotate/reflect as necessary so the 3 adjacent
numbers to this corner are in the adjacent x, y, and z cells in
increasing order of size.

This variation of the basic cube is
simply a roll 90 degrees forward.
What was the face is now
the bottom. What was the back is now the top.
The 4 red numbers in the bottom left of the basic cube are the numbers in
the index, written 1, 15, 17, 23.
Note that the center number remains constant (for
odd orders) when the cube is rotated or reflected. 
For all practical purposes, a cube is just as magic
if it is a variation, rather then the basic cube. The value of basic
position only becomes important when you wish to count or list the
different cubes of a particular order.
Four numbers are required to define any of the 4
basic cubes of order 3. In fact, partly because order 3 is associated, it
is possible to reconstruct the entire cube, if given these 4 initial
numbers!
As the order becomes larger, more cell values will have to be included in
the index string. However, no one yet has determined how many basic cubes
there are even for order 4, so developing an index for higher orders has
not yet become a necessity.
[1] John R. Hendricks, Inlaid
Magic Squares and Cubes, 2nd edition, selfpublished, 2000, 0968470033,
p. 134136
Coordinates

This diagram illustrates the
coordinates of some of the cells in an order 3 cube array.
The coordinates serve as place names for specifying individual
cells.
Examples are shown of lines that must sum correctly, and an example
diagonal, which need not sum correctly.
Note that for rows, columns and pillars, 1 coordinate varies as you
move along the line.
For diagonals, 2 coordinates vary, and for triagonals, 3 coordinates
vary as you move along the line.
The underlined coordinates (in the lower left
corner) are the locations for the 4 numbers of the index string. 
One method of constructing magic cubes is to use
coordinates in conjunction with modular equations.
This is a favored method of John R. Hendricks and is explained in many of
his books. [1]
It uses modular equations to form the m digits of a base m
number for each cell. Each number is then converted to decimal and 1 is
added to get the final value for each cell.
These 3 modular equations will
give an order 3 cube.
D2 ≡ x + y + 2z (mod 3)
D1 ≡ x + y + z + 1 (mod 3)
D0 ≡ x + 2y + z (mod 3) 
These 3 modular equations will
give an order 5 cube.
D2 ≡ x + y + 2z (mod 5)
D1 ≡ x + 4y + 2z + 1 (mod 5)
D0 ≡ 4x +4y + 2z + 1 (mod 5) 
(Your browser may show a ? after the D2, D1, D0. If
so, the symbol should be that used for congruent equations.)
[1] John R. Hendricks, Magic
Squares to Tesseracts by Computer, selfpublished, 1998, 0968470009
Species
Species is a consideration of how even and odd
numbers are placed in order 3 normal magic hypercubes.
There is only 1 order 3 basic magic square, and so only 1 species, with
the even numbers appearing at the 4 corners.
There are 4 basic order 3 magic cubes, but again
there is only 1 species, because the magic cube must have all even numbers
on 2 edges of 3 faces. Or just remember that 2 outside parallel faces
(planes) each have 5 even numbers on 2 edges. The central parallel plane
has the remaining 3 even numbers on one of its diagonals.
Or still another way to look at it. Each of the six faces has 1 odd number
and 3 even numbers.
Species 1 of 1 for magic square and cube 

However, when we look at the 58 tesseracts (4D
hypercubes) we find that there are 3 different ways the even numbers are
arranged. The example shown here, which John Hendricks has labeled species
#1, appears in only 2 tesseracts. Species #2 is found in 24 tesseracts,
with the remaining 32 tesseracts having species #3 .
No one has determined how many species there are for
any of the order 4 hypercubes. A quick scan of order 4 magic squares seems
to indicate that there are always 2 odd and 2 even numbers in the corners,
making just two species; O, E, O, E and O, O, E, E.
[1] John R. Hendricks, Species of
ThirdOrder Magic Squares and Cubes, JRM 6:3,1973, pp190192.
[2] H.D. Heinz and J.R. Hendricks, Magic Square Lexicon: Illustrated,
Selfpublished, 2000, 0968798500
