A
Nasik (perfect) magic
cube is pantriagonal and all of its planes (the magic
squares) are pandiagonal. There are 13 m^{2} lines
that sum correctly
(m^{2} rows, m^{2} columns,
m^{2} pillars, 4m^{2} triagonals and 6m^{2}
diagonals).
These combine to form 9m pandiagonal magic squares of order
m.
Order8 is the smallest possible order perfect magic cube.
On this
page, comments referring to magic cubes should be considered,
by extension, to apply to magic hypercubes of any dimension. 
This is my original page on Perfect Magic Cubes I
have since posted another page that
discusses this subject from a different point of view. The text in the
above box is also a more recent addition.
Introduction 
Why I feel this page is necessary and
what is included to show the need for a revised definition. 
History 
Mr. Hendricks recounts the events in the
development of the modern definition of perfect for magic objects. 
Definitions 
Definitions relevant to
Perfect magic squares, excerpted from Magic Square Lexicon:
Illustrated. 
Interdimensional
Comparisons 
Tables compare features
between hypercubes of different dimensions. 
Examples  Cubes 
The 4 'basic' order3, a
'triagonal' order4, an order8 'perfect' and an order9 'perfect'
magic cube. 
Dimensions 4 and 5 
A 'basic' order 3 and a 'quadragonal'
order4 magic tesseract, and information on two 'perfect' magic
tesseracts. 
Conclusions and References 
A summary of this page,
some relevant references, and some relevant links. 
Introduction
In discussions regarding magic
cubes, there seems to be much confusion about the term ‘perfect’.
Pandiagonal magic squares have long
been referred to as being perfect, presumably because they had additional
lines (of m numbers) that summed correctly.
Over the last 125 years or so, as work with magic cubes progressed and
more features were discovered, they have often been referred to as
‘perfect’.
Over the last 25 years or so, John
Hendricks and others have been doing more and more work with magic objects
of 4 (tesseracts), 5, and higher dimensions. When features between these
different dimensions were compared by Mr. Hendricks, it quickly became
evident that the term ‘perfect’ should be redefined.
This he preceded to do, and since his retirement has been actively
investigating and refining his ideas.
In an effort to publicize his work and the new definitions, he has self
published a number of books.
I received an email from Mr. Hendricks
on February 27, 2002 on the history of this definition. This was after I
had mentioned to him about the large number of inquiries on the subject I
was receiving through my Web pages.
The result is this page about perfect
magic hypercubes, with emphasis on hypercubes of dimension 3. I have
included the modern definition for these cubes and other associated
definitions. Also shown are examples of orders 8 and 9 perfect cubes, and
tables that compare features common to hypercubes of the different
dimensions.
The section closes with a short discussion of order 16 and order 32
perfect magic tesseracts. These will further reinforce the relationship
between ‘perfect’ hypercubes of different dimensions.
Addendum: After doing a survey
of published magic cubes, I have published another page on perfect cubes.
It serves as a supplement to this page, which I have left unchanged.
Addendum2 (Jan. 2005) I have
added several definitions. Required because of the discovery of a sixth
class of magic cube.
Addendum3 (May 2006) Expanded
material on the term Nasik and emphasizing monagonal (or 1agonal) as the
preferred term to replace irow.
This page uses m for order of
the magic object and n for the dimension being referred to.
History
From an email attachment of March 2, 2002 from John Hendricks:
General History
Sixty years ago when we were kids you were lucky to
have a magic cube in the first place. There were a few around, but no
7x7x7 cube existed although they had orders 3 to 10 except it. Andrews &
Co. [1] set out the definition of a magic cube to
sum in rows, columns, pillars and only the four space diagonals. Then they
speculated on what would be considered a perfect cube. They all agreed
that the continuous diagonals of a plane face, as an extra measure, would
do as a bare minimum requirement. And that is about where I came into the
picture. I eventually filled in the 7th order gardenvariety cube and had
it published. Nobody to my knowledge ever dreamt that it would be possible
to get magic squares throughout a cube until Myers did so. (Editors note:
Myers cube contained only simple magic squares.) So the definition changed
for perfect cube to row, columns, pillars, the four triagonals and all the
diagonals in all the planes including the broken ones. This allowed
Collison and others to show an order 7 such cube. These are considered
nowadays as pandiagonal magic cubes.
Meanwhile, working in isolation, apart from the
mathematical community, I felt that such a cube is not equivalent to a
pandiagonal magic square and went searching for the pantriagonal cube. I
did not care about the planar diagonals, but did care about the broken
triagonals, hence I came up with an order 4 such cube and had it
published. Then, the critics advised me that someone else had such a cube
on file at a university in the east and that I was not the first to make
such a cube. It turned out that I was the first to publish such a cube.
Then, I got busy with the meteorological service and
marriage and had no more time for several years to look into such
mathematical matters, During that interval, Capt. Benson
[2] came along and made a perfect 8th order magic cube, wrote a
book and so forever more one would conclude that the matter was settled
and the definition should be immediately updated for perfect cubes for all
time. They referred to this cube (which contains 30 pandiagonal magic
squares) as pandiagonally perfect. They called the Myers cube (which
contains 30 simple magic squares) as perfect. It is now referred to as a
diagonal magic cube.
Back at the drawing board, I concluded that a
perfect magic tesseract would have to have everything working too. I did
publish the first magic tesseract in one diagram and all 57 more, I did
publish the first panquadragonal magic tesseract of order 4. But these
were not perfect.
Eventually, after “retirement” I found the key and
selfpublished the perfect magic tesseract of order 16 and the
5dimensional magic hypercube of order 32 also in the same booklet
[3], I also discounted the Cameron Cube
[4], which was the name given to a special magic
tesseract, because a simple examination of the triagonals of some of the
cubes in the tesseract simply do not add up. I also published a perfect
8th order magic cube,” along with 9th and 11th order ones
[5].
John
Footnotes were added
by the editor
[1] Andrews,
W.S., Magic Squares & Cubes, Dover Publ.,
1960 (original publication Open Court,1917)
[2] Benson,
W. & Jacoby, O., Magic Cubes: New Recreations, ,
Dover Publ., 1981, 0486241408
[3]
J.R.Hendricks, Perfect nDimensional Magic Hypercubes of Order 2^{n},
Selfpublished,1999, 0968470041
[4] Arkin
Arney & Porter, A Perfect 4_Dimensional Hypercube of Order7, JRM:21:2,
1989, 8188
[5] J. R.
Hendricks, Magic Squares to Tesseracts by Computer, Selfpublished 1999,
0968470009
Definitions
To help our understanding of the term ‘perfect’ as
it applies to magic objects, I have included some appropriate definitions.
These are taken from Magic Square Lexicon: Illustrated, but have been
edited for brevity and most illustrations removed.
Compact Plus
When the eight corners of all orders of subcubes contained within a cube
including wraparound, sum to 8S/m where m is the order of
the parent cube. I have adapted this term from Gakuho Abe’s [1]
term ‘compact’ which he used to indicate that all 2x2 squares in an
order4 magic square sum to S.
Kanji Setsuda [2] uses the term ‘composite’ for this
feature in magic cubes but I feel that this can cause confusion with
‘composite’ magic squares.
Complete
Every pantriagonal contains m/2 complement pairs, spaced m/2 apart. Note
that this is a requirement for mostperfect magic squares. Coined by Kanji Setsuda
[2].
[1] Gakuho Abe, Fifty Problems of Magic Squares,
Self published 1950. Later republished in Discrete Math, 127, 1994, pp
313.
[2] Kanji Setsuda’s Compact (composite) and Complete magic Cubes Web pages
may be accessed from here.
http://homepage2.nifty.com/KanjiSetsuda/pages/EnglishP1.html
Diagonal magic cube
A magic cube that has the additional feature that both main diagonals of
all 3m planar squares sum to S. Because all rows and columns sum
correctly as the original requirement for a magic cube, this means that
all 3m orthogonal planes are simple magic squares. Some, but not
all of these planar squares may be pandiagonal magic. The six oblique
squares will automatically be be magic too. Order 5 is the smallest
possible diagonal magic cube.
For a short time, I used the term 'myers' for this type of magic cube.
However, this name, I believe, is more suitable, and so will be used from
now on. The name diagonal magic cube was suggested by Aale de
Winkel.
This class of cube was missed in Hendricks original unified
classification.
The Myer's diagonal cube was popularized as 'perfect' by Martin Gardner in
Jan. 1976.
ALL singlyeven orders of diagonal magic cubes are proper (see definition
for proper).
Christian Boyer and Walter Trump refer to this class as perfect. Trump has
since started using the term strictlymagic to avoid confusion with
Hendricks perfect.
irow
An irow is a row, column, pillar, file, etc., of an ndimensional
hypercube of orderm. Some authors refer to these as “the orthogonals”
because they are all mutually perpendicular to each other. (An irow is
parallel to an xiaxis where the axes are numbered x1, x2, x3, etc.)
A more modern term for irow is monagonal (or 1agonal). It should be used
instead of this term in future references.
Magic cube
An m x m x m array of cells with each cell containing a number, usually an
integer. These numbers are arranged so that the sum for each row, each
column, each pillar, and the four main triagonals are all the same. Note
that it is not required that the squares in the 3m planes of the
cube have correct diagonals.
These are the minimum requirements for a simple magic cube.
[1][2] All magic cubes later defined have these
features plus the additional required features.
[1] This was first (?)
published in W. S. Andrews, Magic Squares and Cubes, Open Court
Publ., 1908 p 64.
[2] This definition also appears in the better known edition 2 published
in 1917, also on page 64.
Magic hypercube
An ndimensional array of m^{n} cells containing the
numbers1, 2, ..., m^{n} arranged in such a way that all
rows, columns, etc sum the magic sum, as well as the 2n1 nagonals.
While usually used to refer to a higher dimension object, a square or cube
may be considered 2 or 3dimensional hypercube (respectively).
There are 2 classes or ranks of magic hypercubes of dimension two, 6
classes or ranks of magic hypercubes of dimension three, and 18 classes
for dimension four, the tesseract!
Magic Sum
The constant each row, column, etc., sums to is called the magic sum. It
is denoted by S.
For a normal ndimensional magic hypercube of orderm the sum is
m(m^{n} + 1)/2.
Magic tesseract
A magic tesseract is a fourdimensional array, equivalent to the magic
cube and magic square of lower dimensions, containing the numbers 1, 2, 3,
…, m^{4} arranged in such a way that the sum of the numbers
in each of the m^{3} rows, m^{3} columns,
m^{3} pillars, m^{3} files and in the eight
major quadragonals passing through the center and joining opposite corners
is a constant sum S, called the magic sum, which is given by: S = m(m^{4}+1)/2
and where m is the order of the tesseract.
Monagonal
A row, column, pillar, file, etc., of an ndimensional hypercube of
orderm. Some authors refer to these as “the orthogonals” because
they are all mutually perpendicular to each other. (A monagonal (ie
1agonal) is parallel to an xiaxis where the axes are numbered x1, x2,
x3, etc.)
Customarily, a row runs from left to right; a column from front to back; a
pillar runs up and down and a file runs obliquely to the other three in
the projection of a tesseract. There are n(mn1) monagonals
in an ndimensional hypercube of orderm.
If 1agonal is used for rows , columns, etc., the nasik (perfect) magic
hypercube may then be defined as having all (including broken) ragonals
summing correctly for r = 1 ... n.
Nasik cubes
In 1866, A. H. Frost introduced the term Nasik for magic squares having
the property that all monagonals and diagonals (including broken ones),
summed to the magic constant. These magic squares would later be called
pandiagonal or perfect.
In 1905, C. Planck extended the term to to refer to magic objects of any
dimension in which all ragonals summed to the constant.
If all panragonals are correct for r = 1...n, we have a nasik
(perfect) magic hypercube of dimension n.
There is an extensive C. Planck quotation
here.
Nasik is unambiguous, and therefore the preferred
term for the often confusing perfect when referring to magic
hypercube classes. (See perfect magic cube.)
nagonals
nagonal (continuous): A line going from 1 corner, through
the center to the opposite corner, of a magic hypercube. For a cube or
greater dimension hypercube, this is sometimes called a space diagonal.
There are 2^{n} corners and 2^{n1}
nagonals in a magic hypercube.
nagonal (broken ): Lines parallel to a continuous nagonal.
For a twodimensional object (a magic square) these lines will consist of
2 segments totaling length m. For a cube, the line would consist of
2 or 3 segments, etc.
A 1agonal (monagonal) is a line which crosses only
1 dimension. A 2agonal is usually called a diagonal. It crosses two
dimensions. [1]
A 3agonal is usually called a triagonal and crosses three dimensions, a
quadragonal crosses 4 dimensions, etc. The variable r may be used
instead of n to indicate these other values for the agonal. (See
ragonal)
[1] First published mention was John R. Hendricks, The
Pan3Agonal Magic Cube, JRM 5:1:1972, pp 5154
Number of broken nagonals for each continuous one 
n 
2 seg. 
3 segments 
4 segments 
Total 
2 
m1 
0 
0 
m 
3 
3(m1) 
(m1)(m2) 
0 
m^{2} 
4 
2(5m8) 
2(2m^{2}7m+7) 
(m1)(m2)(m3) 
m^{3} 

For each continuous nagonal in
ndimensional space, there are a number of broken nagonals,
depending upon the order of the hypercube.
There are 2 continuous diagonals in a square, 4 continuous
triagonals in a cube, and 8 continuous quadragonals in a tesseract.
So, the numbers in the table must be multiplied by the number of
continuous ones in order to determine how many and of which kind of
nagonals are in a hypercube. 
Pandiagonal
Pandiagonal means “all diagonal”, which signifies that the broken
diagonals are also included. Sometimes pan2agonal is used instead.
Especially in ndimensional space. A 2agonal is described through
space if any two coordinates change while the rest remain constant.
For example in a cube of order 4, one could describe a diagonal through
(1,2,3) by holding y constant while x and z is allowed to change. Such a
set could be:
(1,2,3) ; (4,2,4) ; (3,2,1) ; and (2,2,2)
In this example x is decreasing in increments of one and z is increasing
by increments of one and all coordinates are kept within the modulus 4.
There are N = n!.m^{n1}/(n2)! diagonals in an ndimensional
magic hypercube of order m, including the broken ones.
If all panragonals are correct for r = 1 and 2, we have a
pandiagonal magic square, If r = 1, 2, ... n, a pandiagonal
hypercube of dimension n.
Pandiagonal magic cube
A Pandiagonal Magic Cube has the normal requirements of a magic cube plus
the additional one that all the squares parallel to the sides of the cube
(planar squares) also be pandiagonal.
Remember that an ordinary magic cube does not require even the main
diagonals of these squares to be correct.
There are 9m^{2} + 4 lines that sum correctly (m^{2}
rows, m^{2} columns, m^{2} pillars, 4 main
triagonals and 6m^{2} Diagonals). It contains 3m
pandiagonal magic squares and 6 oblique squares, 0 to 3 of which are
pandiagonal magic and the others simple magic.
Order7 is the smallest possible order of pandiagonal magic cube. It
contains 27 pandiagonal magic squares (3 x 7 + 6).
M. Gardner referred to H. Langman's 1962 pandiagonal
magic cube as a perfect magic cube at the
same time he was calling the Myer's diagonal
cube perfect! Presumably he did not recognize
the difference between the three types of cubes.
Pandiagonal magic square
Also known as Diabolic, Nasic, Continuous, Indian, Jaina or
Perfect. To be pandiagonal, the broken
diagonal pairs must also sum to the constant. This is considered the top
class of magic squares. There are 4m lines that sum correctly (m
rows, m columns and 2m diagonals).
Panquadragonal
Broken quadragonal pairs that are parallel to a quadragonal and that sum
to the magic constant. If all these pairs sum correctly, the magic
tesseract is panquadragonal. It is analogous to a pandiagonal magic square
but instead of moving a row or column from one side to the other and
retaining the magic properties, you move any cube from one side to the
other. When one moves along the panquadragonal, 1 cell at a time, four
coordinates change. See also, Pantriagonal.
Pantriagonal
Sometimes called Pan3agonal.
This term is used for cubes, or high dimensional hypercubes. In ndimensional
space, if any three coordinates are changing while the rest remain
constant, then one describes a triagonal through space, of which most are
broken. The main triagonal is the one which passes through (1,1,1) and has
successive coordinates (2,2,2),…, (m,m,m) in a cube.
In Ndimensional space, the nagonal may be broken into as many as
n segments. For magic cubes there are:
4 continuous triagonals
12(m1) triagonals broken into pairs, and
4(m2)(m1) triagonals broken into 3
sections.
If all the broken Triagonal lines sum correctly, the magic cube is
pantriagonal.
Pantriagonal magic cube
If all triagonal pairs and triplets (pantriagonals) sum correctly, the
magic cube is pantriagonal. [1] It is analogous to a
pandiagonal magic square but instead of moving a row or column from one
side to the other and maintaining the magic properties, you may move any
plane from one side to the other.
There are 7m^{2} lines that sum correctly (m^{2}
rows, m^{2} columns, m^{2} pillars, and 4m^{2}
triagonals). There may be some diagonals in the cube, but they are not
required. There may also be some magic squares, either simple or
pandiagonal, but they also are not required. Order4 is the smallest
possible order pantriagonal magic cube. See also, Pandiagonal magic cube.
[1] First published mention was John R. Hendricks, The
Pan3Agonal Magic Cube, JRM 5:1:1972, pp 5154
Pantriagonal Diagonal
A magic cube that is a combination Pantriagonal and Diagonal cube. All
main and broken triagonals must sum correctly, In addition, it will
contain 3m order m simple magic squares in the orthogonal
planes, and 6 order m pandiagonal magic squares in the oblique
planes.
For short, I will reduce this unwieldy name to PantriagDiag. This is
number 4 in what is now 6 classes of magic cubes. So far, very little is
known of this class of cube. The only ones constructed so far (both by
Nakamura) are order 8 (not associated and associated).
This cube was discovered by Mitsutoshi Nakamura and named by him in 2004
Pathfinder
An orderly and systematic way to find one’s way through ndimensional
space. Through any given element, or cell, there are (3^{n}
1)/2
different paths, or lines. For a square, this means that there are 4
paths, which are a row, a column and two (broken, if needed) diagonal
ways. Through any cell of a cube, there are 13 routes. Through a
tesseract, there are 40. One may travel forwards, or backwards on any
route, or path. The method is found in Magic Squares to Tesseracts by
Computer. Hendricks uses this method to show the numbers contained in his
higher order magic hypercubes (via simple computer programs).
Perfect magic cube
A perfect magic cube is pantriagonal and all of its planes (the magic
squares) are pandiagonal. In a perfect magic cube there are 9m pandiagonal
magic squares. That is, all 3m orthogonal planes, the 6 oblique
planes, and the 6(m1) broken planes parallel to the oblique
planes. There are 13m^{2} lines that sum correctly (m^{2}
rows, m^{2 }columns, m^{2} pillars, 4m^{2}
triagonals and 6m^{2} diagonals).
Order8 is the smallest possible order perfect magic cube. Perfect magic
cubes of orders 10, 14, 18, etc. and 12, 20, etc. are impossible. this was
proved by Rosser and Walker in 1939, and by Stertenbrink and de Winkel in
2004. . See
Pandiagonal Impossibility Proof.
*** The above is a new definition. ***
Examples of an older definition of a Perfect Magic Cube was the
Frankenstein 1875, Myers 1970 order 8 cubes. These cubes contained 3m
simple magic squares and are now referred to as Diagonal magic
cubes. Perfect is now construed to mean that the cube is
pandiagonal and pantriagonal, and all lower order magic objects
(i.e. squares) within it are perfect. This makes the definition
consistent for all dimensions. See the definition (above) for pandiagonal
and pantriagonal magic cubes .
This is also consistent with C. Planck's (1905) revised definition for A.
H. Frost's (1866) term Nasik . (See Perfect magic hypercube)
Mitsutoshi Nakamura uses the terms pan2,3agonal and nasik
instead of perfect to avoid confusion. He also uses the term
strictlymagic for the diagonal class, to avoid confusion over
Boyer's perfect.
Perfect magic hypercube
A hypercube of dimension n is perfect if all pannagonals sum correctly,
and all lower dimension hypercubes contained in it are perfect.
Through every cell on the perfect hypercube there are (3^{n}
1)/2 different routes that must sum the magic sum.
As per Dr. C. Planck (1905), these hypercubes are of type Nasik. The
pandiagonal magic square is a perfect hypercube of dimension 2 (and was
called Nasik by Dr. A.H. Frost in 1866).
If all panragonals are correct for r = 1...n, we have a nasik
(Hendricks perfect) magic hypercube of dimension n.
Because nasik in unambiguous, it is preferred instead of the often
confusing perfect.
Perfect magic square
Another traditional (but now not commonly used) name for Pandiagonal magic
square. However, this name shows the relationship of the highest class of
rectilinear magic figures, the perfect square, perfect cube, perfect
tesseract, etc. See the comment re nasik in perfect magic hypercube.
Perfect magic tesseract
A tesseract is a 4dimensional hypercube. It is perfect if all panquadragonals
are correct, and all the magic squares and magic cubes within it are
perfect. i.e. the magic squares are all pandiagonal and the magic cubes
are all pantriagonal and pandiagonal. There are 40m^{2}
lines that sum correctly. They are m^{3} rows, m^{3}
columns, m^{3} pillars, m^{3} files, 8m^{3}
quadragonals, 16m^{3} triagonals, and 12m^{3}
diagonals.
John R. Hendricks constructed the first perfect magic tesseract (order16)
in 1998. It was confirmed correct by Clifford Pickover in 1999. He later
published the equations for a 5dimensional perfect magic hypercube of
order32. However, as it contains the numbers 1 to 33,554,432, he thought
it impractical to publish the hypercube itself!! The numbers in these two
hypercubes may be inspected using his simple Pathfinder programs.
If all panragonals are correct for r = 1...n, we have a nasik
(perfect) magic hypercube of dimension n.
Because nasik in unambiguous, it is preferred instead of the often
confusing perfect.
Proper
When applied to a magic cube of a particular class, means that this cube
contains only the minimum features required for that class.
For example: a simple magic cube requires that no planes within it
be magic squares. However, some planes may be magic and the cube is still
called simple, because the next classification requires that ALL planes be
simple magic squares.
So if a cube contains NO magic squares, it may then be referred to as a
proper simple magic cube!
Likewise a proper pantriagonal magic cube would contain NO magic squares,
a proper diagonal magic cube would contain NO pandiagonal magic squares,
etc.
This term was suggested by Mitsutoshi Nakamura in an Apr. 15, 2004 email.
He also reported that all singlyeven orders of diagonal magic cubes are
proper!
ragonal
Term used to indicate a group of agonals (diagonal, triagonal, etc).
(See nagonal)
Examples:
If all ragonals are correct for r = 1 and 2, we have the
minimum qualifications for a simple magic square.
If all panragonals are correct for r = 1...n, we have a nasik
(Hendricks perfect) magic hypercube of dimension n.
Simple magic hypercube
A magic square, cube, tesseract, etc., where all orthogonal lines plus all
space diagonals sum correctly.
If all ragonals are correct for r = 1 and 2, we have the
minimum qualifications for a simple magic square. If r = 1 and n,
a simple magic hypercube of n dimensions!
Space diagonal
A line that goes from a corner of a magic hypercube, through the center,
to the opposite corner. See (nagonal)
Triagonal
A space diagonal that goes from 1 corner of a magic cube to the opposite
corner, passing through the center of the cube. There are 4 of these in a
magic cube and all must sum correctly (as well as the rows, columns and
pillars) for the cube to be magic. As you go from cell to cell along the
line, all three coordinates change. In tesseracts this is called a
quadragonal. For higher order hypercubes, this is called an nagonal
or space diagonal. Of course, with these higher dimensions there are more
coordinates. A triagonal is sometimes called a long diagonal. See nagonal
Interdimensional Comparisons
The magic sum for an nDimensional Magic Hypercube
of Order m is given by:
S = m(1 + m^{n})/2
In a magic object, there are many lines that produce
the magic sum. The table below, shows the minimum requirement of the
number of lines for various types of magic hypercubes and is derived from
the following equation:
N = 2^{(r1)}n!m^{(n1)}/[r!(nr)!]
Where:
N is the number of ragonals
n is the dimension of the hypercube
m is the order of the hypercube, and
r is the dimension of the hyperplane.
When r = 1, the number of orthogonals is
given by N. As well, shown is the smallest order for the various
classifications of pandiagonal, pantriagonal, etc. which is known. for
each dimension. Some of the tesseracts are not known yet and some of these
varieties have not been constructed yet.
This table provides the minimum requirements for each category. Usually,
there are some extra lines, which may sum the magic sum, but not a
complete set so as to change the category.




nagonals 

Lines 
Magic Hypercube 
n 
Lowest
Order 
i 
rows 
2 
3 
4 
Total 
Equivelent
order 8 
Square 
2 







Simple 

3 
2m 
2 


2m + 2 
18 
Perfect 

4 
2m 
2m 


4m 
32 
Cube 
3 







Simple 

3 
3m^{2} 

4 

3m^{2} + 4 
196 
Diagonal 

5 
3m^{2} 
6m 
4 

3m^{2}+6m+ 4 
244 
Pantriagonal 

4 
3m^{2} 

4m^{2} 

7m^{2} 
448 
PantriagDiag 

8? 
3m^{2} 
6m 
4m^{2} 

3m^{2}+6m+ 4m^{2} 
452 
Pandiagonal 

7 
3m^{2} 
6m^{2} 
4 

9m^{2} + 4 
580 
Perfect 

8 
3m^{2} 
6m^{2} 
4m^{2} 

13m^{2} 
832 
Tesseract 
4 







Simple 

3 
4m^{3} 


8 
4m^{3 }+ 8 
2056 
Panquadragonal 

4 
4m^{3} 


8m^{3} 
12m^{3} 
6144 
Pandiagonal 

? 
4m^{3} 
12m^{3} 

8 
16m^{3 }+ 8 
8200 
Pantriagonal 

? 
4m^{3} 

16m^{3} 
8 
20m^{3 }+ 8 
10240 
Pan2 + pan4 

? 
4m^{3} 
12m^{3} 

8m^{3} 
24m^{3} 
12296 
Pan3 + pan4 

? 
4m^{3} 

16m^{3} 
8m^{3} 
28m^{3} 
14336 
Pan2 + pan 3 

? 
4m^{3} 
12m^{3} 
16m^{3} 
8 
32m^{3 }+ 8 
16392 
Perfect 

16 
4m^{3} 
12m^{3} 
16m^{3} 
8m^{3} 
40m^{3} 
20480 
Hypercubes – number of correct
summations. [1] 

[1]
H. D. Heinz and J. R. Hendricks, Magic Square Lexicon: Illustrated, 2000,
0968798500, page 165
Comparison  Correct Summations Required 
magic Square
Regular 
Magic Cube
Regular 
Magic Tesseract
Regular 
m rows 
m^{2} rows 
m^{3} rows 
m columns 
m^{2} columns 
m^{3} columns 
2 diagonals 
m^{2} pillars 
m^{3} pillars 

4 3agonals 
m^{3} files 


8 4agonals 
Perfect 
Perfect 
Perfect 
m rows 
m^{2} rows 
m^{3} rows 
m columns 
m^{2} columns 
m^{3} columns 
2m diagonals 
m^{2} pillars 
m^{3} pillars 

4m^{2} 3agonals 
m^{3} files 

6m^{2} 2agonals 
8m^{3} 4agonals 


12m^{3} 3agonals 


16m^{3} 2agonals 
For a normal ndimensional
magic hypercube of orderm,
the sum is
m(m^{n}+1)/2 
Magic
Squares, Cubes and Tesseracts Compared
[2] 
[2]
H. D. Heinz and J. R. Hendricks, Magic Square Lexicon: Illustrated, 2000,
0968798500, page 90

Hyperplanes contained in a hypercube 
Dimension 
irows 
Squares 
Cubes 
Tesseracts 
5D Hyper. 
2 
2m 
1 
0 
0 
0 
3 
3m^{2} 
3m 
1 
0 
0 
4 
4m^{3} 
6m^{2} 
4m 
1 
0 
5 
5m^{4} 
10m^{3} 
10m^{2} 
5m 
1 
6 
6m^{5} 
15m^{4} 
20m^{3} 
15m^{2} 
6m 
7 
7m^{6} 
21m^{5} 
35m^{4} 
35m^{3} 
21m^{2}
[3] 
An ndimensional array of mn cells containing
the numbers1, 2, ..., mn arranged in such a way that all rows,
columns, etc sum the magic sum, as well as the 2n1 nagonals.
Remember that irows are orthogonals only. Correct nagonals
are not shown in this table.
If the hypercube is perfect, all these hyperplanes will also have all the
nagonals summing correctly.
[3]
H. D. Heinz and J. R. Hendricks, Magic Square Lexicon: Illustrated, 2000,
0968798500, page 80
Hypercubes cut by n hyperspaces 



A magic square may be cut by a
1dimensional magic irow in 2 ways 
A magic cube may be cut by a
2 dimensional magic square in 3 ways. 
A magic tesseract may be cut by a
3 dimensional magic cube in 4 ways. 
If you are wondering where the cube in
the fourth direction (in the tesseract) is, it is formed by the center
square planes in the 3 horizontal planes of cubes.
All order3 magic hypercubes are associated magic. The hypercubes they are
cut by are also associated magic with correct nagonals. 
Examples  Cubes
The 4 Basic Order3 Magic
Cubes 




Catalogue numbers are
1151723
2151824
4171826 and
6161726
Each of these four may be viewed in 48 aspects due to rotations and
reflections. There is 1 basic order3 magic square with 8
aspects and 58 basic magic tesseracts, each with 384 aspects
due to rotations and reflections.
The 3 squares that bisect each of these four cubes are also magic
although that is not a requirement of a simple magic cube. 
Pantriagonal magic Cube 

05
58 08 59 28 39 25 38 53 10 56 11 44 23 41
22
52 15 49 14 45 18 48 19 04 63 01 62 29 34
32 35
09 54 12 55 24 43 21 42 57 06 60 07 40 27
37 26
64 03 61 02 33 30 36 31 16 51 13 50 17 46
20 47
(x,
y, 1) (x, y, 2) (x, y, 3)
(x, y, 4)
This is text format. Best used for larger magic cubes. 
Pantriagonal is the lowest class of
magic cube (next to the simple). Lines are required to sum
correctly in 7m^{2} ways: m^{2} rows, m^{2}
columns, m^{2} pillars, and 4m^{2}
triagonals. Note that diagonals are NOT required to sum correctly.
Just as a complete row or column may be moved from 1
side to the other of a pandiagonal magic square, so too can a square plane
be moved to the opposite side of this pantriagonal magic cube without
destroying the triagonals.
J. R. Hendricks, Magic Squares to
Tesseracts by Computer, Selfpublished 1999, 0968470009, p. 70 
An Order8 Perfect Magic Cube
A perfect magic cube must add up in all
possible ways. There are 13 such ways through each element.
Looking at it the other way, there are a total of l3m^{2}
paths that sum correctly.
m^{2} rows
m^{2} columns
m^{2} pillars
4m^{2} triagonals
6m^{2} diagonals
Order8 is the smallest possible magic cube that can
be perfect, with 832 lines summing to 2052. This one by J. R. Hendricks is
an example of such a cube.
Horizontal plane I  Top II
88 185 240 449 408 377 304 1 395 350 307 38 75 158 243 486
300 5 84 189 236 453 404 381 242 487 394 351 306 39 74 159
403 382 299 6 83 190 235 454 73 160 241 488 393 352 305 40
234 455 402 383 298 7 82 191 309 36 77 156 245 484 397 348
81 192 233 456 401 384 297 8 398 347 310 35 78 155 246 483
301 4 85 188 237 452 405 380 247 482 399 346 311 34 79 154
406 379 302 3 86 187 238 451 80 153 248 481 400 345 312 33
239 450 407 378 303 2 87 186 308 37 76 157 244 485 396 349
III IV
65 152 249 496 385 344 313 48 422 331 286 51 102 139 222 499
317 44 69 148 253 492 389 340 223 498 423 330 287 50 103 138
390 339 318 43 70 147 254 491 104 137 224 497 424 329 288 49
255 490 391 338 319 42 71 146 284 53 100 141 220 501 420 333
72 145 256 489 392 337 320 41 419 334 283 54 99 142 219 502
316 45 68 149 252 493 388 341 218 503 418 335 282 55 98 143
387 342 315 46 67 150 251 494 97 144 217 504 417 336 281 56
250 495 386 343 314 47 66 151 285 52 101 140 221 500 421 332
V VI
112 129 216 505 432 321 280 57 435 358 267 30 115 166 203 478
276 61 108 133 212 509 428 325 202 479 434 359 266 31 114 167
427 326 275 62 107 134 211 510 113 168 201 480 433 360 265 32
210 511 426 327 274 63 106 135 269 28 117 164 205 476 437 356
105 136 209 512 425 328 273 64 438 355 270 27 118 163 206 475
277 60 109 132 213 508 429 324 207 474 439 354 271 26 119 162
430 323 278 59 110 131 214 507 120 161 208 473 440 353 272 25
215 506 431 322 279 58 111 130 268 29 116 165 204 477 436 357
VII VIII  Bottom
121 176 193 472 441 368 257 24 414 371 294 11 94 179 230 459
261 20 125 172 197 468 445 364 231 458 415 370 295 10 95 178
446 363 262 19 126 171 198 467 96 177 232 457 416 369 296 9
199 466 447 362 263 18 127 170 292 13 92 181 228 461 412 373
128 169 200 465 448 361 264 17 411 374 291 14 91 182 227 462
260 21 124 173 196 469 444 365 226 463 410 375 290 15 90 183
443 366 259 22 123 174 195 470 89 184 225 464 409 376 289 16
194 471 442 367 258 23 122 175 293 12 93 180 229 460 413 372
J. R. Hendricks, Magic Squares to Tesseracts by Computer, Selfpublished 1998, 0968470009, pp 7780
An Order9 Perfect Magic Cube
Top II
52 442 588 339 495 233 704 158 274 494 227 707 157 277 46 444 591 342
104 256 7 469 624 402 522 215 686 627 405 521 209 689 103 259 1 471
170 713 140 319 34 451 606 348 504 28 453 609 351 503 164 716 139 322
384 567 197 695 122 265 16 406 633 121 268 10 408 636 387 566 191 698
433 615 366 513 179 650 149 301 79 173 653 148 304 73 435 618 369 512
283 25 415 570 393 549 242 677 131 396 548 236 680 130 286 19 417 573
659 86 310 61 478 597 375 531 188 480 600 378 530 182 662 85 313 55
558 224 722 113 292 43 424 579 330 295 37 426 582 333 557 218 725 112
642 357 540 206 668 95 247 70 460 671 94 250 64 462 645 360 539 200
III IV
271 48 447 594 341 488 230 706 160 335 491 229 709 154 273 51 450 593
688 106 253 3 474 630 404 515 212 477 629 398 518 211 691 100 255 6
497 167 715 142 316 30 456 612 350 318 33 459 611 344 500 166 718 136
639 386 560 194 697 124 262 12 411 700 118 264 15 414 638 380 563 193
75 438 621 368 506 176 652 151 298 509 175 655 145 300 78 441 620 362
133 280 21 420 576 395 542 239 679 575 389 545 238 682 127 282 24 423
185 661 88 307 57 483 603 377 524 60 486 602 371 527 184 664 82 309
332 551 221 724 115 289 39 429 585 109 291 42 432 584 326 554 220 727
465 648 359 533 203 670 97 244 66 202 673 91 246 69 468 647 353 536
V VI
156 276 54 449 587 338 490 232 703 590 337 493 226 705 159 279 53 443
214 685 102 258 9 476 623 401 517 8 470 626 400 520 208 687 105 261
347 499 169 712 138 321 36 458 605 141 324 35 452 608 346 502 163 714
413 632 383 562 196 694 120 267 18 190 696 123 270 17 407 635 382 565
303 81 440 614 365 508 178 649 147 364 511 172 651 150 306 80 434 617
676 129 285 27 422 569 392 544 241 416 572 391 547 235 678 132 288 26
526 187 658 84 312 63 485 596 374 315 62 479 599 373 529 181 660 87
578 329 553 223 721 111 294 45 431 723 114 297 44 425 581 328 556 217
72 467 641 356 535 205 667 93 249 538 199 669 96 252 71 461 644 355
VII VIII
708 162 278 47 446 589 340 487 228 445 592 334 489 231 711 161 272 50
514 210 690 108 260 2 473 625 403 254 5 472 628 397 516 213 693 107
607 349 496 165 717 144 323 29 455 720 143 317 32 454 610 343 498 168
11 410 634 385 559 192 699 126 269 561 195 702 125 263 14 409 637 379
153 305 74 437 616 367 505 174 654 619 361 507 177 657 152 299 77 436
237 681 135 287 20 419 571 394 541 23 418 574 388 543 240 684 134 281
376 523 183 663 90 314 56 482 598 89 308 59 481 601 370 525 186 666
428 580 331 550 219 726 117 296 38 222 729 116 290 41 427 583 325 552
251 65 464 643 358 532 201 672 99 352 534 204 675 98 245 68 463 646
IX
234 710 155 275 49 448 586 336 492
399 519 216 692 101 257 4 475 622
457 604 345 501 171 719 137 320 31
266 13 412 631 381 564 198 701 119
656 146 302 76 439 613 363 510 180
546 243 683 128 284 22 421 568 390
595 372 528 189 665 83 311 58 484
40 430 577 327 555 225 728 110 293
92 248 67 466 640 354 537 207 674
This cube is also by John Hendricks
[1].
Order 9 is the lowest order normal cube that can be perfect and also
associated (although this cube is not associated).[2]
[1] J. R. Hendricks, Magic
Squares to Tesseracts by Computer, Selfpublished 1999, 0968470009, pp
8184.
[2] W. S. Andrews, Magic Squares & Cubes, 2nd edition, Dover Publ. 1960, 9
(1917) Dr. Planck, p 366.
Dimensions
4 and 5
A magic tesseract is a dimension4 hypercube. To be
simply magic, it is required only that all rows, columns, pillars, files,
and the 8 quadragonals sum correctly.
Basic Order3 Magic
Tesseract MT#9 (Index # 54) 

One of the authors (Hendricks)
has found and all 58 basic magic tesseracts of order3. He lists and
displays illustrations of each of them in his book All ThirdOrder
magic Tesseracts using the following indexing method:
 Identify the lowest of the 16 corner
numbers.
 Take the adjacent number to this corner in
each of the four lines.
 Rearrange these four numbers (if necessary)
in ascending order and write them after the corner number.
In this figure, the lowest corner number is 12
and the four numbers adjacent to it are 52, 61,62, and 76. Taking
them in order; row, column, pillar and file they are already in
ascending order, and, because the lowest corner is in the bottom
left position we realize this tesseract is in the standard position.
This definition is consistent with that of the Basic magic cube.
This tesseract (and each of the 57 others) can
be displayed in 383 other aspects (orientations). These are not
considered unique solutions.
John refers to this solution as MT#9 because
it is the ninth order3 tesseract he constructed. 
H. D. Heinz and J. R. Hendricks, Magic Square Lexicon:
Illustrated, 2000, 0968798500, page 14
An Order4 Quadragonal Magic Tesseract
A PAN4AGONAL MAGIC
TESSERACT OF ORDER 4
The ability to translocate a
row, or column of a magic square from one side of the square to the
other side without destroying the diagonals is the hallmark of being
pandiagonal.
Pantriagonal magic cubes also have this feature, but instead of
translocating a single row, or column, whole planes of numbers can
be moved.
With the panquadragonal magic tesseract, you will find entire cubes
of number can be translocated. In the table below, the set of
squares on the lefthand side, designated (4,x,y,1) to (1,x,y,l)
constitute a cube. They may be translocated to the righthand side
as a group (to the top in the above diagram) and the “new” tesseract
is magic. None of the main quadragonals has lost its magic property,
even though the numbers in them are different.
J. R.
Hendricks, Magic Squares to Tesseracts by Computer, Selfpublished
1999, 0968470009, page 126
J. R. Hendricks, The Pan4Agonal Magic Tesseract, American
Mathematical Monthly, Vol. 75, No. 4, April 1968, page 384. 
NOTE that the squares and cubes mentioned in the
above quotation are themselves not magic. The diagonals of the squares sum
incorrectly, as do the triagonals of the cubes. Only the rows, columns,
pillars and files and the 8 quadragonals are required to sum correctly for
a simple magic tesseract. This is exactly the equivalent of a
simple magic cube which requires only that the rows, columns, pillars and
4 main triagonals.
The quadragonal magic tesseract is the next higher
class after the simple magic tesseract.
Required to sum correctly for this class of magic tesseract :
m^{3} rows
m^{3} columns
m^{3} pillars
m^{3} files
8m^{3} quadragonals
Equals 12m^{3} lines that sum correctly (other lines may
sum correctly but are not required).
Text format for the above Order4 Quadragonal
Magic Tesseract 
239 
116 
30 
129 

56 
154 
197 
107 

210 
77 
35 
192 

9 
167 
252 
86 
153 
199 
108 
54 
79 
36 
190 
209 
168 
250 
85 
11 
114 
29 
131 
240 
34 
189 
211 
80 
249 
87 
12 
166 
31 
132 
238 
113 
200 
106 
53 
155 
88 
10 
165 
251 
130 
237 
115 
32 
105 
55 
156 
198 
191 
212 
78 
33 
4, x, y, 1 
4, x, y, 2 
4, x, y, 3 
4, x, y, 4 
138 
229 
123 
24 
109 
51 
160 
194 
183 
220 
70 
41 
84 
14 
161 
255 
52 
158 
193 
111 
218 
69 
43 
184 
13 
163 
256 
82 
231 
124 
22 
137 
71 
44 
182 
217 
164 
254 
81 
15 
122 
21 
139 
232 
157 
195 
112 
50 
253 
83 
16 
162 
23 
140 
230 
121 
196 
110 
49 
159 
42 
181 
219 
72 
3, x, y, 1 
3, x, y, 2 
3, x, y, 3 
3, x, y, 4 
19 
144 
226 
125 
204 
102 
57 
151 
46 
177 
223 
68 
245 
91 
8 
170 
101 
59 
152 
202 
179 
224 
66 
45 
92 
6 
169 
247 
142 
225 
127 
20 
222 
65 
47 
180 
5 
171 
248 
90 
227 
128 
18 
141 
60 
150 
201 
103 
172 
246 
89 
7 
126 
17 
143 
228 
149 
203 
104 
58 
67 
48 
178 
221 
2, x, y, 1 
2, x, y, 2 
2, x, y, 3 
2, x, y, 4 
118 
25 
135 
236 
145 
207 
100 
62 
75 
40 
186 
213 
176 
242 
93 
3 
208 
98 
61 
147 
38 
185 
215 
76 
241 
95 
4 
174 
27 
136 
234 
117 
187 
216 
74 
37 
96 
2 
173 
243 
134 
233 
119 
28 
97 
63 
148 
206 
1 
175 
244 
94 
235 
120 
26 
133 
64 
146 
205 
99 
214 
73 
39 
188 
1, x, y, 1 
1, x, y, 2 
1, x, y, 3 
1, x, y, 4 
The colored cells indicate the corners of the
tesseract,
The Perfect Magic Tesseract

Required to sum correctly for
this highest class of dimension 4 magic hypercube are:
 m^{3} rows
 m^{3} columns
 m3 pillars
 m^{3} files
 8m^{3} quadragonals
 16m^{3} triagonals
 12m^{3} diagonals
Equals 40 m^{3} ways (lines)
that sum correctly (this is all the possible paths through any given
point).
Corner values are shown for the order 16
Perfect magic tesseract.
Colors indicate the ends of the 8 main quadragonals. 
The order16 is the smallest possible perfect magic
tesseract.
John R. Hendricks constructed the first one in 1998.
It contains the consecutive numbers from 1 to 65,536 and has the magic sum
of 1,048,592 which is obtained in the following 163,840 ways.
49,152 diagonals
65,536 triagonals
32,768 quadragonals
16,384 rows columns pillars and files
It contains 64 perfect magic cubes and 1 536 perfect (pandiagonal) magic
squares.
This perfect tesseract and the order 32 perfect
dimension5 hypercube he later constructed are obviously too large to
publish here. The numbers in these two hypercubes may be inspected using
his simple Pathfinder programs.[1]
More information on this order 16 and an order 6
Inlaid magic tesseract may be seen on
John R. Hendricks
archived web site.
[1] J. R. Hendricks, Magic Squares to Tesseracts by Computer,
Selfpublished 1999, 0968470009, Appendix C.
A 5Dimensional Perfect Magic Hypercube of
order32
This magic hypercube was constructed by John
Hendricks in 2000.
33,554,432
5,242,880
327,680
10,240
160 
Consecutive numbers, resulting
in:
irows (rows, columns, etc)
perfect magic squares
perfect magic cubes
perfect magic tesseracts
Which translates to: 
5,242,880
20,971,520
41,943,040
41,943,040
16,777,216
126,877,696 
irows (monagonal)
diagonals
3agonals (triagonals)
4agonals (quadragonals)
5agonals
total lines, each summing to 536,870,928 
J. R.
Hendricks, Perfect nDimensional Magic Hypercubes of Order 2^{n},
Selfpublished,1999, 0968470041, page 25
Pathfinders
Obviously, magic objects with this many numbers
cannot be printed out for visualization or study. Mr. Hendricks has come
up with the concept of pathfinders. These are relatively simple computer
programs [1][2] that generate the numbers that
appear in any magic line through the hypercube. Conversely, the programs
may be used to find the coordinate position of a given number in the magic
hypercube.
Using the tesseract as an example: To show any row,
column, or whatever you want. Pick any starting position (w,x,y,z) and any
route from 1 to 40 and enter the values into the computer. Out come the
numbers and the sum. If you want the reverse direction enter the route as
a negative number (–5 instead of 5) and the numbers will all be reversed.
For coordinates, 0 is the same as 16. For routes, zero is not a route.
Some of the routes might be:
 Route #1 is a pillar. Zchanges.
 Route #3 is a column. Only y changes.
 Route #9 is a row. Only x changes.
 Route #27 is a file. Only w changes.
 . . .
 Route #40 is the main quadragonal.
[1] J. R. Hendricks, Magic Squares to Tesseracts by Computer,
Selfpublished 1999, 0968470009, Appendix C.
[2] J. R. Hendricks, Perfect nDimensional Magic Hypercubes of
Order 2^{n}, Selfpublished,1999, 0968470041, pages 2736.
Conclusions and
References
On this page I have attempted to explain the meaning
and reasoning behind the term ‘perfect’ when applied to magic hypercubes.
In an attempt to avoid confusion, I have included definitions for related
terms.
I have expanded on this idea with illustrations and tables, including 2
orders of perfect magic cubes.
Much of the information here appears also on other pages, but has been
included here for convenience.
Much of the material for this page was obtained from
the two books below by John Hendricks. These books, however, go into much
greater detail and should be referred to if a more indepth explanation is
desired.
Over many years, Mr. Hendricks has contributed much
to the study of higher order magic hypercubes. However, He has also done a
lot of original work with the lowly magic square, especially in the
development of unique variations.
I encourage you to see more of his work via the link
below. Also much of his work is featured on my
J. R. Hendricks page. Also, on my newer section (of this site) on
Tesseracts.
J. R. Hendricks, Perfect nDimensional
Magic Hypercubes of Order 2n, Selfpublished,1999, 0968470041.
36 pages of theory, equations and discussion on the definition of perfect
as related to magic objects.
Also pathfinder basic programs for order16 and 32 perfect dimension 4 and
5 hypercubes
J. R. Hendricks, Magic Squares to
Tesseracts by Computer, Selfpublished 1999, 0968470009
212 pages of text, theory, appendices, diagrams , etc, including basic
program listings.
H. D. Heinz and J. R. Hendricks,
Magic Square Lexicon: Illustrated, 2000, 0968798500
239 definitions, most with illustrations, dealing with a large variety of
magic objects.
More information and how to order the book is
here.
Clifford A. Pickover, The Zen of Magic
Squares, Circles, and Stars, Princeton Univ. Pr., 2002, 0691070415
A great new book destined to become a classic on magic squares, cubes,
etc.
Specifically mentioned on page 101 is the new requirements for a cube to
be considered 'perfect'.
W. H. Benson and O. Jacoby, Magic
Cubes: New Recreations, Dover, 1981, 0486241408
A great book, but terminology and examples are not too clear. He refers to
'perfect' and 'pandiagonal perfect' cubes
but these are our definition for pandiagonal and pantriagonal magic cubes.
Martin Gardner, Time Travel and Other
Mathematical Bewilderments, W. H. Freeman & Sons, 1988, 071671924X.
This includes a chapter on the first 'perfect' magic cubes, but these are
all by the old definition.
This is mostly from his Scientific America, Jan. 1976 column. He includes
references to some old works.
W. S. Andrews, Magic Squares & Cubes,
Dover Publ., 1960 (original publication Open Court,1917)
The 'bible' on the subject of magic objects. Of course, it is very much
out of date but indicates how much was known 100 years ago.
Here is a more extensive
bibliography of magic hypercube related
literature.
