Magic Cube Definitions

I have chosen to write this page in narrative form. The six sections each include a number of terms, which I show in boldface. The meaning of these terms, if not specifically stated, can be inferred from the contents.
Most of these terms are also be explained, using different wording, on my perfect.htm and perfect-2.htm pages.
For a more comprehensive list of definitions for magic objects, go to my glossary.htm page .
And for still more definitions, refer to the book Magic Square Lexicon: Illustrated. [1]
Finally, for a very comprehensive list of concise definitions, see Mitsutoshi Nakamura's http://homepage2.nifty.com/googol/magcube/en/terms.htm.
These definitions for the most part, are consistent with those in the above sources, but expressed in a logical, concise format and include many tesseract classes as well. Especially note the section Conditions on (pan-)r-agonals.

To simplify my explanations, I will use the general term hypercube when referring to magic objects that may be a square, a cube, a tesseract, etc.

I will use m to indicate the order of the hypercube, and n to indicate the dimension.

[1] For information about the Lexicon see booksale.htm.

Aspect

An aspect is an apparently different but in reality only a disguised version of the magic square, cube, tesseract, etc. It is obtained by rotations and/or reflections of the basic figure.
Once one has a hypercube of any dimension, through mirror images (reflections) and rotations one can view the hypercube in many ways.

There are:  A = (2n) n! ways of viewing a hypercube of dimension n.

I will illustrate aspect by first using a magic square, then a magic cube. These examples of aspect will continue into the section on associated.

This order-3 magic square (fig. a) is the smallest possible with primes in arithmetic progression. [1] The second magic square (fig. b) seems to be a different one. However, closer inspection reveals that it is figure a reflected around a horizontal axis. What was the top is now the bottom, and what was the bottom is now the top. It turns out that there are 7 variations of the same original magic square. Normally, when counting magic hypercubes, the aspects are not included, but are assumed to be variations of the same hypercube. Figure # 1 - Basic is one of 4 basic cubes of order 3. As mentioned above, each of these has 48 variations (aspects) due to rotations and /or reflections.

It is called a basic cube because it cannot be transformed into another basic cube by rotations and reflections. Which of the 48 aspects is designated the basic one is rather arbitrary. John Hendricks [2] defined the standard position as the one with the lowest corner in the bottom left position, and the 3 numbers adjacent to that corner in increasing order in the x, y and z directions. Here I have shown these four index numbers in red. The second cube shown is a rotation and reflection of cube #1. Again I have shown the four index numbers in red. However, obviously this cube is not in the standard position. In the next section, I show still another aspect of this same magic cube. Notice that in all cases, the center cell is the same. This will always be the case for odd order magic objects that are different aspects of the same construction. [1] H. E. Dudeney, Amusements in Mathematics, Dover Publ. 1958, p. 246 [2] John R. Hendricks, The Third-Order Magic Cube Complete, JRM 5:1:1972, pp 43-50

Associated

A magic hypercube where all pairs of cells diametrically equidistant from the center of the hypercube equal the sum of the first and last terms of the series, or m2 + 1 for a pure magic square, is called associated. These number pairs are said to be complementary. The series used is consecutive and starts at 1 in a normal magic hypercube. However, even if the series consists of non-consecutive numbers, the complement of a particular number is found by subtracting it from the sum of the first and last numbers in the series. This type of magic square is often referred to as a symmetrical magic square.

All order 3 hypercubes are associated. The center cell of odd order associated magic hypercubes is always equal to the middle number of the series.
However, an odd order magic hypercube that is not associated, may also have the middle number of the series in the center cell!

I will show some example associated magic squares, then an associated magic cube.

	Luoshu-1 is the smallest magic square and the only one of order 3 (not counting the different aspects) and is associated. It uses the series of numbers from 1 to 9. The middle number, 5, appears in the center of the square. To obtain the complement of each number, we subtract it from 10, which is equal to 1 + 9 (and also m2 + 1). We do this with each number in the square to obtain the complementary magic square (Luoshu-2). This is a different aspect of the same Luoshu-1 square and could also have been obtained by rotating the original 180 degrees. The fact that the compliment of the magic object is a different aspect of the original is called self-similar. This happens with all associated magic hypercubes because they are center-symmetric. If the magic hypercube is horizontal-symmetric (symmetric across a horizontal line) or vertical-symmetric, the object is also self-similar. Note that only center-symmetric hypercubes are called associated! However, if the magic object is not associated, or is not symmetric as described above, the resulting magic hypercube will be a completely different one! This order-3 magic square (fig. c) is the smallest possible with primes in arithmetic progression. While it looks like a new magic square, it is actually a complement of figure a in the section on aspects. So it is another example of a self-similar magic square and is a 180 degree rotated aspect of the original. Because the numbers used are prime numbers they obviously are not consecutive. The sum of the smallest number and the largest number is 2078. Each of the four pairs of numbers that surround the center of the square sum to 2078, making this an associated magic square. We expected this because this is an order 3 hypercube! If the magic hyper cube is an even order, it may still be associated. In this case there is no center cell. The complement pairs are diametrically opposite the center point of the magic object. This order 4 magic square (fig. d) is associated. The number series used is 1 to 16. To complement this square we subtract each number from 17 i.e. 16 + 1 (or m2+1). And again, we will obtain a different aspect, rotated from the original by 180 degrees. This cube is associated, because, as we stated previously, all order 3 hypercubes are. However, that is confirmed when we notice that the two numbers on each side of the center number in all cases sum to 28 which is 33 + 1. The cube shown here is a third aspect of the cube shown in the section on aspect. By subtracting each number from 28 (which is the sum of 27 + 1), we obtain a fourth aspect of the same cube. The central partitions of all odd order associated magic hypercubes are themselves associated magic. Examples: The 3 central square arrays in an odd order magic cube are associated magic squares. The 4 central cube arrays in an odd order associated magic tesseract (order 4 hypercube) are themselves associated magic cubes. Each of these cubes, in turn, have 3 central associated magic squares.

Point of interest. Any of the four basic order 3 magic cubes can be reconstructed from just the four index numbers, because the cubes are associated and contain only three numbers per line.

NOTE: To find out more about even order symmetric (not associated) magic cubes see the self similar page.

Constant (S)

The sum produced by each row, column, and n-agonal (space diagonal) of a magic hypercube (and possibly other arrangements). This value is also called the magic sum.

The order (m) is the size of the hypercube. It indicates how many cells for holding numbers, that each line consists of.

For a normal magic square, S = m(m²+1)/2.
Minimum requirement is m rows, m columns and 2 diagonals

For a normal magic cube,    S = m(m³+1)/2.  Minimum requirements for these classes.
    Simple magic ------------- m² rows, m² columns, m² pillars and 4 triagonals 
    Pantriagonal magic------ m² rows, m² columns, m² pillars and 4m² pantriagonals
    Diagonal magic------------m² rows, m² columns, m² pillars, 6m diagonals and 4 triagonals
    Pantriagonal Diagonal---m² rows, m² columns, m² pillars, 6m diagonals and 4m² pantriagonals
    Pandiagonal magic------- m² rows, m² columns, m² pillars, 6m² pandiagonals and 4 triagonals
    Perfect magic------------- m² rows, m² columns, m² pillars, 6m² pandiagonals and 4m² pantriagonals    

For a tesseract (simple magic)                     S = m(m⁴+1)/2.
Minimum requirement is m rows, m columns, m pillars, m files and 8 quadragonals

In general; for a n-dimensional simple magic hypercube S = m(mⁿ+1)/2.

The above equations all assume the hypercube is using a series of consecutive numbers that start with 1.
There is no reason this must be the case but it is so for a normal or true magic square, cube, etc.

If the hypercube uses consecutive numbers that do not start with 1. the equation is S = (mn+1) + m)/2 + m(a-1),
where m = order, n = dimension, and a = starting number.

Magic ratio

These terms were defined by Walter Trump in January, 2004. Their value is mainly for cubes that are almost magic.
They are also of value for cubes that are simple magic but not quite diagonal magic (magic ratio).
Also for measuring magic cubes against a perfect cube (panmagic ratio).

Magic cube ratio:
The magic ratio is the number of correct monagonals, diagonals, and triagonals
divided by the highest possible for a diagonal magic cube, which is 3m2 monagonals + 6m diagonals + 4 triagonals.

Panmagic cube ratio:
The panmagic ratio is the number of correct monagonals, pandiagonals, and pantriagonals divided by the highest possible, which is 3m²  monagonals + 6m² pandiagonals + 4m² pantriagonals.

Compact and Complete

These are terms that are relatively new and have not seen much usage to date. As we study and compare cube features, I will be using these terms quite often.

I will describe the terms generally, then will attempt to clarify the explanation with a diagram.

Compact:
This term was originally defined by Gahuko Abe [1] in 1990 as a feature of certain even order magic squares. Compact magic squares consist of 2x2 subsquares where the 4 cells sum to S.

Kanji Setsuda [2] extended the concept to magic cubes of order 4 where the four cells of all 2z2 squares, parallel to the sides of the cube sum to 130 (which is S for order 4 cubes).

However, Setsuda used the term composite instead of compact. Composite, though, is an old term for magic squares that are composed from a group of small magic squares arranges as a magic square.

For that reason, I am reverting back to Abe’s original term ‘compact’, to avoid confusion.

CompactPlus:
I have expanded on the concept of compact by proposing the term compactplus for the following. If the corners of all sub-cubes of a given order within the cube sum to 8S/m where m is the order of the parent cube, then that cube is compactplus.

By this definition, an order 4 cube that is compact may be considered compactplus because two parallel 2x2 squares, in fact, form a 2x2x2 cube. The corners of that cube are 8*130/4.

However, compactplus is really meant for higher even-order cubes where all sub-cubes of one , or more, orders sum correctly. For example, it seems that all perfect (nasik) cubes of order 8 have all sub-cubes of all orders (2 to 8) correct. Not all higher even-order nasik are compactplus!
Because 'all' includes wrap-around, there are m3 sub-cubes of each order from 2 to m, contained within a cube.
Of course, because of wrap-around, many of these m3 sub-cubes will just be different arrangements of the same 8 numbers!

Complete:
Also defined by Kanji Setsuda [2]. To be complete, every pantriagonal must contain m/2 complement pairs, and the two members of each pair must be spaced m/2 apart. (Point-of-interest. This is one requirement of a most-perfect magic square). As with the compact cubes, the cube must be of even order. Also, the cube must be pantriagonal or perfect (which implies pantriagonal).

Proper:
Refers to a cube that contains exactly the minimum requirements for that class of cube. i.e. a proper simple or pantriagonal magic cube would contain no magic squares, a proper diagonal magic cube would contain exactly 3m plus 6 simple magic squares, etc.
NOTE that NO odd order associated magic hypercube can be proper because all of these contain 3 central associated magic hypercubes of lower dimensions!
This term was coined by Mitsutoshi Nakamura in an email of April 15, 2004

This example cube is pantriagonal, compact (and compactplus), complete, and proper.

Demonstration of compact.  Three 2x2 squares parallel to the cube faces 7, 50, 16, 57 are parallel to the sides of the cube. 56, 41, 22, 11 are parallel to the front of the cube. 49, 48, 25, 8 are parallel to the top of the cube. Demonstration of compactplus. 56, 41, 22, 11 and 1, 32, 35, 62 together are the 8 corners of a 2x2x2 sub-cube. These 8 corners sum to 260, which is 8/4 of 130.

Demonstration of complete. Because in the example cube, the number series used is 1 to 64, the sum of the two complement members must be 1 + 64 = 65. I will show 3 example pantriagonals. 21, 2, 44, 63   is a 1 segment pantriagonal (a main triagonal). 36, 59, 29, 6   is a 2 segment pantriagonal. 20, 55, 45, 10 is a 3 segment pantriagonal. Demonstration of proper Because no diagonals are correct, there are NO magic squares in this cube, so it meets just the minimum requirements for a pantriagonal magic cube.

No order 3 are proper
No order 3 magic cube is proper because all contain magic squares. In fact, this is true for order-3 rectilinear objects of all higher dimensions.

[1] Gakuho Abe, Fifty Problems of Magic Squares, Self published 1950. Later republished in Discrete Math, 127, 1994, pp 3-13. The last 10 problems deal with magic cubes.
[2] Kanji Setsuda’s excellent Compact (composite) and Complete magic Cubes Web pages may be accessed from. http://homepage2.nifty.com/KanjiSetsuda/pages/EnglishP1.html

Cube parts

Rows, columns, pillars, files, diagonals, pandiagonals, triagonals, pantriagonals, orthogonal planes, diagonal (oblique) planes, broken diagonal planes. For simplicity, I am using an order 3 cube as an example (see fig. a). Order 3’s only features are that it is simple and associated. Additional features require higher orders. Only the rows, columns, pillars and triagonals are required to sum correctly for a simple magic cube. More parts must sum correctly for the more advanced cubes.

Note that two diagonals go from corner to opposite corner of each of the 3m orthogonal planes (2-D figures). Four triagonals go from corner, through the center, to opposite corner of the one cube (3-D figure).
Pandiagonals are the m-1 two-segment lines that parallel each diagonal.
Pantriagonals are the m²-1 two or three-segment lines that parallel each triagonal.

Orthogonal planes are those parallel to a face of the magic cube. This is one of 3m such planes in a magic cube. Rows and columns must sum correct for the cube to be magic. The 2 diagonals do not have to be correct. However, if both are correct in a particular plane, then that plane is a magic square. In this case they are, because the plane shown is one of the 3 central planes of an associated magic cube. As such, it must be an associated magic square.

Diagonal planes (I usually call them oblique planes) are those that go from 1 edge of the cube, through the center, to the opposite edge. There are 6 of them joining the 12 edges of the cube. In all 6 of these planes, both main diagonals must always be correct. That is because the total of 12 diagonals are actually the 4 main triagonals of the cube. In each plane, all rows or all columns must be correct (because they are the rows or columns of the orthogonal planes. If all rows and all columns in a particular plane are correct, then that plane is a magic square.

Broken diagonal planes are the 2-dimensional equivalent of the broken diagonal pairs in a pandiagonal magic square. There are m-1 of these parallel to each of the six diagonal planes, The only time these are of concern is in a perfect (nasik) magic cube. In that case, each of these broken plane pairs automatically form a pandiagonal magic square.

Classifications

This classification method is comprehensive, coordinated, and may be consistently expanded for ever greater dimensions. It was developed over the last 20 years or so by John Hendricks to avoid the ambiguity over the term ‘perfect’ as applied to magic cubes. [1]

Shown here are the 6 classes of dimension-3 hypercubes, A reminder that there are only 2 classes of dimension-2 hypercubes; the simple magic square and the pandiagonal (nasik) magic square. There are 18 classes of dimension-4 hypercubes (the magic tesseract). They are listed here.

A. H. Frost published pantriagonal, pandiagonal and perfect magic cubes in 1866 and 1878. He referred to them as Nasik cubes, and did not differentiate between the three types. [2].

Another early source on this subject is C. Planck's Theory of Paths Nasik published in 1905. See a direct quote from his paper at the end of this page.

For more information on perfect magic cubes and some history on the confusion over this name, go to my cube_perfect.htm and cube_perfect-2.htm

Simple magic:

m² rows, m² columns, m² pillars and 4 triagonals = S
Many simple magic cubes have some correct diagonals which means some magic squares. Some of the oblique planes may have all correct rows and columns, meaning that they also are magic squares. The diagonals of the oblique planes are already correct, because they are actually the main triagonals of the cube!

Mitsutoshi Nakamura suggested (April 15, 2004) the use of the term 'proper' for a cube that contains only the minimum features required for it's class.
A simple magic cube with NO magic squares for example, would thus be referred to as a proper simple magic cube.

Pantriagonal magic:

Pantriagonal Diagonal:

m² rows, m² columns, m² pillars, 6m diagonals and 4m2 pantriagonals = S
The above sentence indicates that all 3m orthogonal arrays are order m simple magic squares. All six oblique planes are also magic squares.
In addition, all 4m² pantriagonals must sum correctly. This is a combination Pantriagonal and Diagonal magic cube.

This new class of magic cube was discovered by Mitsutoshi Nakamura in 2004 and named by him. 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 known one being order 8 (and unassociated) [5]. More about this on the Update-3 page.

Pandiagonal magic:

m² rows, m² columns, m² pillars, 6m² pandiagonals and 4 triagonals = S
The above sentence indicates that all 3m orthogonal arrays are order m pandiagonal magic squares. All six oblique planes are also magic squares, usually simple, but it is possible that some of them are pandiagonal.

Order 7 is the smallest pandiagonal magic cube possible.
I have seen 12 of these cubes (Dec. 2003) All were order 7, and all but one were associated.
In March , 2004, I received 7 more pandiagonal magic cubes, of odd orders 9 to 19. Five of these were not associated.

Perfect (nasik) magic:

m² rows, m² columns, m² pillars, 6m² pandiagonals and 4m² pantriagonals = S
The above sentence indicates that all 3m orthogonal arrays are order m pandiagonal magic squares. As well, all six oblique planes are pandiagonal magic squares.

Furthermore, because all pantriagonals are correct, there are m-1 broken pandiagonal magic squares parallel to each of the 6 oblique pandiagonal magic squares. [3][4]. Each of these broken magic squares consist of two segments. These oblique planes may be transposed as horizontal or vertical planes to form a different magic cube.

Order 8 is the smallest perfect magic cube possible.
Perfect magic cubes of orders 10, 14, 18, etc. and 12, 20, etc. are impossible. this was proved by Rosser and Walker in 1939 [4], and by Stertenbrink and de Winkel in 2004. . See Pandiagonal Impossibility Proof.

Above is Hendricks definition. Because of the confusion over the definition of perfect, I suggest adopting the term nasik.  See my reason in The Theory of Paths Nasik.

[1] H. D. Heinz & J. R. Hendricks, A Unified Classification system for Magic Cubes, JRM, 32:1, pp. 30-36, 2003

[2] A. H. Frost, Invention of Magic Cubes. Quarterly Journal of Mathematics, 7, 1866, pp 92-102
      A. H. Frost, On the General Properties of Nasik Cubes, QJM 15, 1878, pp 93-123 plus plates 1 and 2.
[3] F. Liao, T. Katayama, K. Takaba, Technical Report 99021, School of Informatics, Kyoto University, 1999. Available on the Internet at
      http://www.amp.i.kyoto-u.ac.jp/tecrep/TR1999.html
[4] B. Rosser and R. J. Walker, Magic Squares: Published papers and Supplement, 1939, a bound volume at Cornell University, catalogued as QA 165 R82+pt.1-4.
[5] Mitsutoshi Nakamura has a good Web site at http://homepage2.nifty.com/googol/magcube/en/

Classification of cubes by included magic squares

A cube is considered 'proper' if it contains only the minimum features to qualify for the classification (suggested by Mitsutoshi Nakamura).

Simple magic cubes
May have some orthogonal or oblique magic squares, either simple or pandiagonal.

Pantriagonal magic cubes
May have some orthogonal or oblique magic squares, either simple or pandiagonal.
Only 2 of the 36 pantriagonal magic cubes of orders 4 to 11 that I tested, had any oblique magic squares.

Diagonal magic cubes
All 3m orthogonal and 6 oblique planes are simple magic squares.
An order 8 diagonal magic cube, for example, contains 30 order 8 simple magic squares. That is, 3m planar, and 6 oblique magic squares.
For a short time, I referred to this type of cube as a 'myers' cube. The term 'diagonal', suggested by Aale de Winkel, is much more descriptive, however.
On Nov. 13, 2003, Trump and Boyer discovered an order 5 diagonal magic cube. It contains 21 simple magic squares. There are now diagonal cubes for all orders greater then 4!
NOTE that Trump and Boyer refer to this class as a `perfect` cube.

Pantriagonal Diagonal
is a  diagonal magic cube that is also a pantriagonal magic cube. First discovered in late 2004 by Mitsutoshi Nakamura. the only known such cube (Jan. 2005) is order 8.

Pandiagonal magic cubes
All 3m orthogonal are pandiagonal magic squares. The 6 oblique planes are simple magic squares (several of these may be pandiagonal).
An order 7 pandiagonal magic cube, for example, the smallest possible, contains 21 or 22 order 7 pandiagonal magic squares (and 6 or 5 simple magic squares).

Perfect (nasik) magic cubes
Are a combination of pantriagonal and pandiagonal magic cube features.
All 3m orthogonal and 6m oblique planes are pandiagonal magic squares. (All but 6 of the oblique planes consist of two segments)
An order 8 perfect magic cube, the smallest possible, contains 72 order 8 pandiagonal magic squares.

Order 9 is the smallest possible perfect associated magic cube. The precedent for this is already set for perfect (pandiagonal) magic squares. Order 4 is the smallest of these, but order 5 is the smallest pandiagonal associated.

Higher Dimensions
A tesseract is a 4-dimensional hypercube. It is perfect (nasik) if all pan-quadragonals 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 40m³ lines that sum correctly. They are m³ rows, m³ columns, m³ pillars, m³ files, 8m³ quadragonals, 16m³ triagonals, and 12m³ diagonals.

A magic hypercube of ANY dimension n is perfect and nasik if all pan-r-agonals sum correctly, and all lower dimension hypercubes contained in it are perfect!
The r in pan-r-agonal indicates that we are talking about all the agonals (orthogonal, diagonal, triagonal, quadragonal, etc). Pan indicates all, i.e. main and broken.

Confusion over the term Perfect Magic Cube - Historically, the term perfect has been used to describe any cube with unusual features.

• In late 2003, Christian Boyer and Walter Trump defined `perfect` to mean what I call a Diagonal magic cube. They group the higher classes also under the term perfect, but with enhancements.
   
• In late 2004, Mitsutoshi Nakamura adopted the term Pan-2,3-agonal to replace the term perfect on his website. In this way, he is avoiding the misunderstandings and ambiguities of the past.
   
• I, Harvey Heinz, will continue to support John Hendricks definition of Perfect, and the coordinated system of magic cube class definitions. I feel this system has real merit because it is easy to understand, is logical, and is consistent for all dimensions and orders of hypercubes.
   
• Recently, Heinz filled two gaps in Hendricks coordinated hypercube definition by defining two new types of magic cubes. It is now understood that as we go to higher dimensions, more classes will appear. However, they will appear as gaps between the recognized classes. The perfect class will always be the hypercube with the highest number of features (and it's definition will not change).

By the end of 2005, this classification system is gaining in popularity!
In the last few years, the term `nasik` is being used more frequently as a substitute for Hendricks term `perfect`, in an effort to avoid confusion (see the next section).

Theory of Paths Nasik

Nasic, pan-r-agonal, and perfect.

Comments by Dr. C. Planck in 1917 (originally 1905) on the features of a perfect magic hypercube.

This is a direct quote by Dr. C. Planck from W. S. Andrews, Magic Squares and Cubes. 1917, Pages 365,366.

If the Nasik criterion be applied to 3-dimensional magics what does it require? We must have 3 magic directions through any cell parallel to the edges, (planar contact), 6 such directions in the diagonals of square sections parallel to the faces (linear contact), and 4 directions parallel to the great diagonals of the cube (point contact), a total of 13 magic directions through every cell. It has long been known that the smallest square which can be Nasik is of order 4, or if the square is to be associated, (that is with every pair of complementary numbers occupying cells which are equally displaced from the center of the figure in opposite directions), then the smallest Nasik order is 5. Frost stated definitely that in the case of a cube the smallest Nasik order is 9. Arnoux was of the opinion that it would be 8, though he failed to construct such a magic. It is only quite recently that the present writer has shown that the smallest Nasik order in k dimensions is always 2^k, (or 2^k+1 if we require association).

It is not difficult to perceive that if we push the Nasik analogy to higher dimensions, the number of magic directions through any cell of a k-fold must be 1/2(3^k-1), for we require magic directions from every cell through each cell of the surrounding little k-fold of order 3. In a 4-fold Nasik, therefore, there are 40 contiguous rectilinear summations through any cell.

This subject is discussed in much greater detail in [2]. In the introduction to that paper he says

Analogy suggests that in the higher dimensions we ought to employ the term nasik as implying the existence of magic summations parallel to any diagonal, and not restrict it to diagonals in sections parallel to the plane faces. The term is used in this wider sense throughout the present paper.

It must be pointed out here that Frost used the term nasik for both his order 7 pandiagonal and order 8 pantriagonal magic cubes in his 1866 paper. [1]
Planck, in the above quote is using it for the perfect magic cube (which is a combination of the pantriagonal and pandiagonal cubes). [2][3]

By Frost's definition:

• There is one nasik magic Square. Hendricks (and many others) call it pandiagonal OR perfect.
• There are two nasik (Frost) magic cubes. Hendricks called them pantriagonal and pandiagonal.
  The combination of these 2 cubes is nasik (by Plancks definition) Hendricks called it perfect.
• There are three nasik (Frost) magic tesseracts. Hendricks called them pantriagonal, pandiagonal, and panquadragonal.
  The combination of these 3 tesseracts is nasik (by Plancks definition) Hendricks called it perfect.

(Actually, there are also tesseracts with combination of two of the three pan-r-agonals. These are not Nasik.)

To eliminate confusion between Hendricks and Boyer/Trump 'perfect', I suggest Planck's term nasik for hypercubes where all lines sum correctly.

[1]  A. H. Frost, Invention of Magic Cubes, Quarterly Journal of Mathematics, 7, 1866, pages 92-102.  See page 99, para. 23 and page 100, para. 26.
[2] C. Planck, The Theory of Path Nasiks, Printed for private circulation by A. J. Lawrence, Printer, Rugby (England),1905
         (Available from The University Library, Cambridge).
[3]   W. S. Andrews, Magic Squares and Cubes. Open Court Publ.,1917. Pages 365,366, by Dr. C. Planck.
         Re-published by Dover Publ., 1960 (no ISBN); Dover Publ., 2000, 0486206580; Cosimo Classics, Inc., 2004, 1596050373

	This page was originally posted February 2003 It was last updated December 09, 2009 Harvey Heinz   harveyheinz@shaw.ca Copyright © 1998-2009 by Harvey D. Heinz