Introduction Magic Squares
Magic Cubes Magic Tesseracts
Introduction
Magic hypercubes may be
classified by reference to which ragonals sum correctly to the
magic constant. For magic squares and magic cubes, this has been discussed
previously. [1]
[2] [3]
[4]
A brief review will be presented on this page, using a different
prospective. Then I will present the classes for the magic tesseract. Note
that within these classes, the hypercube may have additional features,
such as associated, compact complete, inlaid, multiply, etc.
On this page, the variable
r will range from 1 to m, where m indicates the order. n,
as usual on my pages, will indicate the dimension of the hypercube.
Another name for 1agonal is orthogonal lines (those parallel to the edges of
the hypercube). Only 1 coordinate change when moving along the line.
Another name for 2agonal is diagonal. 2 coordinates change when moving along
the line.
Another name for 3agonal is triagonal. 3 coordinates change when moving along
the line.
Another name for 4agonal is quadragonal. 4 coordinates change when moving along
the line.
A particular hypercube may have some, but not all of the correct ragonals
that would qualify it for a higher class.
The prefix pan
indicates all of that ragonal, both 1segment and multisegment
(broken).
A pandiagonal magic square
may be transformed to another pandiagonal magic square by moving a row or column
from one side of the square to the opposite side. Similarly, a pantriagonal
magic cube may be transformed into another pantriagonal magic cube by moving a
plane from one side of the cube to the other! Furthermore, a panquadragonal
magic tesseract may be transformed to another one by moving a cube from one side
to the other! Etc.
Magic
squares (n = 2)
There are only two classes of
magic squares.
Class name 
Minimum requirements 
Minimum correct summation 
Lowest
order 
Simple 
All 1agonals and the
two main 2agonals sum correctly. 
2m +
2 
3 
Pandiagonal (nasik) 
All 1agonals and all
2agonals sum correctly.
This class has been referred to historically as being Perfect, and
has been so called in Hendricks Universal Classification System. Frost
called it Nasik.
To avoid confusion in the higher dimensions for this highest possible
class, I have started using the term nasik in place of perfect.
[5] 
4m 
4 
Magic
cubes (n = 3)
There are six classes of
magic cubes.
Class name 
Minimum requirements 
Minimum correct summations 
Lowest
order 
Simple 
all 1agonals and the
four main 3agonals sum correctly. 
3m^{2} + 4 
3 
Pantriagonal 
all 1agonals and all
3agonals sum correctly. 
7m^{2} 
4 
Diagonal 
all 1agonals and the
four main 3agonals sum correctly. In addition, the two main diagonals
(2agonals) of each orthogonal plane sum correctly. 
3m^{2}+6m+4 
5 
Pantriagonal
diagonal 
all 1agonals and all
3agonals sum correctly. In addition, the two main diagonals (2agonals)
of each orthogonal plane sum correctly. 
7m^{2}+6m 
5 
Pandiagonal 
all 1agonals and the
four main 3agonals sum correctly. In addition, all 2agonals of each
orthogonal plane sum correctly. 
9m^{2} + 4 
7 
Nasik 
this is a combination
Pantriagonal and Pandiagonal cube, so all 1agonals and all 3agonals sum
correctly, and all 2agonals of each orthogonal plane sum correctly.
Hendricks calls this top class (all possible lines sum correctly)
perfect. Nakamura calls it pan2,3agonal. 
13m^{2} 
8 
Magic
tesseracts (n = 4)
There are
2 classes of magic squares and 6 classes of magic cubes. So it is to be expected
that there will be many more classes of magic tesseract. In fact, there are 18.
Names of these are arbitrarily chosen to be descriptive, rather then concise. As
in the case for the square and the cube, these classes are listed in order of
increasing number of correct lines. In late 2007, Mitsutoshi Nakamura had
constructed a tesseract in most of these classes. [4]
Class name 
Minimum requirements 
Minimum
correct summations 
Lowest
order 
Simple 
All 1agonals and the
eight main (1segment) 4agonals sum correctly.
This is a basic requirement for classes of tesseract to be magic. 
4m^{3} + 8 

Triagonal 
Basic + all main
(1segment) 3agonals sum correctly. 
4m^{3} + 16m + 8 
4 
Diagonal 
Basic + all main
(1segment) 2agonals sum correctly. 
4m^{3 }+ 12m^{2}
+ 8 
4 
Diagonal + Triagonal 
Basic + all main
2agonals and 3agonals sum correctly. 
4m^{3} + 12m^{2}
+ 16m + 8 
8? 
Panquadragonal 
Basic + all 4agonals sum
correctly. 
12m^{3} 
4 
Triagonal + Pan4 
Basic + all 1segment
3agonals + all 4agonals. 
12m^{3}
+ 16m 
4 
Diagonal + Pan4 
Basic + all 1segment
2agonals + all 4agonals. 
12m^{3}
+ 12m^{2} 
8? 
Diagonal + Triagonal +
Pan4 
Basic + all 1segment
2agonals + all 1segment 3agonals + all 4agonals. 
12m^{3}
+ 12m^{2} + 16m 
8? 
Pandiagonal 
Basic + all 2agonals. 
16m^{3} + 8 
9? 
Triagonal + Pan2 
Basic + all 1segment
3agonals + all 2agonals. 
16m^{3}
+ 16m + 8 
? 
Pantriagonal 
Basic + all 3agonals.

20m^{3} + 8 
4 
Diagonal + Pan3 
Basic + all 1segment
2agonals + all 3agonals. 
20m^{3}
+ 12m^{2} + 8 
? 
Pan2 + Pan4 
Basic + all 2agonals +
all 4agonals. 
24m^{3} 
13? 
Triagonal + Pan2 +
Pan4 
Basic + all 1segment
3agonals + all 2agonals + all 4agonals. 
24m^{3} + 16m 
16? 
Pan3 + Pan4 
Basic + all 3agonals +
all 4agonals. 
28m^{3} 
4 
Diagonal + Pan3 + Pan4 
Basic + all 1segment
2agonals + all 3agonals + all 4agonals. 
28m^{3}
+ 12m^{2} 
8 
Pan2 + Pan3 
Basic + all 2agonals +
all 3agonals. 
32m^{3} + 8 
15? 
Nasik
[5] [6]
[7] [8] 
Basic + all 2agonals +
all 3agonals + all 4agonals.
Hendricks calls this top class (all possible lines sum correctly)
perfect.
Nakamura calls it pan2,3,4agonal [4]. 
40m^{3} 
16 
Statistical
information on these classifications are shown in tables on my
Hypercube Math page.
I have not personally checked most of the tesseracts mentioned in this table.
Footnotes
[1] Heinz & Hendricks, A Unified Classification
System for Magic Hypercubes, Journal of Recreational Mathematics, 32:1,
20032004, pages 3036
[2] H. D. Heinz, The First (?) Perfect Magic
Cubes, JRM, 33:2, 20042005, pages 116119
[3] My
Six Classes of Cubes
article
[4] Mitsutoshi Nakamura’s
http://homepage2.nifty.com/googol/magcube/en/classes.htm
[5] Nasik from Frost’s original term for pandiagonal
magic squares, and Planck’s subsequent expansion of the definition to include
the highest class of all dimensions of magic hypercubes. See [6][7][8] and my
Nasik article. Less ambiguous then perfect!
[6]
A. H. Frost,
Invention of Magic Cubes, Quarterly Journal of Mathematics, 7, 1866,
pages 92102. See page 99, para. 23 and page 100, para. 26.
[7] 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).
[8] W. S. Andrews, Magic Squares and Cubes. Open
Court Publ.,1917. Pages 365,366, by Dr. C. Planck. Republished by Dover Publ.,
1960, Pages 365,366 (no ISBN); Dover Publ., 2000, 0486206580; Cosimo Classics,
Inc., 2004, 1596050373
