

This page is the
gateway to 45 pages about magic cubes.
Index to this page
Introduction
Several years ago John R. Hendricks introduced a
coordinated set of definitions for magic cubes. It included a new
definition for the ‘perfect’ magic cube, which is applicable for magic
hypercubes of any dimension. This inspired me
to investigate the different definitions of ‘perfect’ magic cubes that had
appeared over the years. The result was a new page that discussed the
subject. However, I ended up with more questions then when I started, so
developed a series of spreadsheets to investigate many characteristics of
magic cubes. After looking closely at over 200
published magic cubes of order 3 to 17, I am amazed at how few cubes I
found that had all identical features. Considering the large number of
possible combinations available to form a magic cube of a given order, it
is not surprising I found very few duplicate cubes. 
However, features such as number, type, and location
of included magic squares, feature variations in the oblique squares,
etc., were found to be extremely varied. The result is this new series of pages, which
explores the subject in some depth. With one or two possible exceptions,
all magic cubes shown on these pages will have different features (or at
least will be different orders). As is usual
with the other pages on this site, I intend to keep the discussions simple and
will not normally go into methods of construction. Methods, and involved
mathematics, will be left to others that are more qualified to present them.
These pages will be more concerned with basic principles and a survey of the
history and variety of magic cubes. Navigation
This site is simple to navigate. The pages are listed in sequence in the index
(although, of course, they do not have to be read in that order).
At the bottom of each page are left and right pointing arrows that link to the
previous and next pages. The 'top' button goes to the top of the page. The 'up'
button will return you to this page.
The buttons at the top of each page go to:
'Home" goes to the main Gateway page for this entire web site of over 130 pages.
The 'square', 'star', and '#' go to the start page of each major division of
that site.
The 'cube' button returns you to this page.
The button bar now also includes a link to a map of the entire site. From there
you can go to any desired page. As usual with my Web
pages, I welcome comments, both laudatory and critical. Magic cubes covers a
wide field and I am sure everyone may not agree with everything I have said on
these pages. Also, some may feel I have put too much emphasis on certain
subjects and not enough on others. I can only respond that this is how I see it.
I would like to thank some of those who helped me with my
research of magic cubes. In no particular order they are Christian Boyer, Walter
Trump, Aale de Winkel, John Hendricks, Abhinav Soni, and Mitsutoshi Nakamura.
Links to many of their sites are on my links page.
Some others who helped in a lesser degree are Paul Vaderlind, Brian Alspach,
Mark Swaney, Vladimír Karpenko, Jacques Sesiano, Rich Schroeppel. I apologize
for any I may have missed. Thanks to all of you. It
is almost inevitable that despite the utmost care, a work of this size will
contain some errors. I apologize for any and appreciate them being brought to my
attention. December 30, 2003. I now consider this
site on magic cubes complete. However, I intend to keep updating it as new
material becomes available. Please refer to my Summary page where I show a consolidation of what has
been accomplished in this field
6 Classes of Cubes
The following definitions will be used throughout this web
site. They are presented here simply as a concise introduction to the subject.
Examples and further explanation will be presented where appropriate.
See especially: Perfect magic Cubes,
The Road to Perfect, and
Magic Cube
Definitions.
NOTE: In January, 2005, a 6th class was added to the previous
5 classes. Pantriagonal Diagonal of PantriagDiag for short.
Mitsutoshi Nakamura has an excellent site on magic hypercubes, and has
extensively researched their classes. His definitions page is at
http://homepage2.nifty.com/googol/magcube/en/terms.htm
Magic cubes:
Minimum requirements are: All rows, columns, pillars, and 4 triagonals must sum
to the same value. Nasik:
An unambiguous term that may be used in place of the term perfect (which has
differing meanings). See (perfect). Simple:
Contains NO, or less then 3m orthogonal magic squares.
Pantriagonal:
All 4m^{2} pantriagonals must sum correctly (that is 4 onesegment,
12(m1) twosegment, and 4(m2)(m1) threesegment). There
may be some simple AND/OR pandiagonal magic squares, but not enough to satisfy
any other classifications.
The pantriagonal magic cube is similar to a pandiagonal magic square in this
respect. 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.
Diagonal:
All 3m planar arrays must be 'simple' magic squares (some may be
pandiagonal). i.e. all planar diagonals must sum correctly.
The 6 oblique squares will then automatically be magic. The smallest normal
diagonal magic cube is order 5.
These squares were referred to as ‘Perfect’ by Gardner and others! At the same
time he referred to Langman’s 1962 pandiagonal cube as ‘Perfect’.
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 are order 8 (not associated and
associated).
This cube was discovered in 2004 by Mitsutoshi Nakamura.
Pandiagonal:
ALL 3m planar arrays must be ‘pandiagonal’ magic squares. The 6 oblique squares
are always magic. Several of them may be
pandiagonal magic.
Gardner also called this (Langman’s pandiagonal) a ‘perfect’ cube, presumably
not realizing it was a higher class then Myer’s cube.
Nasik:
ALL 3m planar arrays must be ‘pandiagonal’ magic squares. In addition, ALL
pantriagonals must sum correctly. These two conditions combine to provide a
total of 9m pandiagonal magic squares. When Hendricks devised this
classification system, he called this perfect. However, because there are
different definitions of perfect, Nasik is a better choice.
Generalized:
A hypercube of dimension n is perfect (nasik) if all panragonals
sum correctly. Then all lower dimension hypercubes contained in it are also
perfect!
Through every cell on a perfect hypercube of dimension n there are (3n1)/2
different routes that must sum the magic sum. A. H.
Frost (1866) referred to all but the simple magic cube as Nasik! See a
quotation by C. Planck, in which he redefined
nasik to mean a Hendricks perfect hypercube only.
Nasik is an unambiguous term that should be used in place
of the term perfect.
Index to Magic Cube Pages on
this Site
1. Magic Cubes 
Basics 
Magic cube parts, associated, basic cube
and aspects, coordinates, species.

2. Magic Cube
Definitions 
A discussion, with examples, of terms
relating to magic cubes.

3. Perfect Magic
HyperCubes 
My original page on perfect (nasik)
magic cubes.

4. Cubes  the
Road to Perfect 
The progression to perfect (nasik)
cubes. Presented is 1 cube each of orders 3 to 11.

5. The Early Cubes 
15 different cubes from Fermat's 1640
order 4 to Worthington's 1910 order 6.

6. A. H. Frost`s
Cubes 
8 assorted cubes published by Rev.
Frost in 1866 and 1878.

7. Barnard
Perfect Cubes 
His perfect (nasik) orders 8 and 11. And
other magic objects from his 1888 paper.

8. Order3 Magic
Cubes 
The only four order 3 basic cubes, and
some variations. Other material.

9. Order4 Magic
Cubes 
A number of order4 cubes with differing
characteristics.

10. Order5 Magic
Cubes 
8 different cubes, (1876 to 2001).The
Trump/Boyer diagonal order 5 cube of 2003.

11. Order6 Magic
Cubes 
A variety of 7 cubes, published between
1838 and 1999.

12. Order7 Magic
Cubes 
A variety of 7 cubes, published between
1922 and 2001.

13. Order8 Magic
Cubes 
A variety of 6 cubes, published between
1908 and 2001.

14.
Order9 Magic Cubes 
Three simple magic cubes, all with
slightly different features.

15.
Order10 Magic Cubes 
Three simple magic cubes, one of them
with an order 6 inlaid cube.

16.
Order11 Magic Cubes 
A simple, a pantriagonal, and 2 perfect
order 11 cubes.

17.
Order12 Magic Cubes 
A pantriagonal, a diagonal, and a
simple, but inlaid order 12 magic cube.

18.
Order13 Magic Cubes 
A perfect cube, an unusual pantriagonal
cube, an example of a broken plane.

19.
Large Magic Cubes 
Some order 15, 16 and 17 magic cubes.
Most notably Gabriel Arnoux's perfect magic cube of 1887. 
20.
Arnoux Magic Patterns 
Arnoux demonstrated a multitude of magic
patterns in his order 17 perfect magic cube. Investigation reveals
that these patterns are common in all types and orders of magic
hypercubes. 
21.
Modulo Magic Cubes 
Seven order 5 cubes that are magic
because all relevant line sums are evenly divisible by the same
number i.e. 2, 3, 5, 10, 31, 62. 
22.
Multimagic Cubes 
Presenting the world's first Bimagic and
Trimagic cubes.

a. Monster Cubes 
A paper by Christian Boyer announcing
advances in multimagic cubes and tesseracts. 
b. Boyer16 
The complete listing Boyer's bimagic
order 16 cube of Jan. 23, 2003.

c. Boyer32 
The top horizontal plane of Boyer's
bimagic order 32 cube of Jan. 27, 2003.

23.
Order4 Magic Cube Groups 
Dudeney groups I to VI magic squares and
their magic cube equivalents.

24.
Prime Number Magic Cubes 
Two order 3 prime cubes. An order 4
simple cube, and an order 4 pantriagonal.

25.
Multiply Magic Cubes 
Three different types of order 3
multiply cubes. An order 4 and an order 5 cube.

26.
Composition Magic Cubes 
An order 9 cube consisting of 27 order 3
cubes, an order 12 cube with 27 order 4 cubes, and a new method
using multiplication for another order 9 cube. 
27.
Hendricks Inlaid Magic Cubes 
Order 8 cubes with 1, 8, and 27 inlaid
order 4. Order 12 with 8 order 4 pantriagonal magic cubes and 48
order 4 pandiagonal magic squares. 
28.
Heinz X6 Magic Cube 
Description, pictures, and listings of
my model of 6 order 4 cubes in one.

29.
Selfsimilar Magic Cubes 
Different types of symmetrical cubes.
Thanks Walter Trump!

30.
Pan and Semipan Cubes 
A short description of the
characteristics of pandiagonal and semipandiagonal magic squares
and their counterpart in the pantriagonal and semipantriagonal
magic cubes. 
31.
Unusual Magic Cubes 
About 15 cubes that are not magic in the
ordinary sense, but are unusual! 
32.
Mostperfect Magic Cubes 
Discussion and examples of the
3dimensional equivalent of the mostperfect magic square. 
a. Order16 Perfect Cubes 
Listings of two order 16 perfect magic
cubes. Only one is mostperfect. 
33.
Summary of this cube section 
Concluding remarks, new advances in
magic cube knowledge, and some challenges! 
34.
Cube Update1 
Material that I received in January,
2004. (Heterocube, Purely Pan cube, Magic ratio, etc.) 
35.
Cube Update2 
Information I received to April 30,
2004. (Cubes (1757), Order 6 Projection cube, etc.). 
36.
Cube Update3 
Information I received to the end of
2004. Nested order 16, New class, etc.

37.
Cube Update4 
Aug. 2005. More on Panmagic ratios,
Semidiagonal magic order 4, The Leibniz cube, Prime magical cubes. 
38.
Cube Update5 
May 2007. Magic cuboids, Magic Knight
Tours, transform associated to pantriagonal.
Also miscellaneous items and links. 
39.
Cube Update6 
Feb. 2010. Frost Order9 model. More on
Compact & Complete. Multiply order4. etc. 
40.
Timeline 
92 references to cubes and tesseracts
16402009. Also first cube in each class and order. 
The
Tests
By the use of Excel spreadsheets, I examined the
characteristics of about 320 (Oct./09) published magic cubes. I limited
the tests to orders 3 to 17 because of the scarcity of larger published
cubes, and the increased effort required to enter the larger cubes into
the spreadsheets.
Also, that is about at the practical limit using this spreadsheet
approach. The file size for order 16 is over 2 MB!
A different spreadsheet design was required for each
order, but they all had the following features in common.
 An area at the top for file name, title, and a bit of
relevant information
 An m x m square array of cells for each
of the m horizontal planes
 Both of these areas were unprotected to admit input.
The rest of the spreadsheet was protected to prevent accidental overwriting
because all necessary information was automatically copied from these
horizontal arrays.
 An m x m square area of cells for each of
the m vertical planes parallel with the front of the cube
 An m x m square area of cells for each of
the m vertical planes parallel with the side of the cube
 An m x m square array of cells for each
of the six oblique squares
 The above m x m arrays were automatically
filled from the contents of the horizontal arrays, and the row, column and
pandiagonal totals computed.
 An m x m square array of cells for each
of the four directions of triagonals. Each cell of these arrays contained the
sum of the broken triagonal pair or triplet originating at that cell. The cube
was pantriagonal if all m x m cells in each of the 4 arrays
contained the magic constant.
 Supplementary tests for orders 4x.
Features the spreadsheet
looked at:
 Number of planar squares magic
 Number of planar squares pandiagonal magic
 Center plane in each of 3 orthogonal directions magic
(odd orders)
 Number of planar squares with all pandiagonals in 1
direction correct
 Number of oblique squares with rows and columns sum
correct (square is magic)
 Number of oblique squares with rows only sum correct
 Number of oblique squares with columns only sum correct
 Number of oblique squares with all pandiagonals. in 1
direction correct
 Number of oblique squares with all pandiagonals correct
 Number of directions with all pantriagonals correct
 Compact  All 2 x 2 squares (3 orientations) sum
correct (order4),
 Compactplus # of orders (2 to m) of cubes with all
cubes have corners summing O.K. (orders 8,12,16)
 Complete  Every pantriagonal contains m/2 complement
pairs spaced m/2 apart (orders 4x).
These features were tabulated in a Word document (CubeComparison.doc
available for downloading) for each cube in the
collection.
For each additional cube within an order, I simply made a
copy of the spreadsheet, then pasted or typed in the horizontal plane numbers.
Features of this magic cube
site:

Explanation of basic principles,
features, and definitions.

A large number of cube examples, but
all within an order will have different characteristics

Subject matter is differentiated by
separate pages

References, where applicable, will be
listed at the bottom of the section.

With only one or two exceptions, every
cube shown on these pages will be unique (i.e. no cube shown twice).

I will use m on these pages to indicate
order of the cube, and n to indicate dimension (where required).
