Summary of this Magic Cube site

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Introduction

The time has come to write finis to what has become a labor of love!

In late 2001, I decided to write a page about John Hendricks new magic hypercube definitions. After some preliminary research into old ‘perfect’ magic cubes, this page was posted to my magic squares (formerly Geocities) site on March 10, 2002.
I then decided to do additional research on early magic cubes. The end result, as they say, is history!
I became hooked on the subject when I discovered the large variety of cubes.

On December 14, 2002, I started a new Web site devoted to magic cubes. I used the same page design as used on my magic squares site. Also, as on that site, I have acknowledged the work of known contributors. In order to make my pages simpler for the casual browser to read, I have not included details of construction methods (except in a few rare occasions). This information is better handled on specialized sites.

The original intent was to eventually incorporate it into my magic squares site, but I have decided to keep it independent. However, on the top of each page are buttons that will link you directly to the introductory pages of the three major divisions of that site.

Now, one year later, this site has grown to 38 pages, and I feel like it is a good time to summarize what has been accomplished.
During this past year (2003) much consolidation of past knowledge and many advances have been made in the field of magic cubes!
These are brought to your attention with special acknowledgements, in the New Developments and the Summary Tables toward the end of this page.
I have included lots of links, so you may find that this is a convenient method of browsing through these cube pages.

Of course, I will continue to maintain these pages, and add new material when appropriate.

Acknowledgements and Thanks.

What has been accomplished on this site is due in no small measure to help I have received from many sources.

I thank my local library Interlibrary Loans department and many universities and other institutions for locating and providing me with old documents.

I also thank the many magic square and cube hobbyists and readers of my pages for suggestions, information, and contribution of new material. I hope you continue to review my pages periodically, and continue to offer constructive criticism, suggestions, and new material.

Recent Developments

A very special thanks to each of the following persons. Here are some of the new discoveries they made.

I cannot acknowledge here every person who has contributed to these pages (although they are credited where their material is located).
However, there is a small group of dedicated magic cube fans, who have become as enthusiastic about the subject as I have. Throughout my endeavor, they have provide a constant stream of advise, suggestions, and constructive criticism. In addition, during 2003 they have made many significant advances in magic cube knowledge! This has added tremendously to the amount of material I have been able to post. It also, of course, has made the whole subject of magic cubes, that much more interesting.

Here I list five friends, who I have known for some years, including links to their home pages. I will mention their major contributions and provide links to my relevant pages.

Christian Boyer (France)
Christian seems to have limitless time and energy because he seems to get so much accomplished. From the start of my project, he has been constantly available for advice and suggestions. In addition, he has:

Helped in locating and obtaining old French papers. Fermat, Huber, Violle, Sauveur, Arnoux and Leibniz are some examples.
Made amazing advances in multimagic cubes.

Discovered an order 9 cube with planar diagonals all magic, and collaborated with Walter Trump on the order 5 diagonal cube. He calls them perfect magic (a traditional definition). I call them diagonal magic, as part of a new coordinated system of hypercube definitions.

John Hendricks (Canada)

John has been a good friend and collaborator since the early 1990’s. Of those hobbyists mentioned here, he is the only one that lives close enough that I can visit with in person. He lives about 3 ½ hours away so we manage to get together about once a year.

The set of magic hypercube definitions he formulated is what inspired me to publish a magic cube site. These definitions are summarized on my index page but discussed in depth on the Perfect and Perfect 2 pages. The Inlaid Magic cubes page describes 4 cubes and a tesseract.

I also have a page on my magic-squares site that displays a variety of his work.

 John is a prolific investigator and writer. He has published over 50 articles and books on the general subject of magic hypercubes. Also numerous papers on statistics and miscellaneous mathematics. His contributions to magic cube knowledge are scattered throughout both of my sites.

Walter Trump (Germany)

Walter also seems to have boundless energy. His contributions to magic cube knowledge include:
Discussion of 4 types of symmetry in magic cubes. All are self-similar.

Investigation into Magic cube groups I, II, and III as well as cubes were all planar arrays are magic squares of groups I, II, and III. These last cubes do not fit into any order 4 Dudeney classes.
Discovery of orders 7, 6, and 5 cubes with planar diagonals all magic.
Finding 4 simple order 5 magic cubes, with all line sums (including planar diagonals) magic modulo 2, 10, 31, 62.
He also found an order 5 simple bordered (it contains an order 3 central magic cube) magic cube that is diagonal magic modulo 3.
All Trump orders 5 and 7 cubes are unique because none of the 4 directions have ALL pantriagonals correct.
In fact of the 94 different odd order normal magic cubes in my collection (Dec. 17, 2003), ALL have all pantriagonals correct in at least one direction except for the following:

Aale de Winkel (The Netherlands)

Over 6 or so years that I have known Aale, he has been a great help to me. He seems to have an infinite source of patience in explaining mathematical procedures that I am having trouble comprehending!
He and I have collaborated in several investigations, most notable quadrant magic squares and 3_D magic stars.

Aale was a great help to me throughout the 2 years I worked on this project. He offered suggestion, advised of typographical errors, etc.
In addition, he:

• attempted to find a cube that is NOT magic because no orthogonal lines are correct BUT all pantriagonals are. He found a cube with rows and columns all incorrect, with only pillars (and all pantriagonals) correct! I show it on my Unusual Cubes page.

• constructed two order 5 magic cubes, simple and pantriagonal, that are diagonal magic modulo 5

• supplied me with a number of cubes to help fill in my collection.

• was a source of inspiration and ideas for all in our group.

Guenter Stertenbrink (Germany)

Guenter came on the scene only recently (October, 2003), but provided me with a copy of Planck’s 1905 paper, and reconstructed Planck’s order 15 perfect magic cube ( Planck had published only construction details).
Guenter also provided an order 4 pantriagonal magic cube that is a closed knight tour, and later, his own version of the order 15 perfect magic.

Guenter does not have his own web pages yet.

Mitsutoshi Nakamura (Japan)

Mitsutoshi also came on the scene recently. He came to my attention when he quickly filled in some vacant spots in my summary table. Since then he has created an excellent web site and has produced a large variety of magic hypercubes.

He established the fact that there are 18 classes of magic tesseracts and has constructed examples of 17 of those (July 2008).
In addition, he:

• has compiled and listed an excellent group of magic hypercube definitions.

• has published (on his site) several proofs pertaining to magic hypercubes.

• created a number of algorithms for constructing various types and orders of magic hyercubes

Harvey Heinz (Canada)

I include myself in this list because of some additions I have contributed to magic cube knowledge. I have:

• Collected over 300 cubes (2005) from orders 3 to 17 and analyzed them by comparing over 15 features.

• Constructed an order 4 six-in-one magic cube model and several composition magic cubes.

• Collaborated with Walter Trump on the Dudeney Group IV to VI and identified transformations.

• Found that the patterns suggested by G. Arnoux are very general in ALL magic squares and cubes!
  (It seems strange that no one in the 115+ years since Arnoux published this that this is a source of many magic patterns
  in virtually every magic square and cube!)

• Collaborated with Aale de Winkel in the discovery, investigation, and naming of odd-order Quadrant magic squares

• Defined and provided examples of semi-pantriagonal magic cubes.

• Defined the ‘diagonal' magic cube. The name was suggested by Aale de Winkel.
  This cube type fills a hole accidentally left in John Hendricks ‘simple’, ‘pantriagonal’, ‘pandiagonal’, ‘perfect’ list of magic cube classes.

• Helped John Hendricks with the definitions and publicizing of the new magic cube definitions which now (2005) consists of 6 main classifications.

• Co-authored with John Hendricks The Magic Square Lexicon: Illustrated (2000).

Two Summary Tables

In the first of these two tables I will show the first cube for each order and class that I have actually seen and tested.
The second table lists cubes that are unusual for one reason or another.
In each case I will provide a link to take you to that cube, complete with reference, on my site.
I have included a footnote for cubes not shown on my site. Note [6] and higher are for cubes added to this table after it was first posted.

January 2005. This table is now incomplete. A 6th class of magic cubes has been discovered, by Mitsutoshi Nakamura. It is a combination Pantriagonal and diagonal magic cube. So far the only known cube is order 8. It is called a Pantriagonal Diagonal magic cube, or PantriagDiag for short. More on Definitions and Update-3 pages.

First cube of each class for each order

[1]  Frost published a Perfect order 9 in 1878, but it was not normal. It used numbers in the series from 1 to 889.
[2]  Planck provided instructions only. The cube was constructed by Stertenbrink in November, 2003.
[3]  This cube is shown also in Special Cubes table because it is composite.
[4]  This cube is shown also in Special Cubes table because it is bimagic.
[5]  The first normal perfect magic cube?
[6]  In February, 2004. I generated 3 additional cubes for the above table, using a program supplied by Abhinav Soni.
[7]  I received a Simple (?) order 17 cube from Abhinav Soni on Feb. 11, 2004.
      This cube must be classified as 'simple', but actually contains 36 pandiagonal and 2 simple magic squares.
[8]  Benson & Jacoby, Magic Cubes New Recreations, Dover,1981, 0-486-24140-8, pp 105-115 and pp 116-126.
[9]  I received these 5 (plus an order 19) pandiagonal magic cubes from Abhinav Soni on March 9, 2004.
[10]  Proved by Stertenbrink and de Winkel . See Pandiagonal Impossibility Proof. This was previously proved by 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.
[11]  I received these 3 Diagonal magic cubes from Mitsutoshi Nakamura on March 23, 2004.
        These cubes are also unusual. Only 4 planes in each orthogonal are simple magic squares. All others and all 6 oblique squares
        are pandiagonal magic.
        NOTE: Nakamura suggested the term 'proper' for cubes that have only the minimum features required for their class.
           These cubes (and those of [7] ) would not be 'proper'! [8], [12], [13], [14], [16] are some that are proper.
[12] I received these two Pandiagonal cubes from Abhinav Soni on March 29, 2004.
[13] I received these 2 Pantriagonal cubes from  Mitsutoshi Nakamura  on Apr. 11, 2004.
[14] I downloaded Li Wen's Order 10 cube from Christian Boyer's page. (He (and some others) still use the old definition of 'Perfect').
[15] From Akira Hiriyama & Gakuho Abe, Researchs in Magic Squares, Osaka Kyoikutosho, p. 154. (Nakamura email Apr. 18, 2004)
[16] I received this proper Diagonal cube from  Mitsutoshi Nakamura  on Apr. 18, 2004.
[17] I received this cube from Mitsutoshi Nakamura  on  Apr. 28, 2004. He reported that he had help from Abhinav Soni on this one.
That is fitting because the two of them filled the 18 cells in the above table that were vacant when I posted the table at the beginning of 2004.
Thanks and congratulations Mitsutoshi and Abhinav ! Read more about the unusual order 15 diagonal cube on Update-2.

 Mitsutoshi Nakamura is a 40 year old computer programmer living in Morioka, Japan. He majored in mathematics at university, but has studied magic cubes
only since 2000. He is unmarried and is an admirer of Yoshihiro Kurushima (? – 1757). See his new Website on magic cubes.
Abhinav Soni is a graduate student in the Bachelor of Technology degree at the Indian Institute of Technology in Roorkee, India. His interest in mathematics led him to write a program to generate magic cubes.

[18] Due to confusion with the term perfect, Nasik is now the preferred title for a hypercube with all possible line summing correctly.
[19] Trump now also refers to the diagonal type as strictly magic, again to lesson the confusion over the term perfect.

Special (unusual) Cubes

Challenges

Magic cubes - unanswered questions

 Pantriagonal magic cubes

Are all normal order 4 pantriagonal magic cubes either compact or complete (or both)?
No. As of Dec. 15/03 I had 19 pantriagonal order 4 cubes. 12 were both compact and complete, 4 were complete only, 2 were compact only and Stertenbrink’s (Nov. 9/03) Knight Tour cube was neither!
Compact = Every 2x2 square sums to 130; Complete = Every pantriagonal contains m/2 complement pairs spaced m/2 apart.

All order 7 pandiagonal magic cubes I examined have at most, just 1 pandiagonal oblique magic square (the other 5 are simple magic squares).
Is it possible for more than one of the 6 oblique squares in an order 7 pandiagonal magic cube to also be pandiagonal magic?
Guenter Stertenbrink (Nov. 29/03) thinks the answer is no, as a result of counting the number of path directions possible.

The 11 normal pandiagonal magic cubes I have seen (to Dec./03) are all associated!.
Are there NO pandiagonal magic cubes that are not associated?
Yes. On Sept. 12/03, Aale de Winkel sent me a normal pandiagonal magic cube that was not associated.
Guenter Stertenbrink (Nov. 29/03) made a common mistake by stating “All are NOT associated, because you can move planes from 1 side of the cube to the other to destroy the association.” But no, shifting planes works with the equivalent of a pandiagonal magic square, which is a pantriagonal magic cube! Shifting planes in a pandiagonal magic cube destroys the triagonals.

A pandiagonal magic cube consists of 3m orthogonal pandiagonal magic squares AND NOT all pantriagonals are correct (or it would be a perfect cube).
Are all pandiagonal magic cubes order 7? Who will be the first to construct one of a different order?
Abhinav Soni constructed ones for orders 8, 9, 11, 13, 15, 16, and 17 in March 2004!

Is an order 10 pandiagonal or perfect cube possible?
Not a normal (consecutive numbers) perfect one. This has been proved many times for singly even pandiagonal magic squares.
Guenter Stertenbrink (Nov. 29/03) does not think a pandiagonal magic cube of this order (also order 12) is possible either.
Aale de Winkel (Jan. 2004) posted on his encyclopedia site a summary of George Chen and Guenter Stertenbrink impossibility proofs.

Perfect magic cubes

Corners of all orders 2, 3, 4, 5, 6, 7 and 8 cubes within all four of the order 8 perfect cubes I have seen sum correctly to 2052.
Is this feature (called compact) always present in order 8 perfect cubes? In all 4x perfect cubes (x>1)?
Addendum: My Soni perfect order 16 has only corners of sub-cubes 2, 4, 5, 6, 8, 9, 10, and 16 summing correctly. (HDH Nov. 29/03)

 All 4 order 8 perfect cubes also have another feature in which every pantriagonal contains m/2 complement pairs spaced m/2 apart.
Is this also common to all order 8 perfect cubes? To all 8x perfect cubes?
Addendum: The Soni perfect order 16 cube (NOT associated) also has this feature (it is called complete). (HDH Nov. 29/03)
Addendum2: The Nakamura perfect order 16 cube is associated and does NOT have this feature, so the answer to the last question is NO.
This cube has corners of sub-cubes 3, 5, 7, 11, and 15 summing correctly. (HDH Mar. 18/04)

 Prime number magic cubes:

There has not been much work done with magic cubes consisting of prime numbers.
What is the smallest possible (i.e. has the smallest constant) prime number magic cube?
What is the smallest possible magic cube consisting of consecutive primes?

 Order 4 magic cube groups:

So far I have seen no magic cubes that correspond to magic square groups 7 to 12.
When one is found that fit within group 7, 8, 9, or 10, the other three will be available by swapping planes. Likewise for groups 11 and 12.
Who will be the first to find cubes for some of these groups?

Walter Trump has found examples of 4 cubes that do not fit into any of the 12 Dudeney groups. It is not surprising that such cubes exist, given the fact that cubes are so much more complicated than squares.
Who will be the first to find cubes belonging to additional groups?
Who will be the first to find order 4 magic cubes with all horizontal planes that are magic squares of groups 4,or greater?

 Miscellaneous cube challenges

I have not yet seen a combination Add/Multiply magic cube.  Is it possible to construct one?

It is possible to have a number square where all pandiagonals , but NO rows or columns are correct.
In 2002, such a square was found. It is an order 4 square with all pandiagonals, but no rows or columns, summing to 34, [1]
[1] Peter D. Loly, A Purely Pandiagonal 4*4 Square..., Journal of Recreational Mathematics, Vol. 31, No. 1, 2002-2003, pp 29-31.
Is it possible to construct a cube with no correct orthogonal lines, but all pantriagonals are correct?
Is it possible to construct such a cube, but with all pandiagonals correct as well?
Yes. I received such a cube from Guenter Stertenbrink on Jan. 4, 2004.

 A heterosquare has all line sums different. A subset of this is the antimagic square, with all line sums different but consecutive.
Who will be the first to construct a heterocube or an antimagic cube?
On Jan. 9, 2004, I received an order 3 heterocube from Peter Bartsch  that was almost an antimagic cube!
On Jan. 12 and 13, 2004 I received prime number heteromagic cubes from Peter.

Please send me Feedback about my Web site!
Harvey Heinz   harveyheinz@shaw.ca
This page last updated November 02, 2009
Copyright © 2004 by Harvey D. Heinz