Cube Timeline

Introduction

This timeline is assembled by referring to my large library of books and published papers on the subject as well as over 350 magic cubes I have assembled over the years.
I have also received help from Magic cube fans around the world. I thank them all, but a special thanks to Christian Boyer (France) who located information on Arnoux, Sauveur, and Leibniz that I would otherwise not have been able to access.
Also, special thanks are due Mitsutoshi Nakamura (Japan), and Abhinav Soni (India) who helped me complete a ‘First Cubes’ table by finding the 18 cubes of different orders and classes, that I was missing

Of course, no collection is ever complete, and there are possibly still historical cubes waiting to be unearthed. With that in mind, and mistakes I may have made (for which I apologize), I invite comments, suggestions for improvement, and new material.

I have included most, if not all of the cubes published before 1900 (that I am aware of). After that date, I have just included highlights.

After the main section dealing with magic cubes, I have included a short section on magic tesseracts and higher dimension magic hypercubes.

Logically, I should have included a section on dimension 2 magic hypercubes (the squares). However, these go back to ancient times and precise dates are not known. Other reasons I have decided not to include such a list are; not nearly the variety of magic squares as there is of magic cubes, and the large amount of literature on magic squares makes it difficult to summarize.

Citations have been included with most entries. They have been repeated, as a group, at the end of this page.
I have also included a link with most entries to a section on my site (or another Internet site), that has more detail (again, where available).

For convenience, I have included a Table of First Cubes for each class and each order up to 17. This is a copy of the table on my Cube Summary page.

The Timeline

1640   Pierre Fermat showed what may have been the first magic cube in a letter to Marin Marsenne.
·        It was an order 4 with 8 of the 12 planar squares simple magic. No triagonals were correct.

[1]  Edouard Lucas, L’Arithmétique amusante (Amusing Arithmetic), Gauthier-Villars, 1895
http://members.shaw.ca/hdhcubes/cube_early.htm

1683   Yoshizane Tanaka published a semi-magic order-4 cube with the same features as Fermat’s.

[2]  Akira Hirayama and Gakuho Abe, Researches in Magic Squares, 1983, Osaka Kyoikutosho.
http://members.shaw.ca/hdhcubes/cube_update-2.htm

1686   Adami A. Kochanski, Published some pages in Latin.
·    One page contains two problems about magic cubes... but without the solutions.

[3]  Adami A. Kochanski, Considerationes quaedam circa Quadrata & Cubos Magicos, Acta Eruditorum, 1686, vol. 5, pages 391-395.

1710   J. Sauveur designed an order 5 cube which seemed to set the style for the next 200 years.
·        All 15 5x5 arrays sum to 1575 but rows and columns summed incorrectly.
·        Diagonals of all 21 square arrays summed correctly, and 3 of the 6 oblique arrays were simple magic squares.
·        All four triagonals sum correctly.

[4]  Mémoires de l'Académie Royale des Sciences of 1710. Notes from Christian Boyer because of restrictions on photocopying.
http://members.shaw.ca/hdhcubes/cube_early.htm

1715   G. Leibniz sent an order-3 cube to the Académie des Sciences in Paris in November 1715.
·        It is credited to him, but was actually constructed by Father Augustin Thomas de Saint Joseph.
·        It has exactly the same features of the Sauveur order-5 cube of 1710.
·        Christian Boyer found this missing letter in the Hanover Library, Germany.

[5]  Christian Boyer, Le plus petit cube magique parfait (and Inédit - Le cube magique de Leibniz est retrouvé), La Recherche, issue number 373, March 2004, pages 48-50, Paris, 2004
[6]  http://www.multimagie.com/
http://members.shaw.ca/hdhcubes/cube_update-4.htm

1757  Yoshihiro Kurushima (?-1757) published two simple order-4 magic cubes.
·        Both are fully magic (rows, columns pillars and triagonals all sum correctly).
·        Are these the first magic cubes by present day standards?

[7]  Akira Hirayama and Gakuho Abe, Researches in Magic Squares, 1983, Osaka Kyoikutosho.
http://members.shaw.ca/hdhcubes/cube_update-2.htm

1838   Violle published a huge book on recreational mathematics, in which he illustrated 4 magic cubes.
·        Order 4. All 18 planes (3 x 4 + 6) sum to 4 x 130. Both diagonals of each of these squares and therefore all 4 triagonals are also correct. Rows and columns sum incorrectly.
·        Order 5. All 21 planes (3 x 5 + 6) sum to 5 x 315 as do all 4 triagonals. Pandiagonals of all 15 planar arrays sum correctly. Rows and columns sum incorrectly
·        Orders 6 and 7 have the same features as 4 and 5.
·        All of these as per J. Sauveur’s definition of a magic cube

[8]  Par B. Violle, Traité complet des Carrés Magiques, 1837, (French) This book is available on the Internet at http://gallica.bnf.fr.as scanned pages.
http://members.shaw.ca/hdhcubes/cube_early.htm

1866   A. H. Frost introduces the term Nasik for cubes where some or all planes have correct pandiagonals.
·        The first examples I can find of cubes with all orthogonal lines and the 4 main triagonals correct (except for the Kurushima cubes).
·        Frost does not differentiate between what we now call pantriagonal, pandiagonal and perfect cubes.
·        He describes a method of constructing magic cubes and shows an order 7 pandiagonal and an order 8 pantriagonal magic cube (the first published of this type, for this order).

[9]  A. H. Frost, Invention of Magic Cubes. Quarterly Journal of Mathematics, 7, 1866, pp 92-103
http://members.shaw.ca/hdhcubes/cube_frost.htm

1875   G. Frankenstein publishes an order 8 cube that is magic by present definition.
·        It contains 30 simple magic squares and is what we now call a diagonal cube.
·        This is the first published example of a diagonal magic cube.

[10]  F.A.P. Barnard, Theory of Magic Squares and Magic Cubes, Memoirs of the National Academy of Science, 4,1888,pp. 209-270. Construction details of the "Frankenstein" cube is described in a lengthy footnote on pages 244-248.
[11]  W. Benson & O. Jacoby, Magic Cubes: New Recreations, Dover Publ. 1981, pp 32-33
http://members.shaw.ca/hdhcubes/cube_early.htm

1876   T. Hugel publishes an  order-3 simple magic cube that conforms to present day definition.
·        It is a disguised version of the order 3 index # 1 cube
·        He also showed an order 5 simple magic cube
·        These cubes are the first odd order magic cubes published

[12]  Theodore Hugel, Das Problem der magishen Systeme, 1876, Verlag von A. H. Gottschick, 70pp. (German).
 http://members.shaw.ca/hdhcubes/cube_early.htm
http://members.shaw.ca/hdhcubes/cube_5.htm

1878   Frost expands on the previous paper.
·        He presents two order 3 simple magic cubes, one with incorrect triagonals, and an order 4, also with incorrect triagonals. This    30 years before W. S. Andrews defined a magic cube as requiring the space diagonals be correct.
·        He presents an order 4 pantriagonal magic cube (first one published)
·        He presents an order 7 pandiagonal magic cube.
·        He presents an order 9 perfect magic cube with non-consecutive numbers. He explains that order 11 is the smallest permitting consecutive numbers. See also Howard [43] who mentioned the same thing( This is true only for this method of construction. In fact, about the same time Frost converted his non-normal order-9 to a normal one that ended up as a model in the Whipple museum. See update-6)

[13]  A. H. Frost, On the General Properties of Nasik Cubes, QJM 15, 1878, pp 93-123
[14]  W. S. Andrews, Magic Squares & Cubes, Open Court, 1908, p. 64
[15]  W. S. Andrews, Magic Squares & Cubes, 2^nd edition, Dover Publ. 1960 (reprint of 1917, Open Court) p. 64
http://members.shaw.ca/hdhcubes/cube_frost.htm

1887   Gabriel Arnoux constructs the first normal perfect magic cube (first one on record).
·        An order 17 cube using the numbers 1 to 4913, S = 41769.
·        He deposits a 26 page handwritten paper showing the complete cube on April 17, 1887 in l'Académie des Sciences, Paris.]

[16]  Cube Diabolique de Dix-Sept, was deposited in l'Académie des Sciences, Paris, France, April 17, 1887.
http://members.shaw.ca/hdhcubes/cube_big.htm

1888   F. A. P. Barnard publishes an important paper on magic squares and cubes.
·        Included are the first to be published orders 8 and 11 normal perfect magic cubes.
·        Also included are examples of "inlaid" magic squares and other magic objects.

[17]  F.A.P. Barnard, Theory of Magic Squares and Magic Cubes, Memoirs of the National Academy of Science, 4, 1888,pp. 209-270.
[18]  W. Benson & O. Jacoby, Magic Cubes: New Recreations, Dover Publ. 1981, pp 32-33
http://members.shaw.ca/hdhcubes/cube_barnard.htm

1889   W. Firth constructed the first(?) order 6 magic cube before 1889.
·        Dr. Planck writing in 1908 “…Firth in the 80’s constructed what was, almost certainly, the first correct magic cube of order 6.”

[19]  W. S. Andrews, Magic Squares & Cubes, 2^nd edition, Dover Publ. 1960 (reprint of 1917, Open Court) p. 373
http://members.shaw.ca/hdhcubes/cube_early.htm

1894   C. Planck constructed the first order 10 simple magic cube (also an order 6).

[20]  W. S. Andrews, Magic Squares & Cubes, 2nd edition, Dover Publ. 1960 (1917) , pages 310, 311, 314.
http://members.shaw.ca/hdhcubes/cube_10.htm

1898   H. Schubert publishes an order 4 associated magic cube.

·        He also publishes an order 5 associated magic cube with the same features.

[21]  Hermann Schubert, Mathematical Recreations and Essays, Open Court 1899.
http://members.shaw.ca/hdhcubes/cube_early.htm

1899   E. Fourrrey published an order 4 cube with exactly the same characteristics as Pierre Format’s cube of 1640.
·        He also published an order 5 cube with exactly the same characteristics as Joseph Sauveur’s cube of 1710.
·        Neither of these cubes are magic (by today’s standards).

[22]  E. Fourrey, Recréations Arithmétiques, (Arithmetical Recreations) 8^th edition, Vuibert, 2001, 261+ pages (edition 1, 1899).
http://members.shaw.ca/hdhcubes/cube_early.htm

 

 

1905   C. Planck published the first order 9 normal perfect magic cube.
·        Frost published an order 9 perfect magic cube in 1878, but it used the numbers from 1 to 889 (instead of 1 to 729)
·        He also provided instructions for the first perfect order 15. It was assembled by Guenter Stertenbrink in November, 2003.
·    Planck also published the second order 8 perfect magic cube.

[23]  From C. Planck, The Theory of Paths Nasik. Printed in 1905 for private circulation.
http://members.shaw.ca/hdhcubes/cube_9.htm

1908   Andrews presents the modern definition of a simple magic cube.
·        All orthogonal lines and 4 main (space diagonals) must sum to the constant.
·        He shows an order-8 simple magic cube (first published for this order)

[24]  W. S. Andrews, Magic Squares & Cubes, 1908), page 64 ( This is reproduced as the first 188 pages of [25].
[25]  W. S. Andrews, Magic Squares & Cubes, 2^nd edition, Dover Publ. 1960), page 64 (reprint of 1917, Open Court
http://members.shaw.ca/hdhcubes/cube_8.htm

1910   H. Sayles publishes an order-6 with a unique sub-cube feature.
·        Worthington publishes an order-6 cube that contains a magic square on each of the 6 faces.

[26]  H. A. Sayles, A Magic Cube of Six, The Monist, 20, 1910, pp 299-303
[27]  J. Worthington, A Magic Cube of Six, The Monist, 20, 1910, pp 303-309
[28]  W. S. Andrews Magic Squares & Cubes 1960 (1917) pps. 197, 202 .
http://members.shaw.ca/hdhcubes/cube_early.htm

1913   H. Sayles publishes the first multiply magic cubes.
·        An order-3 with P=27,000 and an order-4 with p=57,153,600.

[29]  H. A. Sayles, Geometric Magic Squares and Cubes, The Monist, 23, 1913, pp 631-640
[30]  W. S. Andrews Magic Squares & Cubes 1960 (1917) pps. 283-294.

1917   First mention of pandiagonal and perfect magic cube.
·        Kingery states “It is not easy, perhaps not possible, to make an absolutely perfect cube of order 3” (p.352).
·        Dr. Planck writes “This last term (perfect) has been used with several different meanings by various writers on the subject.” (p.364-365).
·        Dr. Planck also writes “ …the smallest Nasik order in k dimensions is always 2^k, or 2^k + 1 if we require association.”

[31]  W. S. Andrews, Magic Squares & Cubes, 2^nd edition, Dover Publ. 1960 (reprint of 1917, Open Court), p. 352-366

1939   Rosser and Walker, in unpublished papers define (now perfect) cubes as Diabolic.
·        They mention that their definition is more stringent then Frost’s. (They are talking only about perfect cubes, not all cubes with pandiagonal like properties as Frost’s do.)
·        They prove that there are 9m diabolic (pandiagonal) magic squares in such a cube.
·        They prove that such cubes exist for all orders 8x, and all odd orders greater then 8.
·        They show no actual cubes, and their papers are very technical.

[32]  B. Rosser and R. J. Walker, Magic Squares: Published papers and Supplement, a bound volume at Cornell University, catalogued as QA 165 R82+pt.1-4.

1943   R. Heath constructed a 6-in-1 order-4 magic cube.
·        This was a virtual model where each cell was represented by a small cube. A number was placed on each of the 6 faces of each cube. Corresponding faces represented each of the six cubes.
·        All six cubes were simple magic, but all had the same sum (S=770) and all had 52 correct lines.
·        In 2002, H. Heinz (the author) constructed an actual wooden model based on this idea.
·        His six cubes are all pantriagonal magic so each one has 112 correct lines. However, all cubes are also compact, adding another 192 combinations to each cube. However, each cube has a different constant (760, 764, 768, 772, 776, 780).

[33]  R.V. Heath, A Magic Cube With 6n^3 Cells, American Mathematical Monthly, Vol. 50, 1943 p.288-291
http://www.magic-squares.net/ms_models.htm#Six%20magic%20cubes%20in%20One

1948   G. Abe constructed the first(?) order 6 pantriagonal magic cube.

[34]  From Mutsumi Suzuki's Web site at http://mathforum.com/te/exchange/hosted/suzuki/MagicSquare.html
http://members.shaw.ca/hdhcubes/cube_6.htm

1972   John Hendricks discusses the pantriagonal magic cube and reason for using that term.
·        This term would become the second of a set of 6 coordinated definitions (the first definition is simple).
·        This type of cube was commonly referred to as pandiagonal by other writers.
·        He published the first example I had seen of an order-5 pantriagonal magic cube.
·        He had actually shown a pan-4-agonal (panquadragonal) magic tesseract in 1968, but hadn’t yet consolidated his definition.

[35]  John R. Hendricks, The Pan-3-agonal Magic Cube, JRM 5:1:1972, pp 51-54
[36]  John R. Hendricks, The Pan-3-Agonal Magic Cube of Order 5, JRM 5:3:1972, pp 205-206
[37]  John R. Hendricks, Pan-n-agonals in Hypercubes, JRM 7:2, 1974, pp 95-96.
[38]  J.R. Hendricks, The Pan-4-agonal Magic Tesseract, American Mathematical Monthly,75:4 April 1968, p. 384.
http://members.shaw.ca/hdhcubes/cube_perfect-2.htm

1973   J. Hendricks publishes the first order-7 pantriagonal magic cube.
·        It is associated, and contains 3 simple magic squares.

[39]  J. R. Hendricks, Magic Cubes of Odd Order, JRM 6:4, 1973, pp 268-272 and Magic Square Course, 1991, p. 366
http://members.shaw.ca/hdhcubes/cube_7.htm#Hendricks

1975   Bayard Wynne publishes an order 7 ‘pandiagonally perfect’ magic cube.
·        Note: this was the name given for a ‘perfect magic cube in 1981, [30]. The 21 orthogonal squares of this cube are pandiagonal magic. The Wynne cube is now referred to as a ‘pandiagonal’ magic cube.
·        Another example of the need for a universal coordinated classification system.

[40]  Bayard E. Wynne, Perfect Magic Cubes of Order Seven, JRM 8:4, 1975-76, pp 285-293
[41]  W. Benson & O. Jacoby, Magic Cubes: New Recreations, Dover Publ. 1981, page 63
http://members.shaw.ca/hdhcubes/cube_7.htm#Wynne

1976   Gardner publishes the Myers cube in 1976 and called it perfect.
·        Designed in 1970 by 16 year old Richard Myers, Jr., this cube features 30 ‘simple-magic’ squares. By the new definitions, this is a diagonal magic cube

[42]  Martin Gardner, Mathematical Games, Scientific American, Jan. 1976.
[43]  Martin Gardner, Time Travel and Other Mathematical Bewilderments, W. H. Freeman, 1988.
http://members.shaw.ca/hdhcubes/cube_perfect-2.htm

1976   Ian Howard publishes instructions for constructing a true order 11 perfect magic cube.
·        This cube contains 39 pandiagonal magic squares (all 33 orthogonal squares and all 6 oblique squares.). Each cell in the cube is a part of 13 magic lines.
·          Howard mentions “so-called ‘perfect’ magic cubes” published by others, but he calls his cube pandiagonal or Nasik
·        By Hendricks universal classification system, this is indeed a perfect cube (the highest possible classification).

[44]  Ian P. Howard, Pan-diagonal Associative Magic Cubes (Letter to the Editor), JRM 9:4, 1976, pp276-278.
[45]  W. S. Andrews, Magic Squares & Cubes, 2^nd edition, Dover Publ. 1960 (reprint of 1917, Open Court), page 366
http://members.shaw.ca/hdhcubes/cube_perfect-2.htm

1977   Seimiya publishes orders 9 and 11 true perfect magic cubes.
·        Both cubes contain 3m + 6 pandiagonal magic squares (plus the 6m-6 broken oblique planes.

[46]   Mathematical Sciences (Japanese language) Magazine Dec. 1977, p. 45-- Special issue on puzzles.
http://members.shaw.ca/hdhcubes/cube_perfect-2.htm

1977   Akio Suzuki constructed two order-3 magic cubes consisting of prime numbers. Magic constants were 4659 and 3309.
·        In 2003 A. Johnson, Jr. confirmed that this cube has the smallest possible sum for an order 3 prime magic cube using distinct digits.
·        Also in 1977, Gakuho Abe produced an order-4 prime number magic cube (with S= 4020). All three of these cubes were simple magic (the order-3s of course were associated).
·        In 1985, A. Johnson, Jr. published an order-4 prime magic cube consisting of all 4-digit primes. S was a much larger 19740 (but the cube is pantriagonal).

[47]  Gakuho Abe, Related Magic Squares with Prime Elements, JRM 10:2 1977-78, pp.96-97. Akio Suzuki order-3 and 4 cubes.
[48]  A. W. Johnson, Jr., Solution to Problem 2617, JRM 32:4, 2003-2004, pp. 338-339
[49]  A. W. Johnson, Jr., An Order 4 Prime Magic Cube, JRM 18:1, 1985-86, pp 5-7
http://members.shaw.ca/hdhcubes/cube_prime.htm

1978   K. Leeflang surveys past cube features and discusses confusing cube terminology.
·        However, he introduces still more. For instance ‘two-sided orthogonal pandiagonality’.
·        By the new terminology, this is still a simple magic cube, even though it contains10 simple and 1 pandiagonal orthogonal magic squares, and 4 diagonal (oblique) simple magic squares.

[50]  K. W. H. Leeflang, Magic Cubes of Prime Order, JRM 11:4, 1978-79, pp 241-257
http://members.shaw.ca/hdhcubes/cube_5.htm

1981   B. Alspach and K. Heinrick define a perfect magic cube.
·        as one where all the 3m squares are simple magic
·        They then go on to cite Howard, Schroeppel, and Wynne as examples. These cubes all contain 3m pandiagonal magic squares, which they say is “clearly also a perfect magic cube”.
·        Another example of confusing terminology!

[51]  Brian Alspach & Katherine Heinrich, Perfect Magic Cubes of Order 4m, The Fibonacci Quarterly, Vol. 19, No. 2, 1981 pp 97-106
http://members.shaw.ca/hdhcubes/index.htm#Introduction
http://members.shaw.ca/hdhcubes/cube_perfect-2.htm

1981   Benson and Jacoby mention the Frankenstein Cube (1875) as being perfect.
·        They reproduce this cube, which features 30 ‘simple-magic’ squares (similar to Myers cube). 30 = 3m + 6, so it is a diagonal magic cube by the Hendricks classification.
·        They show an order 8 which they call a ‘pandiagonal perfect magic cube’.
This is a perfect magic cube by the new definition. The first even order perfect magic cube published in 76 years? (Rosser and Walker described such cubes, but didn't show examples of any.)
·        The produce order-12 and order-14 diagonal  magic cubes as well (the first such published).

[52]  W. Benson & O. Jacoby, Magic Cubes: New Recreations, Dover Publ. 1981, page 63
http://members.shaw.ca/hdhcubes/cube_12.htm

1984   B. Golunski published an order 9 simple magic cube (the first published).
·        It is associated, and contains 3 magic squares.

[53]  Published in "Młody Technik" (Young Technican) magazine No. 6/1984  (www.golunski.de/)
http://members.shaw.ca/hdhcubes/cube_9.htm

1988   Li Wen constructed the first order-10 diagonal cube.

[54]  It may be downloaded from http://www.multimagie.com/

1993   Hendricks publishes the first Inlaid Magic Cube.
·        An order 8 simple magic cube, but each of the 8 octants are order 4 pantriagonal magic cubes.
·        This seems to be a composite magic cube, but the numbers are arranged differently.
·        Some years later Hendricks published a book with a wide variety of cubes with inlaid cubes and squares.

[55]  John R. Hendricks, An Inlaid Magic Cube, JRM 25:4, 1993, pp 286-288.
[56]  John R. Hendricks, Inlaid Magic Squares and Cubes, 2^nd edition, self-published, 2000, 0-9684700-3-3, 250+ pages. Edited and illustrated by Holger Danielsson.
http://members.shaw.ca/hdhcubes/cube_inlaid.htm

1996   The order–4 projection cube was proposed by K. Brown and Solved by D. Cass
·        Peter Manyakhin reported on April 28, 2004 that he had already found over 200,000 order 6 Projection cubes.
·        The original idea was proposed by K. S. Brown and answered by Dan Cass.
·        This cube is not magic in the normally considered sense of the word.
·        In 2004, Peter Manyakhin produced some order 5 and 6 cubes of this type. [59]

[57]  H.D. Heinz and J.R. Hendricks, Magic Square Lexicon: Illustrated, Self-published, 2000, 0-9687985-0-0,  page 25.
[58]  The Sci.math newsgroup Dec. 2, 1996 and Dec. 10, 1996.
[59]  See http://members.shaw.ca/hdhcubes/cube_update-2.htm#Order%206%20Projection%20Cubes
http://members.shaw.ca/hdhcubes/cube_unusual.htm

1998   Hendricks defines pandiagonal, pantriagonal and perfect magic cubes.
·        He shows examples of the smallest orders possible for each type.

[60]  John R. Hendricks, Magic Squares to Tesseracts by Computer, self-published, 1998, 0-9684700-0-9, pps55-56.
http://members.shaw.ca/hdhcubes/cube_perfect-2.htm

1999   F. Liao and Associates published the first order-13 perfect magic cube.

[61]  From F. Laio, T. Katayama and K. Takaba, On the Construction of Pandiagonal Magic Cubes, Kyoto Univ. Technical Report # 99021, 1999
http://members.shaw.ca/hdhcubes/cube_13.htm

1999   F. Poyo published an order-12 simple magic cube
·        It contains no magic squares, but the diagonals of each planar square array sums to the same value.

I obtained this from the now defunct Suzuki magic square site.
http://members.shaw.ca/hdhcubes/cube_12.htm

2000   Heinz and Hendricks describe in depth, the system of new definitions.
·        Shown with support of various tables, are relationships between types of hypercubes and consistency between dimensions.

[62]  H.D. Heinz and J.R. Hendricks, Magic Square Lexicon: Illustrated, Self-published, 2000, 0-9687985-0-0
http://members.shaw.ca/hdhcubes/cube_perfect-2.htm

2000   John Hendricks publishes an order 25 bimagic cube
·        S₁ = 195,325. S₂ = 2,034,700,525. This is the world’s first bimagic cube.

[63]  John R. Hendricks, A Bimagic Cube of Order 25, self-published, 2000, 0-9684700-7-6, 18 pages.
[64]  Holger Danielsson, editor, Printout of a Bimagic Cube of Order 25, self-published, 2001, 36 pages.
http://members.shaw.ca/hdhcubes/cube_multimagic.htm

2000   Mutsumi Suzuki published an order-11 pantriagonal magic cube on his large website.
·        It was not associated, but contained 1 simple magic square.

http://members.shaw.ca/hdhcubes/cube_11.htm

2001  A. Soni produced the first order 7 simple magic cube that I had seen.
·        Actually, once the order 5 cube had been constructed, it was just a matter of using the same method to construct an order-7.
·        Soni also constructed an order-11 simple associated magic cube that contained 26 magic squares
·        Soni also constructed two order-9 pantriagonal magic cubes with different features.

[65]  Soni’s cube generator is available at http://www.hypermagiccube.cjb.net/
http://members.shaw.ca/hdhcubes/cube_11.htm

2002   M. Trenkler publishes the first order-5 multiply magic cube
·        This is 89 years after Sayles publishes the orders 3 and 4 cubes!
·        He also published orders 3 and 4 multiply magic cubes (with the same magic product as Sayles

[66]  Marián Trenkler, Additive and Multiplicative Magic Cubes., 6th Summer school on applications of modern math. methods, TU Košice 2002, 23-25
[67]  See also Obzory matematiky, fyziky a informatiky 1/2002, 9-16
www.multimagie.com/

2003   Aale de Winkel published the first orders 12 and 16 pantriagonal magic cubes.
·        The order-16 cube contained no magic squares, but corners of sub-cubes of orders 2,4,6,8,9,10,12,14, and 16 (including wrap-around) sum to ½ S.

http://members.shaw.ca/hdhcubes/cube_12.htm     http://members.shaw.ca/hdhcubes/cube_big.htm

2003   Christian Boyer publishes via the Internet, order 16 bimagic cubes and orders 64 and 256 trimagic cubes.

·        Jan. 20, He shows an order 16 bimagic cube with 36 bimagic squares. This cube is also the first published order-16 simple magic cube that I had seen.

·        Jan. 23, He shows an order 16 bimagic cube with 3m orthogonal planes and the 6 diagonal planes all bimagic squares. This is a diagonal cube by new classification (Christian calls it perfect).

·        Feb. 1, He shows an order 64 trimagic cube with all 192 orthogonal planes bimagic squares.

·        Feb. 3, He shows an order 256 trimagic cube with all 768 orthogonal plus the 6 diagonal planes trimagic squares making this also a diagonal cube.

·        May 13, announcement of 7 more multimagic cubes with differing features, including 2 tetramagic cubes!

·        He continues to work on this subject (along with others), striving for higher orders, lower products, or lower maximum number used, so review the Updates on his page.

[68]  C. Boyer’s Multimagic site is at www.multimagie.com/ (click on multimagic cubes)
http://members.shaw.ca/hdhcubes/cube_multimagic.htm

2003   W. Trump and C. Boyer find an order-5 diagonal magic cube

·        On September 1, 2003, Walter Trump reports finding an order-6 diagonal cube.

·        Until then, the only known cubes of this type were two order-8s, and one order-12.

·        Two days later, Walter Trump reports finding an order-7 cube of this type.

·        The same day, Christian Boyer reports finding an order-9 diagonal cube.

·        With five computers now running, Walter reports finding an order-5 cube!

·        What Hendricks defines as diagonal, is considered to be perfect by Boyer and Trump, and was reported to the media as such. (They include diagonal and pandiagonal as being equal.). Trump now also refers to this type as strictly magic, and Heinz refers to Hendricks definition of perfect as nasik.

[69]  This news was published in over 20 magazines and columns. Best source is http://www.multimagie.com/  and click on Perfect magic cubes.
http://members.shaw.ca/hdhcubes/cube_5.htm

2003   Bogdan Golunski provided me with an order-13 pantriagonal magic cube.

·        This cube is unusual in that it contains 29 pandiagonal and 4 simple magic squares.

http://members.shaw.ca/hdhcubes/cube_13.htm

2003   The author (H. Heinz) decides to rename the myers type cube (name previously coined by him) to diagonal cube.
·        This name was suggested by Aale de Winkel (July, 2003) as being more descriptive of the cube.
·        In this cube, the two main diagonals of each orthogonal (planar) array sum correctly to S. these arrays are thus simple magic squares because rows and columns are already magic as per basic requirements of a magic cube.
·        This type of cube must not be confused with a pandiagonal magic cube. In that case, all planar arrays are pandiagonal magic squares!
·        This class was overlooked by Hendricks when he defined his simple, pantriagonal, pandiagonal, and perfect classes.

http://members.shaw.ca/hdhcubes/cube_perfect-2.htm

2003   The author constructed the first(?)order-15 simple magic cube.
·        Again, this is no major accomplishment, because all odd order magic cubes can be constructed using the same algorithms.

http://members.shaw.ca/hdhcubes/cube_big.htm#Order%2015

2003   The author constructed the first composition magic cubes.
·        The above order-15 cube is unique because it is a composite magic cube. It consists of 27 order 5 magic cubes, placed as per the numbers in an order 3 magic cube.
·        At the same time, I constructed an order-9 and an order-12 composition cube.
·        These are not difficult to do, but I had never seen that type of cube published.

http://members.shaw.ca/hdhcubes/cube_compos.htm

2003   A. Soni constructed the first order-14 simple magic cube that I had seen.
·        He also constructed and order-16 perfect magic cube.

http://members.shaw.ca/hdhcubes/cube_13.htm              http://members.shaw.ca/hdhcubes/cube_big.htm

2003   G. Stertenbrink constructs the first closed knight tour magic cube.
·        It is an order-4 simple magic, not associated, but numbers form a closed knight tour.
·        Such tours had been constructed before, but not as forming a magic cube.
·        In 1918 Czepa published a closed knight tour order-4 cube but many of the orthogonal lines and none of the triagonals lines summed correctly.

[70]  I received this via email on Nov. 9, 2003.
http://members.shaw.ca/hdhcubes/cube_unusual.htm
[71] A. Czepa, Mathematische Spielereien (Mathematical Games), Union Deutsche, 1918,  140 pages, (page 77) (Old German script). There are many magic objects in this small format book but just two magic cubes.
http://members.shaw.ca/hdhcubes/cube_unusual.htm

2003   W. Trump constructs an order-5 bordered magic cube.
·        The 6 surface planes are simple magic squares so this is an S-type cube.
·        Both the order 5 and the order 3 cubes are associated, (all bordered magic squares and cubes are) so the 3 central planes of each cube as simple  magic squares.
·        Bordered (or concentric) magic cubes contain the lowest and highest numbers in the border. Inlaid magic cubes look similar, but the complete number range is distributed throughout both the inner cube(s) and the shell. [73][74][75]

[72]  http://members.shaw.ca/hdhcubes/cube_modulo.htm
[73]  John R. Hendricks, Inlaid Magic Squares and Cubes, 2nd edition, self-published, 2000, 0-9684700-3-3, 250+ pages. Edited and illustrated by Holger Danielsson.
[74]  Benson & Jacoby, New Recreations in Magic Squares, 1976, pp 26-33
[75]  J.L.Fults, Magic Squares, Open Court, 1974

2004   Nakamura discovers a sixth class of magic cubes. He calls it Pantriagonal Diagonal. I call it PantriagDiag for short.
·        This magic cube is a combination Pantriagonal and Diagonal cube, so
·        all main and broken triagonals must sum correctly, and
·        it contains 3m order m simple magic squares in the orthogonal planes, and 6 order m pandiagonal magic squares in the oblique planes.

[76]  The six classes are defined on my Magic Cubes-Introduction page, http://members.shaw.ca/hdhcubes/index.htm
http://members.shaw.ca/hdhcubes/cube_update-3.htm

2004   Mitsutoshi Nakamura supplied me with the following cubes. All of these are the first I had seen of that type and order.
·        Pantriagonal orders 10 and 14.
·        Diagonal orders 11, 13, 15, 16, and 17.

http://members.shaw.ca/hdhcubes/cube_summary.htm

2004   Abhinav Soni supplied me with the following cubes. All of these are the first I had seen of that type and order.
·        Simple order-13.
·        Pantriagonal orders 15 and 17.
·        Pandiagonal orders 8, 9, 11, 13, 15, 16, and 17.

http://members.shaw.ca/hdhcubes/cube_summary.htm

2004  Purely pandiagonal Guenter Stertenbrink found an order 4 cube with
·         ALL diagonals and ALL triagonals are correct.
·         NO monagonals correct, so the cube is not magic (but very unusual).

http://members.shaw.ca/hdhcubes/cube_update-1.htm

2004  Mitsutoshi Nakamura created bordered magic cubes from Orders-6 to 40.
·        Each cube contains bordered magic cubes of all lesser even orders.
·        All cubes except order 4 are diagonal, and all cubes use consecutive numbers.

[77]  See them at http://homepage2.nifty.com/googol/magcube/en/works.htm#cubes_bd2

2004  Christian Boyer located the cube sent by Leibniz to the Académie des Sciences in 1715  -
·        in the  Hanover Library, Germany.

http://members.shaw.ca/hdhcubes/cube_update-4.htm

2006   C. Boyer publishes multiply magic cubes of orders 3 to 11 on his web site.
·        He shows an order-8 perfect magic cube (he calls it pandiagonal perfect) where all possible lines (13 x 8² = 832 give the magic product P = 8951 81838 23250 31429 47225 60000.
·        Also shown are perfect magic cubes of orders 9 and 11.
·        Some other orders include solutions for simple magic and diagonal (he calls these perfect) magic cubes. (These provide 3m² + 4, and 3m² + 6m + 4 correct lines).
·        Shown on his site is a table of best solutions, and cubes available for download.

[78]  www.multimagie.com/ (click on Multiplicative cubes)

 2007   Awani Kumar announces via email 4 new magic knight square discoveries.
·    April 28, 2007, the first Magic Knight Tour of an 8x8x8 cube. (Actually only semi-magic by magic cube standards because 2 triangles sum incorrectly).
·       May 4, 2007, A Magic Knight Tour of the six surfaces of an 8x8x8 cube. http://members.shaw.ca/hdhcubes/cube_update-5.htm
·        May 22, 2007, the first truly Magic Knight Tour of an 8x8x8 cube. All rows, columns, pillars, and the 4 main triagonals sum correctly. This is a re-entrant tour.
·    June 19, 2007, the first truly Magic Knight Tour of an 12x12x12 magic cube.  This knight tour is almost, but not quite, re-entrant (closed).
 

2009   Zwong Ming and Peng Boa-wang construct an order-6 prime number magic cube.
·        A few days later they email me an order-8 Concentric prime number cube containing a simple order-6 and a pantriagonal order-4 magic cube.

]79]  emails to me dated August 31, 2009 (but received on Sept.7) and September 8, 2009. See them at http://members.shaw.ca/hdhcubes/cube_prime.htm

Dimensions Greater then 3

1905   C. Planck illustrates an  order-3 octahedron, using the numbers 1 to 81
·        This is a dimension 4 hypercube, now more commonly called a tesseract.
·        He cites others as working with 4-dimension octahedrons before him as, Frost (1878), Stringham (1880), Arnoux (1894).

[79]  C. Planck, The Theory of Paths Nasik. Printed in 1905 for private circulation.
[80]  W. S. Andrews, Magic Squares & Cubes, 2^nd edition, Dover Publ. 1960 (reprint of 1917, Open Court), pp351-362 (H. Kingery; pp363-375 (C. Planck).

1962    J. R. Hendricks publishes a practical way to represent the 4-D Hypercube, the Tesseract.
·        He shows an order 3 tesseract using the new representation, then a 5-D and 6-D magic hypercube.

[81]  J.R. Hendricks, The Five and Six Dimensional Magic Hypercubes of Order 3, Canadian Mathematical Bulletin, vol.5, No. 2, 1962, pp 171-189
[82]  One of Hendricks order-3 tesseracts is shown at http://www.magic-squares.net/hendricks.htm

1968   Hendricks publishes a panquadragonal magic tesseract

[83]  J.R. Hendricks, The Pan-4-agonal Magic Tesseract, American Mathematical Monthly,75:4 April 1968, p. 384.
[84]  John R. Hendricks, Pan-n-agonals in Hypercubes, JRM 7:2, 1974, pp95-96.
http://members.shaw.ca/hdhcubes/cube_perfect.htm

1998   Hendricks constructs the world’s first perfect magic tesseract.
·        It is order 16. It is confirmed correct by Clifford Pickover of IBM. A year later Hendricks completes the order 32 5-D perfect hypercube.

[85]  John R. Hendricks, Magic Squares to Tesseracts by Computer, self-published, 1998, 0-9684700-0-9, 142++ pages.
[86]  H.D. Heinz and J.R. Hendricks, Magic Square Lexicon: Illustrated, Self-published, 2000, 0-9687985-0-0, 184++ pages.
[87]  John R. Hendricks, Perfect n-Dimensional Magic Hypercubes of Order 2ⁿ, self-published, 2000, 0-9684700-4-1, 36+pages.
[88]  C. A. Pickover, The Zen of Magic Squares, Circles and Stars, Princeton Univ. Pr., 2002, 0-691-07041-5, 404 pages.

1999   J. Hendricks published an order-6 magic tesseract with an order-3 inlaid magic tesseract.
·        The order 6 tesseract uses the numbers from 1 to 4⁶ = 1 to 1296. It is not associated.
·        The order-3 tesseract uses the subset numbers 568 to648. It is associated (as all order-3 magic hypercubes are).
·        The complete tesseract is shown at http://members.shaw.ca/hdhcubes/cube_inlaid.htm

[89]  The complete Inlaid Order 6 Magic Tesseract is supplied as an 8 page chart insert with these books;
[90]  John R. Hendricks, Magic Squares to Tesseracts by Computer, self-published, 1998, 0-9684700-0-9, 142++ pages.
[91]  John R. Hendricks, Inlaid Magic Squares and Cubes, 2^nd edition, self-published, 2000, 0-9684700-3-3, 250+ pages.

1999   J. Hendricks published diagrams of all 58 order-3 magic Tesseracts.
·        Each of these may appear in 384 aspects.
·        It later turned out that Key Ying Lin of Taiwan had published the same results in 1986.

[92]  John R. Hendricks, All Third-Order Magic Tesseracts, self-published, 1999, 0-9684700-2-5, 36++ pages.

References for Magic Cube Timeline

1600-1700
[1]  Edouard Lucas, L’Arithmétique amusante (Amusing Arithmetic), Gauthier-Villars, 1895
[2]  Akira Hirayama and Gakuho Abe, Researches in Magic Squares, 1983, Osaka Kyoikutosho.
[3]  Adami A. Kochanski, Considerationes quaedam circa Quadrata & Cubos Magicos, Acta Eruditorum, 1686, vol. 5, pages 391-395.

1700-1800
[4]  Mémoires de l'Académie Royale des Sciences of 1710. Notes from Christian Boyer because of restrictions on photocopying.
[5]  Christian Boyer, Le plus petit cube magique parfait (and Inédit - Le cube magique de Leibniz est retrouvé), La Recherche,
issue number 373, March 2004, pages 48-50, Paris, 2004
[6]  http://www.multimagie.com/
[7]  Akira Hirayama and Gakuho Abe, Researches in Magic Squares, 1983, Osaka Kyoikutosho.

1800-1900
[8]  Par B. Violle, Traité complet des Carrés Magiques, 1837, (French) This book is available on the Internet at http://gallica.bnf.fr.
as scanned pages.
[9]  A. H. Frost, Invention of Magic Cubes. Quarterly Journal of Mathematics, 7, 1866, pp 92-103
[10]  F.A.P. Barnard, Theory of Magic Squares and Magic Cubes, Memoirs of the National Academy of Science, 4,1888,pp. 209-270. Construction details of the "Frankenstein" cube is described in a lengthy footnote on pages 244-248.
[11]  W. Benson & O. Jacoby, Magic Cubes: New Recreations, Dover Publ. 1981, pp 32-33
[12]  Theodore Hugel, Das Problem der magishen Systeme, 1876, Verlag von A. H. Gottschick, 70pp. (German).
[13]  A. H. Frost, On the General Properties of Nasik Cubes, QJM 15, 1878, pp 93-123
[14]  W. S. Andrews, Magic Squares & Cubes, Open Court, 1908, p. 64
[15]  W. S. Andrews, Magic Squares & Cubes, 2^nd edition, Dover Publ. 1960 (reprint of 1917, Open Court) p. 64
[16]  Cube Diabolique de Dix-Sept, was deposited in l'Académie des Sciences, Paris, France, April 17, 1887.
[17]  F.A.P. Barnard, Theory of Magic Squares and Magic Cubes, Memoirs of the National Academy of Science, 4, 1888,pp. 209-270.
[18]  W. Benson & O. Jacoby, Magic Cubes: New Recreations, Dover Publ. 1981, pp 32-33
[19]  W. S. Andrews, Magic Squares & Cubes, 2^nd edition, Dover Publ. 1960 (reprint of 1917, Open Court) p. 373
[20]  W. S. Andrews, Magic Squares & Cubes, 2^nd edition, Dover Publ. 1960 (1917) , pages 310, 311, 314.
[21]  Hermann Schubert, Mathematical Recreations and Essays, Open Court 1899.
[22]  E. Fourrey, Recréations Arithmétiques, (Arithmetical Recreations) 8^th edition, Vuibert, 2001, 261+ pages (edition 1, 1899).
http://members.shaw.ca/hdhcubes/cube_early.htm

1900-1950
[23]  From C. Planck, The Theory of Paths Nasik. Printed in 1905 for private circulation.
[24]  W. S. Andrews, Magic Squares & Cubes, 1908), page 64 ( This is reproduced as the first 188 pages of [25].
[25]  W. S. Andrews, Magic Squares & Cubes, 2^nd edition, Dover Publ. 1960), page 64 (reprint of 1917, Open Court
[26]  H. A. Sayles, A Magic Cube of Six, The Monist, 20, 1910, pp 299-303
[27]  J. Worthington, A Magic Cube of Six, The Monist, 20, 1910, pp 303-309
[28]  W. S. Andrews Magic Squares & Cubes 1960 (1917) pps. 197, 202 .
[29]  H. A. Sayles, Geometric Magic Squares and Cubes, The Monist, 23, 1913, pp 631-640
[30]  W. S. Andrews Magic Squares & Cubes 1960 (1917) pps. 283-294.
[31]  W. S. Andrews, Magic Squares & Cubes, 2^nd edition, Dover Publ. 1960 (reprint of 1917, Open Court), p. 352-366
[32]  B. Rosser and R. J. Walker, Magic Squares: Published papers and Supplement, a bound volume at Cornell University, catalogued
 as QA 165 R82+pt.1-4.
[33]  R.V. Heath, A Magic Cube With 6n^3 Cells, American Mathematical Monthly, Vol. 50, 1943 p.288-291
[34]  From Mutsumi Suzuki's Web site at http://mathforum.com/te/exchange/hosted/suzuki/MagicSquare.html

1951-2000
[35]  John R. Hendricks, The Pan-3-agonal Magic Cube, JRM 5:1:1972, pp 51-54
[36]  John R. Hendricks, The Pan-3-Agonal Magic Cube of Order 5, JRM 5:3:1972, pp 205-206
[37]  John R. Hendricks, Pan-n-agonals in Hypercubes, JRM 7:2, 1974, pp 95-96.
[38]  J.R. Hendricks, The Pan-4-agonal Magic Tesseract, American Mathematical Monthly,75:4 April 1968, p. 384.
[39]  J. R. Hendricks, Magic Cubes of Odd Order, JRM 6:4, 1973, pp 268-272 and Magic Square Course, 1991, p. 366
[40]  Bayard E. Wynne, Perfect Magic Cubes of Order Seven, JRM 8:4, 1975-76, pp 285-293
[41]  W. Benson & O. Jacoby, Magic Cubes: New Recreations, Dover Publ. 1981, page 63
[42]  Martin Gardner, Mathematical Games, Scientific American, Jan. 1976.
[43]  Martin Gardner, Time Travel and Other Mathematical Bewilderments, W. H. Freeman, 1988.
[44]  Ian P. Howard, Pan-diagonal Associative Magic Cubes (Letter to the Editor), JRM 9:4, 1976, pp276-278.
[45]  W. S. Andrews, Magic Squares & Cubes, 2^nd edition, Dover Publ. 1960 (reprint of 1917, Open Court), p 366
[46]   Mathematical Sciences (Japanese language) Magazine Dec. 1977, p. 45-- Special issue on puzzles.
[47]  Gakuho Abe, Related Magic Squares with Prime Elements, JRM 10:2 1977-78, pp.96-97. Akio Suzuki order-3 and 4 cubes.
[48]  A. W. Johnson, Jr., Solution to Problem 2617, JRM 32:4, 2003-2004, pp. 338-339
[49]  A. W. Johnson, Jr., An Order 4 Prime Magic Cube, JRM 18:1, 1985-86, pp 5-7
[50]  K. W. H. Leeflang, Magic Cubes of Prime Order, JRM 11:4, 1978-79, pp 241-257
[51]  Brian Alspach & Katherine Heinrich, Perfect Magic Cubes of Order 4m, The Fibonacci Quarterly, Vol. 19, No. 2, 1981 pp 97-106
[52]  W. Benson & O. Jacoby, Magic Cubes: New Recreations, Dover Publ. 1981, page 63
[53]  Published in "Młody Technik" (Young Technican) magazine No. 6/1984  (www.golunski.de/)
[54]  It may be downloaded from http://www.multimagie.com/
[55]  John R. Hendricks, An Inlaid Magic Cube, JRM 25:4, 1993, pp 286-288.
[56]  John R. Hendricks, Inlaid Magic Squares and Cubes, 2^nd edition, self-published, 2000, 0-9684700-3-3, 250+ pages. Edited and illustrated
by Holger Danielsson.
[57]  H.D. Heinz and J.R. Hendricks, Magic Square Lexicon: Illustrated, Self-published, 2000, 0-9687985-0-0,  page 25.
[58]  The Sci.math newsgroup Dec. 2, 1996 and Dec. 10, 1996.
[59]  See http://members.shaw.ca/hdhcubes/cube_update-2.htm#Order%206%20Projection%20Cubes
[60]  John R. Hendricks, Magic Squares to Tesseracts by Computer, self-published, 1998, 0-9684700-0-9, pps 55-56.
[61]  From F. Laio, T. Katayama and K. Takaba, On the Construction of Pandiagonal Magic Cubes, Kyoto Univ. Technical Report # 99021, 1999

After 2000
[62]  H. D. Heinz and J.R. Hendricks, Magic Square Lexicon: Illustrated, Self-published, 2000, 0-9687985-0-0
[63]  John R. Hendricks, A Bimagic Cube of Order 25, self-published, 2000, 0-9684700-7-6, 18 pages.
[64]  Holger Danielsson, editor, Printout of a Bimagic Cube of Order 25, self-published, 2001, 36 pages.
[65]  Soni’s cube generator is available at http://www.hypermagiccube.cjb.net/
[66]  Marián Trenkler, Additive and Multiplicative Magic Cubes., 6th Summer school on applications of modern math. methods, TU Košice 2002, 23-25
[67]  See also Obzory matematiky, fyziky a informatiky 1/2002, 9-16
[68]  C. Boyer’s Multimagic site is at www.multimagie.com/ (click on multimagic cubes)
[69]  This news was published in over 20 magazines and columns. Best source is http://www.multimagie.com/  and click on Perfect magic cubes.
[70]  I received this via email on Nov. 9, 2003.
[71] A. Czepa, Mathematische Spielereien (Mathematical Games), Union Deutsche, 1918,  140 pages, (page 77) (Old German script). There are many magic objects in this small format book but just two magic cubes.
[72]  http://members.shaw.ca/hdhcubes/cube_modulo.htm
[73]  John R. Hendricks, Inlaid Magic Squares and Cubes, 2nd edition, self-published, 2000, 0-9684700-3-3, 250+ pages. Edited and illustrated
by Holger Danielsson.
[74]  Benson & Jacoby, New Recreations in Magic Squares, 1976, pp 26-33
[75]  J.L.Fults, Magic Squares, Open Court, 1974
[76]  The six classes are defined on my Magic Cubes-Introduction page, http://members.shaw.ca/hdhcubes/index.htm
[77]  See them at http://homepage2.nifty.com/googol/magcube/en/works.htm#cubes_bd2
[78]  www.multimagie.com/ (click on Multiplicative cubes)
]79]  emails to me dated August 31, 2009 (but received on Sept.7) and September 8, 2009. See them at http://members.shaw.ca/hdhcubes/cube_prime.htm

Dimensions > 3
[79]  C. Planck, The Theory of Paths Nasik. Printed in 1905 for private circulation.
[80]  W. S. Andrews, Magic Squares & Cubes, 2^nd edition, Dover Publ. 1960 (reprint of 1917, Open Court), pp351-362 (H. Kingery;
pp363-375 (C. Planck).
[81]  J.R. Hendricks, The Five and Six Dimensional Magic Hypercubes of Order 3, Canadian Mathematical Bulletin, vol.5, No. 2, 1962,
 pp 171-189
[82]  One of Hendricks order-3 tesseracts is shown at http://www.magic-squares.net/hendricks.htm
[83]  J.R. Hendricks, The Pan-4-agonal Magic Tesseract, American Mathematical Monthly,75:4 April 1968, p. 384.
[84]  John R. Hendricks, Pan-n-agonals in Hypercubes, JRM 7:2, 1974, pp95-96.
[85]  John R. Hendricks, Magic Squares to Tesseracts by Computer, self-published, 1998, 0-9684700-0-9, 142++ pages.
[86]  H.D. Heinz and J.R. Hendricks, Magic Square Lexicon: Illustrated, Self-published, 2000, 0-9687985-0-0, 184++ pages.
[87]  John R. Hendricks, Perfect n-Dimensional Magic Hypercubes of Order 2ⁿ, self-published, 2000, 0-9684700-4-1, 36+pages.
[88]  C. A. Pickover, The Zen of Magic Squares, Circles and Stars, Princeton Univ. Pr., 2002, 0-691-07041-5, 404 pages.
[89]  The complete Inlaid Order 6 Magic Tesseract is supplied as an 8 page chart insert with [90][91]
[90]  John R. Hendricks, Magic Squares to Tesseracts by Computer, self-published, 1998, 0-9684700-0-9, 142++ pages.
[91]  John R. Hendricks, Inlaid Magic Squares and Cubes, 2^nd edition, self-published, 2000, 0-9684700-3-3, 250+ pages.
[92]  John R. Hendricks, All Third-Order Magic Tesseracts, self-published, 1999, 0-9684700-2-5, 36++ pages.

First cube of each class for each order

For convenience, I have copied this table from my summary page.

Links are provided to cubes shown on my site and footnotes included  for cubes not shown on my site.
Note [6] and higher are for cubes added to this table after it was first posted.

A 6th class of magic cubes has been discovered, by Mitsutoshi Nakamura in January 2005. 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.

[1]  Frost published a Perfect order 9 in 1878, but it was not normal. It used numbers in the series from 1 to 889.
       However, about the same time he converted the non-sequential numbers to the series from 1 - 729 for the normal nasik
       cube model now in the Whipple museum. See Cube_update-6.
[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.

Please send me Feedback about my Web site!
Harvey Heinz   harveyheinz@shaw.ca
This page last updated February 05, 2010
Copyright © 2006 by Harvey D. Heinz