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Index to this page
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 See
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. See
cube_update-2.htm for more on early cubes.
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.
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/
and 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. See
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.
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 92-10. See
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. See 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).
See 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).
[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, 2nd edition, Dover Publ. 1960
(reprint of 1917, Open Court) p. 64 See 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.
See 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. See 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, 2nd edition, Dover Publ. 1960 (reprint of 1917, Open Court) p.
373. See 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.
See 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. See
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) 8th edition, Vuibert, 2001, 261+
pages (edition 1, 1899). See 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. See
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, 2nd edition, Dover Publ. 1960),
page 64 (reprint of 1917, Open Court. See 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.
See 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) pages 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 2k, or 2k + 1 if we require association.”
[31] W. S. Andrews, Magic Squares & Cubes, 2nd 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
ms-models.htm#Six magic
cubes in One
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.
See also 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.
See also 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 See 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. See 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.
For more on this see 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, 2nd edition, Dover Publ. 1960
(reprint of 1917, Open Court), page 366. See 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.
See 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. See 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. See 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
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. See
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. See 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, 2nd edition,
self-published, 2000, 0-9684700-3-3, 250+ pages. Edited and illustrated by
Holger Danielsson. See 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 cube_update-2.htm and
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. See
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
See 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. See 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
See 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. See 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.
See 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/. See also 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
and 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.
See cube_12.htm
and 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). See
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. Heinz, (and others) now refer 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.
See also 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.
See 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.
See 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.
See
cube_big.htm
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.
See 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.
See cube_13.htm and
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.
See 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,
magic_cubes_index.htm
See also 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.
See 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.
See 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).
See
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.
See
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
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, 2nd 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
hendricks.htm
and Tesseract Representations
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.
See 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
cube_inlaid.htm# inlaid magic tesseract
[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, 2nd 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, 2nd
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, 2nd
edition, Dover Publ. 1960 (reprint of 1917, Open Court) p. 373
[20] W. S. Andrews, Magic Squares &
Cubes, 2nd 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) 8th
edition, Vuibert, 2001, 261+ pages (edition 1, 1899).
http://members.shaw.ca/hdhcubes/cube_early.htm
Dimensions > 3
[79] C. Planck, The Theory of Paths Nasik. Printed in
1905 for private circulation.
[80] W. S. Andrews, Magic Squares & Cubes, 2nd
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
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, 2nd
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.
|
m |
Simple magic |
Pantriagonal magic |
Diagonal magic |
Pandiagonal magic |
Perfect magic |
|
3 |
Hugel 1876 |
(none possible) |
(none possible) |
(none possible) |
(none possible) |
|
4 |
Kurushima 1757 [15] |
Frost 1878 |
(none possible) |
(none possible) |
(none possible) |
|
5 |
Hugel 1876 |
Hendricks 1972 |
Trump/Boyer 2003 |
(none possible) |
(none possible) |
|
6 |
Firth 1889 |
Abe 1948 |
Trump 2003 |
(none possible) |
(none possible) |
|
7 |
Soni 2001 |
Hendricks 1973 |
Trump 2003 |
Frost 1866 |
(none possible) |
|
8 |
Andrews 1908 |
Frost 1866 |
Frankenstein 1875 |
Soni
2004 [12] |
Barnard 1888 |
|
9 |
Golunski 1984 |
Soni 2001 |
Boyer 2003 |
Soni 2004
[9] |
Planck 1905
[1] |
|
10 |
Planck 1894 |
Nakamura 2004
[13] |
Li Wen 1988
[14] |
(none possible)
[10] |
(none possible)
[10] |
|
11 |
Soni 2001 |
Suzuki 2000 ? |
Nakamura 2004
[11] |
Soni 2004 [9] |
Barnard 1888 |
|
12 |
Poyo 1999 |
de
Winkel 2003 |
Benson & Jacobi 1981[8] |
(none possible) [10]
|
(none possible)
[10] |
|
13 |
Soni 2004
[6] |
Golunski 2003 |
Nakamura 2004
[11] |
Soni 2004
[9] |
Liao and assoc. 1999 |
|
14 |
Soni 2003 |
Nakamura 2004
[13] |
Benson & Jacobi 1981[8] |
(none possible) [10] |
(none possible)
[10] |
|
15 |
Heinz 2003 [3] |
Soni 2004
[6] |
Nakamura-Soni 2004
[17] |
Soni 2004
[9] |
Planck 1905 [2] |
|
16 |
Boyer 2003
[4] |
de Winkel 2003 |
Nakamura 2004 [16] |
Soni
2004 [12] |
Soni 2003 |
|
17 |
Soni 2004
[7] |
Soni 2004
[6] |
Nakamura 2004
[11] |
Soni 2004
[9] |
Arnoux 1887 [5] |
[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.
|
