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The following books and papers represent most of the sources of my research into magic cubes. Because some of the books are written in languages I do not understand, my information from these is limited to the actual cubes only. In several cases, I found a listing for a cube that was not magic in any way I could see. As I was not able to understand the accompanying text, I was forced to ignore these cubes.
Perhaps some readers that are familiar with these languages and have access to the relevant document, may be able to provide me with translations of relevant passages.
Many sources gave conflicting information, especially for definitions, dates, and in some cases authors. My notes reflect my studied opinion, so all readers may not agree with all my conclusions. Also, I found quite a number of typographical errors in magic cube listings. In all cases, my listings show the corrections.
Many thanks to all who provided suggestions or information on magic cubes. I cannot acknowledge everyone by name, but a special thanks is due to Christian Boyer, Marian Trenkler, and Paul Vaderlind, all of which were most generous in their help.
A good source for information on magic squares and cubes was the Strens Collection in the Special Collections Department of the University of Calgary (see the Internet URL below).
Many of the books shown here are in French German or Japanese. I indicate such with the listing.
W. S. Andrews, Magic Squares & Cubes, Open
Court, 1908, 193+ pages.
The first 188 pages of edition 2 is almost exactly the same as this. Differences
are:
W. S. Andrews, Magic Squares & Cubes, 2nd
edition, Dover Publ. 1960, 419+ pages .
This is an unaltered reprint of the 1917 Open Court Publication of the second
edition. The first 188 pages are almost identical to edition 1 published in 1908
(see above). Much of the new material consists of essays first published in The Monist
from 1905 to 1916.
The modern definition for a simple magic cube seems to have been published first by W. S. Andrews in his first edition 1908, page 64. It also appears on the same page of the more familiar second edition, 1917. Page 365 has material on early definitions for cubes, 13 paths through cubes, etc.
G. Arnoux, Arithmetique graphique – les espaces arithmetiques hypermagiques, Gauthier-Villars, 1894,175+ pages. (French). Lots of theory with methods of construction. No actual examples of magic cubes. However, on page 61 he starts a discussion about an order 17 perfect magic cube (Cube Diabolique de Dix-Sept) he deposited in the l’Académie des sciences on April 17, 1887
W. H. Benson & O. Jacoby, Magic Cubes: New
Recreations, Dover Publ. 1981, 0-486-24140-8, 142 pages.
This book provides a valuable contribution to the literature, including an early
perfect order-8 magic cube. They include an account, almost word for word, of
the early cube of G. Frankenstein, that appears as an extensive footnote in F.
Barnard’s 1888 paper. However, I then find it curious that they make no mention
of Barnard’s order 8 and two order 11 perfect cubes, that are extensively
written up in the same paper.
E. Cazalas, Carré magique au degree n, Paris, 1934 (French).
E. Cazalas, A travers les hyperspaces magiques (Through Magic Hyperspace). Sphinx, 1936, 19 pages (French). I took an order 3 magic cube from these pages, but it is not consecutive numbers.
A. Czepa, Mathematische Spielereien
(Mathematical Games), Union Deutsche, 1918, 140 pages.
(Old German script). Many magic objects in this small format book. Just 2 magic
cubes.
Holgar Danielsson, Printout of an Order-25
Bimagic Cube, Self-published, 2000, 36 pp plus covers, flat-stitched, 8.5 x
11
A nicely formatted and printed graphical version of John Hendricks Bimagic Cube
of Order 25.
René Descombes, Les Carrés Magiques (Magic Squares), Vuibert, 2000, 2-7117-5261-5, 494 pages. (French)
E. Fourrey, Recréations arithmétiques,
(Arithmetical Recreations) 8th edition, Vuibert, 2001,
2711753123, 261+ pages. (French). Originally published in 1899. (French)
This was the source for my Saveur order 3 and the Fourrey order 4 cubes.
Martin Gardner, Time Travel and Other Mathematical Bewilderments, W. H. Freeman, 1988, 0-7167-1924-X. Chapter 17, pp213-225. (page 222 has the Myers cube which Gardner called perfect. This type of cube is now called a diagonal cube (the new perfect magic cubes are a much higher class.
A. W. Goodman, The Pleasures of Math, Macmillan, 1965, 224 pages
R. V. Heath, Mathemagic, Dover Publ. 1953, originally published in 1933
H.D. Heinz and J.R. Hendricks, Magic Square Lexicon: Illustrated, Self-published, 2000, 0-9687985-0-0, 184++ pages. Definitions.
John R. Hendricks, Magic Square Course, self-published, 1991, 521+ pages.
John R. Hendricks, Magic Squares to Tesseracts by Computer, self-published, 1998, 0-9684700-0-9, 142++ pages.
John R. Hendricks, Inlaid Magic Squares and
Cubes, self-published, 1999, 0-9684700-1-7, 188+ pages.
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.
John R. Hendricks, Perfect n-Dimensional Magic Hypercubes of Order 2n, self-published, 2000, 0-9684700-4-1, 36+pages.
John R. Hendricks, A Bimagic Cube of Order 25,
self-published, 2000, 0-9684700-7-6, 18 pages.
Also, Holger Danielsson, Printout of a Bimagic Cube of Order 25,
self-published, 2001, 36 pages.
John R. Hendricks, All Third-Order Magic Tesseracts, self-published, 1999, 0-9684700-2-5, 36+ pages.
Theodore Hugel, Das Problem der magishen Systeme,
1876, Verlag von A. H. Gottschick, 70pp. (German). This book is
obtainable over the Internet from
Cornell University Library, Digital Collections at http://historical.library.cornell.edu/math/about.html
Kenneth Kelsey, The Ultimate book of Number
Puzzles, Dorset, 1992, 522 pages, 0-88029-920-7.
This is a
combination of 5 books ( four by K Kelsey & the last one by D. King), all
published in Great Britain 1979-1984 by Frederick Muller Ltd.
It consists of numerical puzzles in the form of magic squares, cubes, stars,
etc. No theory, but lots of examples (some quite original) and lots of practice
material.
Harry Langman, Play Mathematics, Hafner
Publ., 1962
Lots of material on magic objects and number patterns. Pages 70 to 76 are on
magic squares and an order 7 pandiagonal magic cube.
Max Bruno Lehmann, Der geometrische Aufbrau
Gleichsummiger Zahlenfiguren (The Geometric Construction of Magic Figures),
1932, xvi+384 pages.
This book is mostly about magic squares, but includes discussions and examples
of magic cubes and magic stars.
Edouard Lucas, L’Arithmétique amusante (Amusing Arithmetic), Gauthier-Villars, 1895,266+ pages. (French). Fermat magic cube. Looks like an interesting book, but only 1 magic cube, Fermat’s order 4.
C. A. Pickover, The Zen of Magic Squares, Circles and Stars, Princeton Univ. Pr., 2002, 0-691-07041-5, 404 pages.
Jacques Ozanam (1640-1717), Recreations in
the Science and Natural Philosophy. Enlarged by Jean Montucla about 1768.
Translated into English by Dr. C. Hutton in 1803. Finally revised by Edward
Little in 1844. This book is 826 pages but only Part 1 (113 pages) is
concerned with recreational mathematics and only pages 94 to 106 with magic
squares. There is nothing on magic cubes.
This book is obtainable over the Internet from
Cornell University Library, Digital Collections at http://historical.library.cornell.edu/math/about.html
RouseBall & Coxeter, Mathematical Recreations &
Essays, 11 edition, 1939. Chapter VII (pp 193-221 is on magic squares
and cubes.
Editions 12, University of Toronto Press, 1974, 0-8020-6189-9 and
Editions 13, Dover Publ., 1989, 0-486-25357-0 have virtually the identical
Chapter VII, with only minor changes to the early part of the chapter. All
illustrations and page numbers are the same.
NOTE: Edition 10, 1922, has a much different chapter VII, It is at pages
137-161, and contains less on magic squares, nothing on magic cubes and more on
magic stars.
William. L. Schaaf, A Bibliography of Recreational Mathematics, vol. 2, NCTM, 1970.
Hermann Scheffler, Die Magischen Figuren (Magic Figures), Martin S, 1968, 112 pages. (German) Nothing original here.
Hermann Schubert, Mathematical Essays and
Recreations. Translated from German to English by Thomas J. McCormack, Open
Court, 1899. 143+ pages.
The chapter on magic squares and cubes is on pages 39 to 63. The next chapter,
(pp 64-111) deals with the 4th dimension, but no mention of magic
tesseracts.
This book is obtainable over the Internet from
Cornell University Library, Digital Collections at http://historical.library.cornell.edu/math/about.html
Hermann Schubert, Mathematische Mussestunden, (Mathematical Pastimes), Walter de Gruyter, 1940, 245 pages. Originally published 1900? The preface was dated 1897. (German). pp 142-172 was on magic squares. Magic stars from 172-176. No magic cubes.
Hermann Schubert, Mathematische Mussestunden II, (Mathematical Pastimes II), G.J. Goshen’sche, 1909, 247+ pages. (German). This book was date stamped Berlin, 12 Nov. 1900! Although one of the keywords was ‘magic cubes’ there were none in this book.
Walter Sperling, Spiel und Spass furs Ganze Jahr (Fun and Games for all Years), Albert Muller, 1951, 111 pages. (German) Not an awful lot on magic cubes. He shows an order 4 block puzzle.
Walter Sperling, Die Grubelkiste (The Amusement Chest), Albert Muller, 1953, 162 pages. (German). He shows the same order 4 cube that Schubert published.
Par B. Violle, Traité complet des Carrés
Magiques, 1837, 1000+ pages (French). About 100 pages on magic cubes.
The book is available on the Internet at
http://gallica.bnf.fr/ as scanned pages. A selected page may be viewed or
the entire book may be downloaded.
For Non-French readers, Christian Boyer has kindly provided this help:
Weidemann, Ingenieur, Zauberquadrate und andere magische Zahlenfiguren der Ebene und des Raumes, Oscar Leiner, 1922, 83 pages. Translated title is Magic squares and other plane and solid magic figures. (German). This book contains many examples of magic squares, cubes, and geometric figures.
Seimiya, Mathematical Sciences (Japanese) Magazine Dec. 1977, Special issue on puzzles, p. 45-47 orders 9 and 11 perfect magic cubes. Another 10 pages on many magic objects.
Singmaster CD This CD contains many files of essays, bibliographies, chronologies, etc., to do with recreational mathematics. Available from David Singmaster zingmast@sbu.ac.uk
All papers listed here are in the English language (unless indicated differently).
Gakuho Abe, Fifty Problems of Magic Squares, Self published 1950. Later republished in Discrete Math, 127, 1994, pp 3-13. The last 10 problems deal with magic cubes. It also includes the Abe order 6 cube.
A. Adler, & R. Shuo-yen, Magic Cubes and Prouhet Sequences, American Mathematical Monthly, vol. 84(8), 1977, p. 618-627. They show (with quite a bit of mathematics) several methods of forming magic squares from smaller order magic cubes.
Brian Alspach & Katherine Heinrich, Perfect
Magic Cubes of Order 4m, The Fibonacci Quarterly, Vol. 19, No. 2, 1981, pp
97-106
They define a perfect magic cube as one where all the main diagonals sum
to S (We now called these diagonal cubes). They then site examples of
pandiagonal magic cubes.
Gabriel Arnoux, (French) Cube Diabolique de Dix-Sept,
Académie des Sciences, Paris, France, April 17, 1887.
26 handwritten pages contain a perfect (new definition) magic cube. Thanks to
Christian Boyer, who kindly photographed these pages for me (the Academy would
not allow photo-copying).
This cube contains 51 planar, 6 oblique, and 96 2-segment oblique, order 17
pandiagonal magic squares.
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, described in a lengthy footnote on pages 244-248, are quoted, almost verbatim, in Benson and Jacoby (1981). He introduces the first (?) normal perfect magic cubes. An order 8 and two order 11 perfect cubes are shown with full information on how they were constructed. He also shows a magic cylinder and magic sphere.
Brown, P. G.,
The
MAGIC SQUARES of Manuel Moschopoulos, A Translation,
Pure Mathematics Report PM97/22, AMS/01A20/01A75, 32 pages (the original was
written about
1315
A. D.)
Christian Boyer, Les cubes magiques, Pour la
Science, Sept. 2003, No. 311, pp 90 - 95.
A hsitory of magic cubes and a description of his order 8192 quadramagic cube.
(French)
A. H. Frost, Invention of Magic Cubes.
Quarterly Journal of Mathematics, 7, 1866, pp 92-102
He describes a method of constructing magic cubes and shows an order 7
pandiagonal and an order 8 pantriagonal magic cube.
A. H. Frost, Supplementary Note on Magic Cubes. Quarterly Journal of Mathematics, 8, 1867, p 74
A. H. Frost, On the General Properties of Nasik
Squares, QJM 15, 1878, pp 34-49
Construction of pandiagonal magic squares. His cube papers make reference to
this.
A. H. Frost, On the General Properties of Nasik
Cubes, QJM 15, 1878, pp 93-123 plus plates 1 and 2.
He shows two order 3 and order 4 cubes, and one each of orders 7 and 9, with
method of construction. These cubes (in order) are not magic, disguised order 3,
not magic, pantriagonal, pantriagonal and perfect.
A. H. Frost, Description of Plates 3 to 9,
QJM 15, 1878, pp 366-368 plus plates 3 to 9.
Illustrations of an group of 7 interrelated order 7 cubes.
Martin Gardner, Mathematical Games, Scientific American, Jan. 1976, pp118-122. First mention of Myers cube and his definition of a “perfect” magic cube. Cubes with Myers features are now called diagonal magic.
R. V. Heath, A Magic Cube With 6n3 cells, American Mathematical Monthly, Vol. 50, 1943, pp 288-291.
J. R. Hendricks, The Five and Six Dimensional Magic Hypercubes of Order 3, Canadian Mathematical Bulletin, vol.5, No. 2, 1962, pp 171-189
J. R. Hendricks, The Pan-4-agonal Magic Tesseract, American Mathematical Monthly,75:4 April 1968, p. 384. (this cube republished in John R. Hendricks, Magic Squares to Tesseracts by Computer, self-published, 1998, 0-9684700-0-9, 142++ pages (page 130).
Vladimír Karpenko, Two Thousand Years of numerical magic squares, Endeavour, New Series, 18, 4, 1994, pp 147-153. No mention of magic cubes.
Adami A. Kochanski, Considerationes quaedam circa
Quadrata & Cubos Magicos, Acta Eruditorum, 1686, vol. 5, pages 391-395.
Published in Latin. One page contains two problems about magic cubes... but
without the solutions.
H. M. Kingery, A Magic Cube of Six, The
Monist, XIX, 1909, pp 434-441
The author describes the method used to construct an order 6 cube with bent
triagonals. However, the main triagonals are incorrect. This assay was later
republished in essentially the same form in W. S. Andrews, Magic Squares and
Cubes, 2nd edition, 1917.
F. Liao, T. Katayama, and K. Takaba, On the
Construction of Pandiagonal Magic Cubes, Technical Report 99021, School of
Informatics, Kyoto University, 1999. Available on the Internet at
http://www.amp.i.kyoto-u.ac.jp/tecrep/TR1999.html
Heavy mathematics. But they list two order 13 pandiagonal magic cubes.
By the new definition, these are perfect magic cubes.
They also demonstrate that there are m-1 broken pandiagonal magic squares
parallel to each of the 6 oblique pandiagonal magic squares.
Manuel Moschpoulos (about 1265 – 1315), The Magic Squares of Manuel Moschpoulos, Pure Mathematics Report PM97/22, AMS/01A20/01A75. Translated into English by P.G. Brown (date?) from a French translation of P. Tannery in 1886. 33 pages. Different methods of constructing magic squares. No mention of magic cubes.
Mathematical Sciences Magazine (Japanese) Dec. 1977, Special issue on puzzles. pp 43-45 Order 9 and 11 perfect magic cubes plus, another 11 pages with lots of magic objects (stars, circles, triangles, etc.)
Dr. C. Planck, Theory of Paths Nasik, Printed for private circulation by A. J. Lawrence, Printer, Rugby. Includes orders 8 and 9 perfect cubes and instructions for building orders 15 and 105 perfect cubes.
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. All papers are very technical. There are NO diagrams. The bound book contains:
H. A. Sayles, A Magic Cube of Six, The
Monist, XX, 1910, pp 299-303
The author describes the method used to construct an order 6 true magic cube
with special cubelets. This assay was later republished in essentially the same
form in W. S. Andrews, Magic Squares and Cubes, 2nd edition, 1917.
H. A. Sayles, Geometric Magic Squares and
Cubes, The Monist, XXIII, 1913, pp 631-640.
The author describes four different methods for constructing multiply magic
cubes (and squares). He seems to be the originator of the term geometric
as applied to magic squares and cubes. I have seen only one other published
paper on this subject (the one by Trenkler). Sayles describes the construction
of orders 3, 4, 5 and 6 magic squares, and orders 3 and 4 magic cubes. This
assay was later republished in essentially the same form in W. S. Andrews,
Magic Squares and Cubes, 2nd edition, 1917.
H. A. Sayles, General notes on the
Construction of Magic Squares and Cubes with Prime Numbers, The Monist, XXVIII, 1918, pp
141-158.
He shows several order 4 magic squares with all prime numbers. He also shows an
order 3 magic cube that contains 26 primes and 1 composite number.
Marián Trenkler, A construction of magic
hypercubes, published on Internet, (University Tohoku - 1996)
Marián Trenkler, Magic cubes, The Mathematical Gazette
82(1998), 56-61
Marián Trenkler, A construction of magic cubes, The
Mathematical Gazette, 84(2000), 36-41
Marián Trenkler, Magic p-dimensional cubes of order n=/2(mod 4),
Acta Arithmetica 92(2000), 189-194
Marián Trenkler, Magic p-dimensional cubes, Acta Arithmetica
96 (2001), 361-364
Marián Trenkler, Connections - magic squares, cubes and matchings,
in: Applications of modern mathematical methods, Ljubljana 2001
, 191-199
Marián Trenkler, Additive and Multiplicative Magic Cubes., 6th
Summer school on applications of modern mathematical methods, TU Košice 2002, 23-25
English language versions of these and many others of Trenkler's published
or unpublished papers may be downloaded from his site at
http://kosice.upjs.sk/~trenkler/papers.htm
John Worthington, A Magic Cube of Six, The
Monist, XX, 1910, pp 303-309
The author describes the method used to construct two order 6 magic cubes. One
of these cubes has magic squares on the 6 faces, the other has 6 central magic
squares. This assay was later republished in essentially the same form in W.
S. Andrews, Magic Squares and Cubes, 2nd edition, 1917.
Treasury of Folklore – Fantasies in Figures, Mathematic Mysteries and Magic, 1957, newsletter edited by Stanley J. Coleman, 11 legal size typewritten sheets. Page 10 has 3 magic cubes.
I have chosen to list papers from The Journal of Recreational Mathematic separately and in chronological order.
John R. Hendricks, The Third-Order Magic Cube Complete,
JRM 5:1:1972, pp 43-50
John R. Hendricks, The Pan-3-Agonal Magic Cube, JRM 5:1:1972, pp 51-54
John R. Hendricks, The Pan-3-Agonal Magic Cube of Order 5, JRM 5:3:1972,
pp 205-206
P.D. Warrington, Graeco-Latin Cubes, JRM 6:1, 1973, pp47-53.
John R. Hendricks, Species of Third-Order Magic Squares and Cubes, JRM
6:3,1973, pp190-192.
John R. Hendricks, Magic Tesseracts and N-Dimensional
Magic Hypercubes, JRM 6:3,1973, pp193-201
J. R. Hendricks, Magic Cubes of Odd Order, JRM 6:4, 1973, pp 268-272
Charles W. Trigg,
Eight digits on a Cubes Vertices, JRM 7:1,1974, pp49-55
John R. Hendricks, Pan-n-agonals in Hypercubes, JRM 7:2, 1974, pp95-96.
Bayard E. Wynne, Perfect Magic Cubes of Order Seven, JRM 8:4, 1975-76, pp
285-293
Ian P. Howard, Pan-diagonal Associative Magic Cubes (Letter to the
Editor), JRM 9:4, 1976, pp276-278.
Gakuho Abe, Related Magic Squares with Prime Elements,
JRM 10:2 1977-78, pp.96-97.
R. J. Lancaster, Computer Constructed Magic Cubes, JRM 10:3, 1977,
pp202-203.
W. H. Leeflang, Magic Cubes of Prime Order, JRM 11:4, 1978-79, pp 241-257
John R. Hendricks, The Perfect Magic Cube of Order-4, JRM 13:3,1980-81,
pp204-206.
John R. Hendricks, the Pan-3-agonal Magic Cube of Order 4, JRM 13:4,
1980-81, pp274-281
A. W. Johnson, Jr., An Order 4 Prime Magic Cube, JRM
18:1, 1985-86, pp 5-7
John R. Hendricks, Ten Magic Tesseracts of Order Three, JRM 18:2,
1985-86, pp125-131
William L. Schaaf,
Vestpocket Biblio. No. 12: Magic Squares & Cubes, JRM 19:2,1987, pp81-86
John R. Hendricks, A Ninth-Order Magic Cube, JRM 19:2, 1987, pp126-131
A. W. Johnson, Jr. Algebraic Forms for Order 3 Magic Squares/Cubes, JRM
19:3, 1987 pp 213-218
John R. Hendricks, Creating Pan-3-agonal Magic Cubes of Odd Order, JRM
19:4, 1987, pp280-285.
John R. Hendricks
,A Magic Cube of Order 7, JRM:20:1,1988, pp23-25
John R. Hendricks, Some Ordinary Magic Cubes of Order 5,
JRM 20:1, 1988,pp125-134.
John R. Hendricks, Magic Cubes of Odd Order by Pocket Computer, JRM 20:2,
1988, pp92-96.
John R. Hendricks, The Diagonal Rule for Magic Cubes of Odd Order, JRM
20:3, 1988, pp192-195.
R. Ondrejka, The Most Perfect (8x8x8) Magic Cube? (Letter to the Editor), JRM
20:3, 1988, pp207-209
A. W. Johnson, Jr. Normal Magic Cubes of Order 4m+2 (Letter to the
Editor), JRM 21:2, 1989, 101-103.
John R. Hendricks, An Inlaid Magic Cube, JRM 25:4,
1993, pp 286-288.
John R. Hendricks, Property of Some Pan-3-agonal Magic Cubes of Odd order,
JRM 26:2, 1994, pp 96-101.
John R. Hendricks, From Inlaid Squares to Ornate Cube, JRM 30:2,
1999-2000, pp 125-136.
H. D. Heinz & J. R. Hendricks, A Unified Classification system for Magic
Cubes, JRM 32:1, 2003-2004, pp 30-36.
H. D. Heinz , The First (?) Magic Cube, JRM 33:2, 2004-2005, pp 111-115.
H. D. Heinz , The First (?) Perfect Magic Cubes, JRM 33:2, 2004-2005, pp
116-119.
In Memoriam: John Robert Hendricks: Sept 4, 1929- July 7, 2007, JRM 34:1,
2005-2006, page 80
H. D. Heinz, Hypercube Classes - An Update, JRM 35:1, 2006, pp 5-10
H. D. Heinz & M. Nakamura, Magic Tesseract Classes, JRM 35:1, 2006, pp
11-14
Cornell University Library, Digital Collections at http://www.math.cornell.edu/~library/reformat.html
Kanji Setsuda’s Compact (composite) and Complete magic Cubes Web pages may be accessed from here. http://homepage2.nifty.com/KanjiSetsuda/pages/EnglishP1.html
Abhinav Soni's HyperMagicCube.exe program. Obtainable from soni_abhinav@yahoo.com
Marian Trenkler’s Home Page is http://kosice.upjs.sk/~trenkler/
The Eugène Strens Recreational Mathematics Collection at the University of Calgary (lucky me, only a 1 day drive away from home). It is at http://www.ucalgary.ca/library/SpecColl/strens.htm
Singmaster CD available from David Singmaster zingmast@sbu.ac.uk This CD contains many files of essays, bibliographies, chronologies, etc., to do with recreational mathematics.
Eric W. Weisstein’s http://mathworld.wolfram.com/ this is an excellent site. However, some of the cube definitions are incorrect and inconsistent.
Matsumi Suzuki’s excellent site is now available at http://mathforum.com/te/exchange/hosted/suzuki/MagicSquare.html
The first British mathematical journal
was The Cambridge Mathematical Journal, founded in 1839 by D. F.
Gregory and R. L. Ellis; a continuation of this journal was later to become
The Quarterly Journal of Pure and Applied Mathematics, which was edited for
a time by J. J. Sylvester and others. One link to try for this journal
is:
http://www.columbia.edu/cu/lweb/indiv/mathsci/findlist.html
This list was supplied by Christian Boyer.
Moon (Robert), On the theory of magic squares, cubes, etc.
Cambridge and Dublin Mathematical. Journal. 1, 1846, p.160-164
Frankenstein (Gustavus), A magic cube, Cincinnati Commercial, 11-3, 1875
Saccani (F.), Quadrati e cubi magici, Reggio d'Emilia, 1887
Planck (C.), Magic squares, cubes, etc. English Mechanic and World of
Science, 47, 1888, p.60, London
Schlegel (V.), Sur une méthode pour représenter dans le plan les cubes
magiques à n dimensions, Bulletin de la Société Mathématique de France, 20,
1892, p.97
Willis (John), Easy methods of constructing the various types of magic
squares and magic cubes, with symmetric designs founded thereon,
Bradford-London, 1909
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Harvey Heinz harveyheinz@shaw.ca
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