Bibliography

Introduction

The following bibliography consists of books, chapters from books, and articles published during the 20^th century, that deal with magic squares, cubes, stars, etc. Because it contain only material that I am personally acquainted with (except for this first section), it is obviously not complete. However, it does contain about 300 items.

For 18^th and 19^th century books on the subject see
Early Books on Magic Squares William L. Schaaf JRM:16:1:1983-84:1-6
For a list of articles published before about 1970 see
A Bibliography of Recreational Mathematics (4 volumes), published. by National Council of Teachers of Mathematics, 1978
For a list of articles published since about 1970 see
Vestpocket Bibliographies No. 12: Magic Squares and Cubes....William L. Schaaf JRM:19:2:1987:81-86

Some books on magic squares published prior to that time are
Agrippa           De Occulta Philosophia (II, 42) 1510
Bachet              Problems plaisans et delectables 1624
Prestet              Nouveaux Elemens des Mathématiques 1689
De la Loubere  Relation du Royaume de Siam 1693
Frenicle           Des Quarrez Magiques. Acad. R. des Sciences 1693 (this is a posthumous paper, not a book)
Ozonam           Récréations Mathématiques 1697 (3 volumes)
                               (May/02 This book is now available at Cornell Univ. Digital Math Library)
(Falkener, Edward, Games Ancient and Oriental and How to Play Them, Dover Publ., 1961, 0-486-20739-0)

Andrews, W. S., Magic Squares & Cubes, Open Court, 1908, 193+ pages.
The first 188 pages of edition 2 is almost exactly the same as this. Differences are:

• Pages in the two editions are identical to page 128.
• Section on ‘Math Study’ is reduced from pp 129-155 to 129-145 in edition 2.
• Section on ‘M. S. and Pythagorean’ is almost identical (pp 156-172 and 146-162.
• Section on ‘Some Curious…’ is expanded from pp 173-184 to 163-178.
• Section on ‘Notes on Various…’ is expanded slightly from pp 185-193 to 178-188 in edition 2.

 Andrews, W. S., Magic Squares & Cubes, 2^nd edition, Dover Publ. 1960, 419+ pages .
This is an unaltered reprint of the 1917 Open Court Publication of the second edition. It consists of essays by different authors, first published in The Monist from 1905 to 1916. The first 188 pages are almost identical to edition 1 published in 1908 (see above).

Arnoux, G., Arithmetique graphique – les espaces arithmetiques hypermagiques, Gauthier-Villars, 1894,175+ pages. (French). Lots of theory with methods of construction. Mention is made of a paper containing 26 handwritten pages with a perfect (new definition) magic cube. Cube Diabolique de Dix-Sept, was deposited in the Académie des Sciences, Paris, France, April 17, 1887.

  Benson, W. & Jacoby, O., New Recreations with Magic Squares, Dover Publ., 1976, 0-486-23236-0
This book is a serious attempt to bring the theory of magic squares up to date (1976). The authors present a new method of cyclically developing magic squares. They include a listing of all 880 4 by 4 magic squares. A chapter shows how to generate all 3600 5x5 pandiagonal magic squares.

 Benson, W. & Jacoby, O., Magic Cubes: New Recreations, Dover Publ., 1981, 0-486-24140-8
This book provides a valuable contribution to the literature, including an early perfect order-8 magic cube..

 Calter, Paul, Magic Squares, T. Nelson and Sons, 1977,0-8407-6546-0
A mathematical detective story with no actual connection to magic squares. but includes mathematical problems.

 Candy, Albert L., Pandiagonal Magic Squares of Prime Order, self-published, 1940
A small hard-bound book with much theory on the subject.

 Cazalas, Emile, A travers les hyperspaces magiques (Through Magic Hyperspace). Sphinx, 1936, 19 pages (French).

 Danielsson, Holgar, 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.

 Descombes, Rene, Les Carrés Magiques (Magic Squares), Vuibert, 2000, 2-7117-5261-5, 494 pages. (French)

 Farrar, Mark S., Magic Squares, self-published 1996. 72 pp plus 34 pages of appendices.
Analyzes order-3, 4 and 5. Gives lists of combinations that sum correctly. Slanted towards presenting magic squares as entertainment.

Fitting, Prof. Dr. F. , Panmagische Quadrate und magische Sternvielecke, Panmagic squares and magic stars, 1939, 70 pages. Pages 52 to 70 are on magic stars including lots of diagrams.

Fults, John Lee, Magic Squares, Open Court Publ., 1974, 0-87548-197-3
This book contains a wealth of information on all types of magic squares. It is written as a text book and includes exercises at the end of each chapter.

 Heinz, H.D. and Hendricks, J. R., Magic Square Lexicon: Illustrated. Self-published, 2000, 0-9687985-0-0.
239 terms defined, about 200 illustrations and tables, 171 captioned.

 Hendricks, John R., The Magic Square Course, Unpublished, 1991, 554 pages 8.5 " x 11" binding posts.
Written for a high school math enrichment class he conducted for 5 years.

 Hendricks, John R., A Magic cube of Order-10, Unpublished, 1998 23 pages 8.5 " x 11" flat stitched.
…With an inlaid cube of order-6 and adorned with 12 inlaid magic squares of order-6.

 Hendricks, John R., Magic Squares to Tesseract by Computer, Self-published, 1998, 0-9684700-0-9
212 pages plus covers, 8.5" x 11" spiral bound, 100+ diagrams.
Lots of theory and diagrams, new methods and computer programs. 3 appendices.

 Hendricks, John R., Inlaid Magic Squares and Cubes, Self-published, 1999, 0-9684700-1-7
206 pages plus covers, 8.5" x 11" spiral bound, 100+ diagrams.
Lots of theory and diagrams. Includes a list of 46 mathematical articles published in periodicals by the author.

 Hendricks, John R., All Third-Order Magic Tesseracts, Self-published, 1999, 0-9684700-2-5, 36 pages plus covers, 8.5" x 11" flat stitched, 60+ diagrams.
Some theory. Lots of diagrams.

 Hendricks, John R., Perfect n-Dimensional Magic Hypercubes of Order 2n, Self-published, 1999, 0-9684700-4-1. 36 pages plus covers, 8.5" x 11" flat stitched, some diagrams.
Theory with examples for a cube, tesseract and 5-D hypercube.

 Hendricks, John R., Bi-Magic Squares of Order 9, Self-published, 1999, 0-9684700-6-8. 14pp + covers, 8.5" x 11" flat-stitched..
A method of generating these squares using equations and coefficient matrices.

 Hendricks, John R., Bimagic Cube of Order 25, Self-Published 2000, 18pp plus covers, flat-stitched 8.5 x 11.
This remarkable cube is presented in equations. A short computer program (listed) displays the number in any specific location..

 Hugel, Dr. Theod., Das Problem der magischen Systeme, Neustadt a. D. H., 1876  48 pp + 12 plates.
This book is written in German. It contains a lot of mathematics. The plates show a large number of magic squares of different orders and types as well as magic cubes of orders 3, 5 and 8.
The book is available online or custom printed hardcopy from Cornell University at http://www.math.cornell.edu/~library/reformat.html

 Kelsey, Kenneth, The Cunning Caliph, Frederick Muller, 1979, 0-584-10367-0
This is one of the five books (the first one) that make up The Ultimate Book of Number Puzzles.

 Kelsey, Kenneth, The Ultimate Book of Number Puzzles, Cresset Press, 1992, 0-88029-920-7, 522 pages.
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 practise material.

Lehmann, Max Bruno, Der geometrische Aufbrau Gleichsummiger Zahlenfiguren (The Geometric Construction of Magic Figures), 1875, xvi+384 pages.
This book is mostly about magic squares, but includes discussions and examples of magic cubes and magic stars. 

Moran, Jim, The Wonders of Magic Squares, Vantage Books, 1982, 0-394-74798-4
A large format book that is simply written with little theory, but demonstrates a large variety of ways to construct magic squares. Contains a forward by Martin Gardner.

 Ollerenshaw, K. and Brée, D., Most-Perfect Pandiagonal Magic Squares, Cambridge Univ. Press, 1998, 0-905091-06-X
The methods of construction and enumeration of these special doubly-even magic squares.

Philip, Morris, The Morris Philip magic squares, 1986, vi+26 pages. 

Pickover, Clifford A., The Zen of Magic Squares, Circles and Stars, Princeton Univ. Press, 2002, 0-691-07041-5
400 + pages filed with usual and very unusual magic objects. Lots of illustrations. Written in Pickover's usual entertaining and informative style.

Scheffler, Hermann, Die Magischen Figuren  (Magic Figures), Martin S, 1968, 112 pages. (German)

Simpson, Donald C., Solving Magic Squares, 1^st Books, 2001, 0-75960-428-2, 102pp.
Various methods are shown for solving the different orders of magic squares.

Swetz, Frank J., Legacy of the Luoshu, Open Court, 2002, 0-8126-9448-1, 214pp
Discusses the order 3 and other magic squares in early China, India, Arab countries and the Western world. It includes a large bibliography of references.

Violle, Par B, Traité complet des Carrés Magiques, 1837, 1000+ pages (French). About 100 pages on magic cubes. It 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.

 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. 

Ahl, David H., Computers in Mathematics, Creative Computing Pr., 1979, 0-916688-16-X
Contains some theory and Basic language programs to generate magic squares. Pages 111-117

 Berlekamp, E., Conway, J. and Guy, R., Winning Ways vol. II, Academic Press, 1982, 01-12-091102-7
Original material on order-4 magic squares. Also shows a tesseract with magic vertices. Pages 778-783.

Czepa, A., Mathematische Spielereien (Mathematical Games), Union Deutsche, 1918,  140 pages.
(Old German script). Many magic objects in this small format book.

Dudeney, H. E., 536 Puzzles & Curious Problems, Charles Scribner's Sons, 1967, 684-71755-7
This material was first published posthumously in 1926 and 1931 Unusual problems and some theory  for magic squares, stars and other objects. pp 141 - 149 and 344 - 354.

Dudeney, H. E., Amusements in Mathematics, Dover Publ., 1958, 0-486-20473-1.  Originally published in 1917. Order 4 classes
Subtraction, multiplication, division, domino, etc. List of first prime # magic squares, etc. Pages 119-127 and 245-247.

Falkener, Edward, Games Ancient and Oriental and How to Play Them, Dover Publ., 1961, 0-486-20739-0
First published by Longmans, Green & Co. in 1892, this book contains the original text with no changes, except for corrections. A comprehensive discussion of magic squares circa 100+ years ago. Pages 267-356.

Fourrey, Emile, Recréations arithmétiques, (Arithmetical Recreations) 8^th edition, Vuibert, 2001, 2711753123, 261+ pages. (French). Originally published in 1899. It includes several magic cubes.

Gardner, Martin, 2nd Scientific American Book of Mathematical Puzzles and Diversions, Simon and Schuster, 1961, 61-12845
Diabolic hypercube (tesseract), diabolic donut, some history, pages 130-140

Gardner, Martin, Incredible Dr. Matrix, Scribners, 1967, 0-684-14669-X
Anti-magic, multiplication & division, pyramid, etc. Pages 21, 47, 211, 246

Gardner, Martin, Mathematical Carnival, Alfred Knopf, 1975, 0-394-49406-7
Hypercubes, pages 41-54. Magic Stars, pages 55-65

Gardner, Martin, Mathematical Puzzles & Diversions, Simon & Schuster, 1959, 59-9501
Chapter 2, Magic With a Matrix, pages 15-22.

Gardner, Martin, New Mathematical Diversions, Simon and Schuster, 1966, 671-20913-2
Euler's spoilers- order-10 Graeco- Latin squares, order-4 playing card magic square. Pages 162-172

Gardner, Martin, Penrose Tiles to Trapdoor Ciphers, Freeman, 1989, 0-7167-1986-X
Alphamagic, smith numbers, 3x3 properties, pages 293-305

Gardner, Martin, Scientific American Book of Mathematical Puzzles and Diversions,, Simon and Schuster, 1959, 59-9501  
Using magic squares for magic tricks, pages 15-22.

Gardner, Martin, Sixth book of Mathematical Games, Charles Scribner's Sons, 1963, 0-684-14245-7
Magic hexagons, pages 23-25. Consecutive prime s. (using #1) pages 86-87.

Gardner, Martin, Time Travel & Other Mathematical Bewilderments, Freeman Publ., 1988, 0-7167-1924-X
First published enumeration of Order-5 magic squares and information about order-8 magic cubes. Gardner refers to ‘perfect’ magic cubes. This type of cube is  now called a myers cubes (the new perfect magic cubes are a much higher class). Chapter 17 Magic Squares & Cubes. Pages 213-226.
Note that Gardner erroneously stated that all 5 x 5 magic squares with 13 in the center are associated.

Goodman, A. W., The Pleasures of Math, Macmillan, 1965, 224 pages
Chapter 3, pp 40 - 57) are on magic squares.

Heath, Royal Vale, Mathemagic, Dover Publ., 1953.
The author copywrited this material in 1933. Some unusual patterns. Pages 87-123.

Hunter, J. & Madachy, J., Mathematical Diversions, Van Nostrand, 1963,
Theory of magic squares includes a simple method to produce bimagic squares. Pages 23-34.

Kraitchik, Maurice, Mathematical Recreations, Dover Publ., 1953, 53-9354. (orig publ. W.W.Norton, 1942)
Construction methods, multi-magic, Greaco-Latin, border, order-4 theory, definitions, etc. Pages 142-192

 Langman, Harry, Play Mathematics, Hafner Publishing Co, 1962
Pages 70 to 76 are on magic squares and an order 7 pandiagonal magic cube.

 Lucas, Edouard, 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.

Ozanam, Jacques (1640-1717), Recreations in the Science and Natural Philosophy, 1697. 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

 Pickover, Clifford A., Wonder of Numbers, Oxford Univ. Press, 2001, 0-19-513342-0
Chap 101, p 233-239 and frontispiece. These few pages have some real gems.

 Rouse Ball, W. & Coxeter, H., Mathematical Recreations & Essays, 12^th Edition, Univ. of Toronto Pr., 1974, 0-8020-6138-9.
Chapter 7 is on magic squares (pages 189-221 in editions 11, 12 and 13).
This classic work was originally published in 1892. H. S. M. Coxeter brought it up to date with the 1938 publication of the 11th edition, with corrections in 1962 (39-27626), the 12th edition in 1974, and edition 13 (Dover, 0-486-25357-0) in 1987 .
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.

 Schubert, Hermann, Mathematical Essays and Recreations. Translated from German to English by Thomas J. McCormack (1899, Open Court, 1903. 143+ pages. The chapter on magic squares is on pages 39 to 64. It includes orders 4 and 5 magic cubes and other magic figures.
This book is obtainable over the Internet from Cornell University Library, Digital Collections at http://historical.library.cornell.edu/math/about.html

Schubert, Hermann, Mathematische Mussestunden (Mathematical Pastimes)), 1963 reprint of 1900 work. Sections on various subjects. pp172-176 discuss magic stars. (This section was not in the original or 1940 reprint).

Schubert, Hermann, 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.

 Schubert, Hermann, 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.

 Sperling, Walter, 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.

 Sperling, Walter, Die Grubelkiste (The Amusement Chest), Albert Muller, 1953, 162 pages. (German). He shows the same order 4 that Schubert published.

 Singmaster, David, MyCD.003, self-published 2001. A CD containing 126 of his files on Recreational mathematics including extensive bibliographies on magic squares (and other recreational mathematics subjects).

Stein, Sherman K., Mathematics: The Man-made Universe, 1963, W. H. Freeman, 63-7786
Chap. 12, Orthogonal Tables. Discussion of Greaco-Latin squares and magic squares. Pages 155-174

 Spencer, Donald D., Game Playing with Computers, Hayden, 1968, 0-8104-5103-4
Computer programs and magic square theory. Pages 23-107. Card, division, upside down, composite, prime, subtracting, etc. Pages 209-224.

 Spencer, Donald D., Game Playing with Basic, Hayden, 1977, 0-8104-5109-3
Computer programs and magic square theory. Pages 119-141.

 Spencer, Donald D., Exploring Number Theory With Microcomputers, Camelot, 1989, 0-89218-249-0
Computer programs and magic square theory, geometric, talisman, multiplying, heterosquares, prime, etc. Pages 155-180.

Weisstein, Eric W., Concise Encyclopedia of Mathematics, CRC Press, 1999, 0-8493-9640-9
A general mathematical encyclopedia containing more then 14,000 entries so has many on magic square related subjects. However, some of these terms are ambiguous or contradictory. No mention of Hendricks modern concise and comprehensive hypercube classes.

 Weisstein, Eric W., Concise Encyclopedia of Mathematics CD-ROM, CRC Press, 1999, 0-8493-1945-5
Contains all of the material in the book, plus interactive graphics and both internal and external hyperlinks.

Games & Puzzles for Elementary and Middle School Mathematics, Readings from the Arithmetic Teacher Published by National Council of Teachers of Mathematics, 1975, 0-87353-054-3. Pages 69-78, 151-156.

Readings for Enrichment in Secondary School Mathematics, Bordered Magic Squares.
Published by National Council of Teachers of Mathematics, 1988, 0-87353-252-X,. Pages 195-199.

 Treasury of Folklore – Fantasies in Figures, Mathematic Mysteries and Magic, 1965, newsletter edited by Stanley J. Coleman (Meloc?), 11 legal size typewritten sheets.

Published Papers

Abe, Gakuho, 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.

 Adler, Allen & Li, Shuo-yen, Magic Cubes and Prouhet Sequences, The American Mathematical Monthly, Vol. 84, No. 8, Oct. 1977, pp. 618-627.
They show (with quite a bit of mathematics) several methods of forming magic squares from smaller order magic cubes.

Agnew, E., Two Problems on Magic Squares, Mathematics Magazine, 44, 1971, pp13-15.

Ajose, Sunday A., Subtractive Magic Triangles, Mathematics Teacher, 76, 1983, pp 346-347.

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  myers cubes). They then site examples of pandiagonal magic cubes.

Amir-Moez, Ali R., Isomorphisms on Magic Squares, College Mathematical Journal, 14, 1983, pp 48-51.

Anderson, D. L., Magic Squares: Discovering Their History and Magic,
Mathematics Teaching in the Middle School, Vol. 6, No. 8, pp.466-473

Gabriel Arnoux, Cube Diabolique de Dix-Sept, Académie des Sciences, Paris, France, April 17, 1887.
 26 handwritten pages contain a perfect (new definition) magic cube of order 17.
This cube contains 51 planar, 6 oblique, and 96 2-segment oblique, order 17 pandiagonal  magic squares.
Thanks to Christian Boyer, who kindly photographed these pages for me (the Academy would not allow photo-copying).

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.

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.

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.)

Benjamin, A. and Yasudi, K., Magic ‘Squares’ Indeed!, American Mathematical Monthly, 1999,106, pp 152-156.

Bona, Miklos, Sur l'enumeration des cubes magiques, 1993, 316, 633-636.

Caldwell, Janet, Magic Triangles, Mathematics Teacher, Vol. 71, No. 1, 1978, pp. 39-42

Carmony, Lowell A., A Minimathematical Problem: The Magic Triangles of Yates,
Mathematics Teacher, Vol.70, No. 5, 1977, pp. 410-413.

Chen, Yung C.  & Fu, Chin-Mei,  Construction and Enumeration of Pandiagonal magic squares of Order n from Step Method,  ARS Combinatoria 48(1998) pp. 233-244.

Cohen, Martin & Bernard, John, From Magic Squares to Vector Spaces,
Mathematics Teacher, Vol. 75, No. 1, Mar. 1982, pp. 76, 77 and 64.

Euler, Leonard, De Quadratis Magicis, Written in Latin , presented Oct. 17, 1776  to St. Petersburg Academy
This is available in English at http://front.math.ucdavis.edu/   and search for 0408230

Fellows, Ralph, Three Impossible Magic Squares,
Mathematical Spectrum, Vol. 23 No. 2, 2000/01, pp. 28-33.

Frost, Rev. A. H., 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.

Frost, Rev. A. H., Supplementary Note on Magic Cubes. Quarterly Journal of Mathematics, 8, 1867, p 74

Frost, Rev. A. H.,  On the General Properties of Nasik Squares, QJM 15, 1878, pp 34-49
Construction of pandiagonal magic squares.

Frost, Rev. A. H.,  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.

Frost, Rev. A. H.,  Description of Plates 3 to 9, QJM 15, 1878, pp 366-368 plus plates 3 to 9.
Illustrations of a group of 7 interrelated order 7 cubes.
Gerdas, On Lunda Designs and Associated Magic Squares, College Mathematical Journal, 2000, 31, 182-188.

Heath, R. V.,A Magic Cube With 6n³ cells, American Mathematical Monthly, Vol. 50, 1943, pp 288-291.

Hendricks, John R., The Five and Six Dimensional Magic Hypercubes of Order 3, Canadian Mathematical Bulletin, Vol. 5, No. 2, 1962,pp. 171-189.

Hendricks, John R., The Pan-4-agonal Magic Tesseract, The American Mathematical Monthly, Vol. 75, No. 4, April 1968, p. 384.

Henrich, C.J., Magic Squares and Linear Algebra, American Mathematical Monthly, 98, 1991, pp 481-488.

House, Peggy A., More Mathemagic From a Triangle, Mathematics Teacher, 73, 1980, pp 191-195.

Karpenko, Vladimír, Between Magic And Science: Numerical Magic Squares,
Ambix, Vol. 40, No. 3, 1993, pp. 121-128 (Charles Univ., Czech Republic)

Karpenko, Vladimír, Two Thousand Years of numerical magic squares, Endeavour, New Series, 18,4, 1994, pp147-153. No mention of magic cubes.
These 3 papers by Dr. Karpenko are well researched. They contain extensive references, illustrations and photos.

Karpenko, Vladimír, Magic Squares: Numbers and Letters, Cauda Pavonis (Univ. of Washington), Vol. 20, No. 1, 2001, pp 11-19.

Kenney, M., An Art-ful Application Using Magic Squares, Mathematics Teacher, Vol. ,75, No. 1, 1982, pp. 83-89

Lancaster, Ronald J., Magic Square Matrices, Mathematics Teacher, 72, 1979, pp 30-32.

F. Liao, T. Katayama, 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
They define Pandiagonal Magic Cubes as all orthogonal and diagonal arrays are pandiagonal magic squares and show 2 order 13 cubes as an example. 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.

Lyon, Betty Clayton, Using Magic Borders to Generate Magic Squares, Mathematics Teacher, Vol. 77, No. 3, Mar. 1984, pp. 223-226.

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.

McClintock, Emory, On the Most Perfect Forms of Magic Squares,  American Journal of Mathematics, 1897, 19, pp 99-120. (Read before the AMS Apr. 25, 1896).  Early research on most-perfect magic squares.

McInnes, S.W., Magic Circles, American Mathematical Monthly, 60, 1953, pp347-350.

Pasles, Paul C., The Lost Squares of Dr. Franklin, The American Mathematical Monthly, Vol. 108, No. 6, June-July 2001, pp. 489-511.
This well researched paper cites 49 sources and includes a fantastic order 16 pandiagonal magic square.

Planck, C., The Theory of Path Nasiks, Printed for private circulation by A. J. Lawrence, Printer, Rugby, 1905

Reiter, Harold B., Problem Solving With Magic Rectangles, Mathematics Teacher, Vol. 79, No. 4, Apr., 1986, pp. 242-245.

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. All papers are very technical. There are NO diagrams. The bound book contains:

• On the Transformation Group for Diabolic Magic Squares of Order Four, Reprinted from Bulletin of the American Mathematical Society, June 1938.
• The Algebraic Theory of Diabolic Magic Squares, Reprinted from Duke Mathematical Journal Vol. 5, No. 4, Dec. 1939, pp 705-728.
  Section 6 (pp 727-728) has been crossed out.
• A continuation of The Algebraic Theory of Diabolic Magic Squares on typewritten pages numbered 729 – 753. (This starts with a new section 6, renamed from ‘conclusion’ to ‘Latin Squares’). Pages 736 to 753 is about diabolic (perfect) magic cubes and points out there are 3m diabolic magic squares parallel to the faces of the cubes and 6m diabolic magic squares parallel to each of the diagonal planes.
• Magic Squares, Supplement, on typewritten pages numbered –2 to 27. Three of the 9 sections are on the subject of magic cubes.

Schwartzman, Steven, Multiplicative Squares: Magic and Special, Mathematics Teacher, Vol. 80, No. 1, Jan. 1987, pp. 51-54.

Sallows, Lee, The Lost Theorem, (about the 3x3 square of squares),The Mathematical Intelligencer, Vol. 19, No. 4, 1997, pp. 51-54

Sallows, Lee, Three Impossible Magic Squares, Mathematical Spectrum, 33, 2000, pp28-33.

Sayles, Harry A., A Magic cube of Order Six, The Monist, 20, 1910, 299-303.

Sayles, Harry A., Geometric Magic Squares and Cubes, The Monist, 23, 1913, 631-640.

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.

Schroeppel, Richard, Appendix 5: The Order 5 Magic Squares, 1973, 1-16
This is a report of Schroeppels work with enumeration of order 5 magic squares , writeup by M. Beeler, mentioned by Gardner in his Scientific American column Jan. 1976.

Schwartzman, S., Multiplicative Squares, Magic and Special, Mathematics Teacher,80. 1987, pp 51-54.

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.

Sesiano, Dr. Jacques, Islamic Magic Square History, From the author 2002, pp 1-9.

Swetz, Frank, Mysticism and Magic in the Number Squares of Old China, Mathematics Teacher, Vol. 71, No. 1, Jan. 1978, pp. 50-56

Swetz, Frank, If the Squares don't Get You - The Circles Will, Mathematics Teacher, Vol. 73, No. 1, Jan. 1980, pp. 67-72

Trenkler, D & Trenkler, G., Magic squares, Melancholy and the Moore-Penrose Inverse, Image, 27, 2001, pp3-10.

Trenkler, Marián, Characterization of Magic Graphs,
Czechoslovak Mathematical Journal, 1983, Vol. 33 ,No.108, pp.435-438 (printed in English)

Trenkler, Marián, Magic Cubes, The Mathematical Gazette, 82, (March, 1998), 56-61.

Trenkler, Marián, Magic Rectangles, The Mathematical Gazette, 83, 1999, 102-105.

Trenkler, Marián, A Construction of Magic Cubes, The Mathematical Gazette, 84, (March, 2000), 36-41.

Trenkler, Marián, Magic p-dimensional Cubes of order n not congruent to 2 (mod 4), Acta Arithmetica (Poland), 92(2000), 189-194.

Trenkler, Marián, Connections - Magic Squares, Cubes and Matchings, Applications of Modern Mathematical Methods, Univ. of Ljubljana, Slovenia, 2001, pp. 191-199

Trenkler, Marián, Additive and Multiplicative Magic Cubes., 6th Summer school on applications of modern math. methods, TU Košice 2002, 23-25.

Trenkler, Marián, Magic Stars, The IIME Journal, vol. 11, No. 10, Spring 2004, pp549-554.

Widdis, D. B., It’s Magic! Multiplication Theorems for Magic Squares, College Mathematical Journal, 20, 1989, 301-306.

Worthington, John, A Magic Cube of Six, The Monist,20, 1910, pp 303-309.

The following Articles on Magic Squares appear in Recreational Mathematics Magazine

I show these as Title, Author, RMM:issue #:date:pages(s)

The following Articles on Magic Squares appear in Journal of Recreational Mathematics

I show these as Title, Author, JRM:volume #:issue #:date:pages(s)

Literature On Magic Stars

In contrast to the voluminous literature for magic squares spanning 100's of years, there has been very little published on magic stars. The main sources of information I have been able to locate are:

H.E.Dudeney, 536 Puzzles & Curious Problems, Scribner's 1967. Pages 145-147 and 347-352.

Prof. Dr. F. Fitting, Panmagische Quadrate und magische Sternvielecke, Panmagic squares and magic stars, 1939, 70 pages. Pages 52 to 70 are on magic stars including lots of diagrams.

Martin Gardner, Mathematical Recreations column of Scientific American, Dec. 1965, reprinted with addendum in Martin Gardner, Mathematical Carnival, Alfred A. Knoff, 1975. Mostly on order 6, but mention made of total basic solutions for orders 7 & 8 (also corrected number for order-6).

Martin Gardner, Some New Results on Magic Hexagrams, College Mathematical Journal, 31, 4, Sept. 2000. pp 274-280.

Max Bruno Lehmann, Der geometrische Aufbrau Gleichsummiger Zahlenfiguren (The Geometric Construction of Magic Figures), 1875, xvi+384 pages.
This book is mostly about magic squares, but includes discussions and examples of magic cubes and magic stars.

 Reiter, Harold B., A Magic Pentagram, Mathematics Teacher, Vol. 76, No. 3, Apr., 1983, pp.174-177.

Reiter, H. B. and Ritchie, D., A Complete Solution to the Magic Hexagram, College Mathematical Journal, 20, 1989, pp 307-316.

Schubert, Hermann, Mathematische Mussestunden (Mathematical Pastimes)), 1963 reprint of 1900 work. Sections on various subjects. pp172-176 discuss magic stars. (This section was not in the original or 1940 reprint).

Marián Trenkler of Safarik University, Kosice, Slovakia published a paper on Magic Stars.
It is called "Magicke hviezdy" (Magic stars) and appeared in Obzory matematiky, fyziky a informatiky, 51(1998), pages 1-7. (my copy is an English translation)
(Obzory = horizons (or line of sight) of mathematics, physics and informatics.
This material is the subject of my Trenkler Stars <trenkler.htm> page.
A modified version of this paper was subsequently published as
Magic Stars, The IIME Journal, vol. 11, No. 10, Spring 2004, pp549-554.

 Trigg, Charles W., Multiplicative Magic Pentagram, School Science &  Mathematics, ?, pp 673-674

As mentioned at the top of the page, the above lists represent only books or articles that are in my library or that I have actually seen (except for the pre 1900 books mentioned at the start). For this reason, this bibliography should not be considered complete.

This Web site has 28 pages (May 2002) on magic squares. They start at Magicsquare.htm
This Web site has 19 pages (May 2002) on magic stars. They start at Magicstar.htm

For other Internet sites dealing with Magic Squares (and related subjects) see my links page.

Please send me Feedback about my Web site!

Last updated November 20, 2009
Harvey Heinz harveyheinz@shaw.ca
Copyright © 2000, 2001 by Harvey D. Heinz