Magic Cubes and Hypercubes - References

Marián Trenkler


  1. B.Violle:
    Traité complet des Carrés Magiques
    Bachelier, Paris 1838, 616 pp. (French) (elec.version)
  2. Theodor Hugel:
    Das Problem der magischen Systeme
    Gettschick-Verlag 1876 (elec.version)
  3. Hermann Scheffler:
    Die magischen Figuren
    Teubner, Leipzig, 1882; reprinted by Sändig, Wiesbaden, 1981.
    Part III: Die magische Würfel, pp. 88-101
  4. Hermann Schubert:
    Mathematical Essays and Recreations (translated from German)
    Open Court, 1899, 143 pp. (elec.version)
  5. William Symes Andrews:
    Magic squares and cubes
    Open Court 1908, 193 pages, 2nd edition, Dover, New York 1960, 419 pages
    [Mathematical Reviews   22 #5582]
  6. W.H.Benson, O.Jacoby:
    Magic cubes, New recreations
    Dover, New York 1981
  7. Clifford A. Pickover:
    The zen of magic squares, circles, and stars.
    An exhibition of surprising structures across dimensions

    Princeton University Press 2002, 405 p.


  1. Adami A. Kochanski: Considerationes quaedam circa Quadrata & Cubos Magicos
    Acta Eruditorum, 1686, vol. 5, pages 391-395 (Latin)
  2. Robert Moon: On the theory of magic squares, cubes, etc.
    Cambridge and Dublin Mathematical. Journal. 1, 1846, p.160-164
  3. F.Saccani: Quadrati e cubi magici, Reggio d'Emilia, 1887
  4. C.Planck: Magic squares, cubes, etc.
    English Mechanic and World of Science, 47, 1888, p.60, London
  5. V.Schlegel: Sur une méthode pour représenter dans le plan les cubes magiques n dimensions
    Bulletin de la Société Mathématique de France, 20, 1892, p.97
  6. Andrew Hollingworth Frost: Invention of magic cubes
    Quarterly Journal of Mathematics 7(1866) 92-102
  7. A.H.Fros:Supplementary Note on Magic Cubes
    Quarterly Journal of Mathematics, 8(1867), 74
  8. Gustavus Frankenstein: A big puzzle
    The Cincinnati Commercial (newspaper) March 11th, 1875
    more information (see also - Magic Reciprocal by G.Frankenstein)
  9. A.H.Frost: Description of plates 3 to 9
    Quart. J. XV.(1878), 366-368
  10. F.A.P.Barnard: Theory of magic squares and of magic cubes
    Memoirs of the National Academy of Science 4 (1888) 209-270
  11. W.S.Andrews: Magic cubes
    The Monist 16(1906), 388-414
  12. John Willis: Easy methods of constructing the various types of magic squares and magic cubes,
    with symmetric designs founded theorem

    Bradford and London: P. Lund, Humphries & Co., Ltd. Nature 81 (1909), 182-183
  13. H.M.Kingery: A Magic Cube of Six
    The Monist XIX(1909), 434-441
  14. Harry A.Sayles: A Magic Cube of Six
    The Monist XX (1910), 299-303
  15. H.A.Sayles: Geometric Magic Squares and Cubes
    The Monist 23(1913), 631-640
  16. H.A.Sayles: General notes on the Construction of Magic Squares and Cubes with Prime Numbers
    The Monist, XXVIII, 1918, 141-158
  17. Kirtland McDonald: Magic cubes which are uniform step cubes
    Univ. California Publ. Math. 2(1934), 197-215
  18. D.N. Lehmer: On the enumeration of magic cubes
    Bull. Am. Math. Soc. 40(1934), 833-837
  19. R.V.Heath: A magic cube with 6n3 cells
    Amer. Math. Monthly 50(1943), 288-291  
    [Mathematical Reviews - MR 4,210b]
  20. John Robert Hendricks: The Five and Six Dimensional Magic Hypercubes of Order 3
    Canad. Math. Bull. 5(1952), 171-189
  21. Maxey Brooke: How to make a magic tesseract
    RMM 5(1961), 40-44
  22. J.R.Hendricks: The Five and Six Dimensional Magic Hypercubes of Order 3
    Canadian Mathematical Bulletin 5(1962), 171-189
  23. J.R.Hendricks: The Pan-4-agonal Magic Tesseract
    The American Mathematical Monthly 75(1968), 384
  24. J.R.Hendricks: The third-order magic cube complete
    J. Recreational Math (JRM) 5(1972) 43-49   [MR 55 #135]
  25. Dan Raul Ionescu: Researches on antic magic squares and cubes
    Bull. Math. Roumanie 16(1972), 173-190   [MR 49 #12]
  26. J.R.Hendricks: The Pan-3-agonal Magic Cube
    JRM 5(1972), 51-52
  27. J.R.Hendricks: The Pan-3-agonal Magic Cube of Order-5
    JRM 5(1972), 205-206
  28. J.R.Hendricks: Magic Tesseracts & n-dimensional Magic Hypercubes
    JRM 6(1973), 193-201
  29. J.R.Hendricks: Magic Cubes of Odd Order
    JRM 6(1973), 268-272
  30. Joseph Arkin: The first solution of the classical Eulerian magic cube problem of order ten
    Fibonacci Quart. 11 (1973), 174-178   [MR 47 #3199]
  31. Bayard E. Wynne: Perfect magic cubes of order seven
    J. Recreational Math. 8(1975-1976), 285-293   [MR 55 #12546]
  32. P.Brooke: Perfect and pandiagonal magic hyper-cubes
    Math. Spec. 9(1976-77) 82-94
  33. Allan Adler,   Shuo-Yen Robert Li: Magic N-cubes and Prouhet sequences
    American Mathematical Monthly 84(1977), 618-627   [MR 58 #21687]
  34. K.W.H. Leeflang: Magic cubes of prime order
    J. Recreational Math. 11(1978/79), 241-257   [MR 80e:05044]
  35. Dao Qi Chen: A construction of magic hypercubes of order n in Euclidean k-dimensional space (Chinese)
    Zhejiang Daxue Xuebao 4(1979), 123-138   [MR 83i:05021]
  36. Dana Richards: Generalized magic cubes,
    Math. Mag. 53(1980), 101-105   [MR 81f:05047]
  37. J.R.Hendricks: The perfect magic cube of order 4
    JRM 13(1980/81), 204-206   [MR 82c:05025]
  38. J.R.Hendricks: The pan-3-agonal magic cube of order 4
    JRM 1 (1980/81), 274-281   [MR 83e:05031]
  39. Brian Alspach,   Kathrine Heinrich: Perfect magic cubes of order 4m
    Fibonacci Q. 19(1981), 97-106   [MR 82i:05014]
  40. J.M.H.Peters: Magic squares and matrices. II: Inverses and cubes
    Math. Gaz. 65(1981), 253-254 (1981)
  41. P.Azl'ov: An algorithm for the construction of magic squares and cubes of odd order (Bulgarian)
    Mathematics and education in mathematics,
    Proc. 10th Spring Conf. Union Bulg. Math., Sunny Beach 1981, 229-233 (1981)
  42. Cherie A.Avil,   Sid Rachlin: Magic cubes: A total experience
    Mathematical Teacher (1981), 464-472
  43. Keh Ying Lin: Third-order magic hypercubes in four-dimensional space
    Chinese J. of Mathematics 12(1984), 29-43   [MR 85c:05009]
  44. Wolfgang Hintze: Die Verwandten des Zauberwürfels (Relatives of the magic cube) (German)
    VEB Deutscher Verlag der Wissenschaften, Berlin, 1985. 143 pp.   [MR 87h:05002]
  45. Li Li: A fast way of constructing standard magic cubes of the fourth kind of order n, (Chinese)
    Neimenggu Daxue Xuebao 16(1985), 507-512   [MR 87k:05036]
  46. Ken Ying Lih: Magic cubes and hypercubes of order 3
    Discrete mathematics 58(1986), 159-166   [MR 88a:05029]
  47. Li Li: Fast ways of constructing optimum magic cubes of order 16n of the fourth kind (Chinese)
    Neimenggu Daxue Xuebao 17(1986), 195-214   [MR 87k:05046]
  48. George O'Sullivan: The 3x3x3 magic cube,
    Math. Gaz. 71(1987), 46-50
  49. J.R.Hendricks: Creating Pan-3-agonal Magic Cubes of Odd Order
    JRM 19(1987), 280-285
  50. A.R.Hanafi: Some generalizations of magic squares,
    Riazi, J. Karachi Math. Assoc. 9(1987), 9-22
  51. Martin Gardner: Magic Squares and Cubes
    Ch. 17 in Time Travel and Other Mathematical Bewilderments
    New York: W. H. Freeman 1988, 213-225
  52. Li Li: Fast constructions of optimal magic cubes of order 8n (Chinese)
    Neimenggu Daxue Xuebao 19(1988), 195-204   [MR 89m:05031]
  53. Li Li: The fast construction methods of the first class for optimal magic cubes of order 16n (Chinese)
    Adv. in Math. (Beijing) 17(1988), 385-390   [MR 89m:05027]
  54. Li Li: The fast construction methods of the first class for standard magic cubes of order 8n (Chinese)
    Adv. in Math. (Beijing) 17(1988), 319-322   [MR 89m:05031]
  55. Rong Guo Sun: The existence of higher-dimensional magic cubes (Chinese)
    Neimenggu Daxue Xuebao 19(1988), 213-223   [MR 89i:05075]
  56. Rudolf Ondrejka: Letter: The most perfect (8 x 8 x 8) magic cube?
    J. Recreational Math. 20(1988), 207-209
  57. J.R.Hendricks: Some Ordinary Magic Cubes of Order 5
    JRM 20(1988), 125-134
  58. J.R.Hendricks: The Third Order Magic Tesseract
    JRM 20(1988), 251-256
  59. J.R.Hendricks: Creating More Magic Tesseracts of Order-3
    JRM 20(1988), 279-283
  60. Joseph Arkin, David C.Arney, Bruce J.Porter: A Perfect 4-dimensional hypercube of order-7
    JRM 21(1989), 81-88
  61. J.R.Hendricks: A 5-Dimensional Magic Hypercube of Order-5
    JRM 21(1989), 245-248
  62. J.R.Hendricks: The Magic Tesseracts of Order-3 Complete
    JRM 22(1990), 215-26
  63. Lu A Ouyang: A theorem for constructing a class of advanced magic cubes (Chinese)
    Hunan Jiaoyu Xueyuan Xuebao 8(1990), 95-103   [MR 91j:05029]
  64. Xiao Song Chen: Ways of constructing advanced magic cubes of order n when (n,2·3·5)=1 (Chinese)
    Hunan Jiaoyu Xueyuan Xuebao 8(1990), 104-106   [MR 91j:05025]
  65. A.M.Lakhany: Magic cubes of odd order
    Bull. Inst. Math. Appl. 27 (1991), 117-118   [MR 92d:05038]
  66. A.M.Lakhany: Magic cubes of even order
    Bull., Inst. Math. Appl. 28(1992), 149-151   [MR 1 192 409]
  67. Lu Ouyang: Standard magic cubes and Galois fields (Chinese)
    Hunan Jiaoyu Xueyuan Xuebao 10(1992), 1-9
  68. Fu Cheng Zhu: On Nasik magic cubes and hypercubes
    Combinatorics and graph theory (Hefei, 1992), 91-94
    World Sci. Publishing, River Edge, NJ, 1993
  69. Joseph Arkin,   David Arney,   Frank Giordano,   Rickey Kolb,   Paul Smith:
    Reverse digit constructions of perfect, magic, and doubly magic cubes
    Transactions of the Tenth Army Conference on Applied Mathematics and Computing
    (West Point, NY, 1992), 91-101   [MR 1 224 060]
  70. Allan Adler: Magic cubes and the 3-adic zeta function
    Mathematical Intelligencer 14 (1992), 14-23
  71. Miklós Bóna: Sur l'énumération des cubes magiques (On the number of the magic cubes) (French)
    C.R. Acad. Sci., Paris, Sér. I 316(1993), 633-636 (1993)   [MR 93k:05010]
  72. Fucheng Zhu: On Nasik magic cubes and hypercubes
    Combinatorics and graph theory. World Scientific (Singapore) 1993, 91-94
  73. J.R. Hendricks: An Inlaid Magic Cube
    JRM 25(1993), 286-288
  74. Cheng-Xu Xu,   Zhun-Wei Lu: Pandiagonal magic squares
    Computing and combinatorics, Lecture Notes in Comput. Sci., 959,
    Springer, Berlin, 1995, 388-391   [MR 98c:05034]
  75. V.Braun,   M.Spiegel,   A.Hoh: Magic cubes. Magische Wuerfel (German)
    Junge Wissenschaft. Jugend forscht in Natur und Technik 10(1995), 14-18
  76. A.M. Lakhany: Magic hypercubes
    Bul. The Institute of Mathematics and its Applications 31(1995), 179-182   [MR 97a:05039]
  77. Allan Adler,   Lawrence Washington: p-adic L-functions and higher dimensional magic cubes
    J. Number Theory 52(1995), 179-197   [MR 96j:11147]
  78. Rong Guo Sun: The existence of classical pandiagonal magic cubes (Chinese)
    Neimenggu Daxue Xuebao Ziran Kexue 27(1996), 18-20   [MR 97b:05036]
  79. Stefan Klembara, Marián Trenkler: Magická hyperkocka (Magic hypercubes) (Slovak)
    Obzory matematiky, fyziky a informatiky 46(1996), 1-5
  80. Yihui Wen,   Hugo Sun: Note on magic squares and magic cubes on abelian groups
    J. Math. Res. Exposition 17(1997), 176-178
  81. Allan Adler: Magic N-cubes form a free matroid
    The Electronical Journal of Combinatorics 4(1997)   [MR 98d:05031]
  82. Marian Trenkler: Magic cubes
    The Mathematical Gazette 82(March 1998), 56-61
  83. Hao Chen, Chun Rong Liu: A formation rule for magic cubes (Chinese)
    J. Central China Normal Univ. Natur. Sci. 32(1998), 273-277
  84. Gui Fang Xu: Pure magic squares and pure magic cubes of order 4 (Chinese)
    Xi'an Jiaotong Daxue Xuebao 33(1999), 109-110   [MR 1 720 407]
  85. Fucheng Liao, Tohru Katayama, Kiyotsugu Takaba:
    On the construction of pandiagonal magic cubes
    Preprint, School of Informatics Kyoto University, Japan (1999)
  86. T.R.Hagedorn: On the existence of magic n-dimensional rectangles
    Discrete Math. 207(1999), 53-63
  87. J.R.Hendricks: Perfect n-dimensional magic hypercubes of order 2n
    Self-publishing 1999, 36 p,
  88. Marián Trenkler: A construction of magic cubes
    The Mathematical Gazette 84(March 2000), 36-41
  89. M.Trenkler: Magic p-dimensional cubes of order n not equiv 2(mod 4) [PDF-file, 218kB]
    Acta Arithmetica 92(2000), 189-194   [MR 2000m:11012]
  90. M.Trenkler: Konštrukcia magických p-rozmerných kociek (Slovak)
    (A construction of p-dimensional magic cubes)
    Obzory matematiky, fyziky a informatiky 2/2000(29), 19-29
  91. M.Trenkler: Magic p-dimensional cubes [PDF-file, 134kB]
    Acta Arithmetica 96(2001), 361-364   [MR 2001m:11024]
  92. Xiao Song Chen: Ways of constructing optimal magic cube of order n when (n,
    Journal of Central South University of Technology, Hunan 9(2002), 70-72
  93. M.Trenkler: Multiplicative magic squares and cubes (Slovak)
    Obzory matematiky, fyziky a informatiky 1/2002, 9-16
  94. Matthias Beck, Moshe Cohen, Jessica Cuomo, Paul Gribelyuk:
    The number of "Magic" squares, cubes, and hypercubes
    The American Mathematical Monthly. Washington, 110(Oct 2003), 707
  95. M. Ahmed, J. De Loera, R. Hemmecke: Polyhedral Cones of Magic Cubes and Squares
    Algorithm combinat. 25(2003), 25-41
    arXiv:math.CO/0201108, 2002 [PDF-file]
  96. Solomon Gartenhaus: Pandiagonal Latin and magic cubes in three and four dimensions
    arXiv:math.CO/0210275 v1 17 Oct 2002 [PDF-file]
  97. M.Trenkler: Algorithms for composing magic cubes,
    Prace Naukowe Wyzsej Szkoly Pedagogicznej w Czestochowie, Matematyka IX, Czestochowa (Poland) 2003, 103-107
  98. Christian Boyer: Les Cubes Magiques (in French),
    Pour La Science n°311, september 2003, pages 90-95
  99. Jeremiah M.Kermes, Michelle Y.Penner: Magic, Semi-Magic, and Super-Magic Cubes,
    Preprint - DePauw University [PDF-file]
  100. A.Rogers, P.Loly: The inertia tensor of a magic cube
    American Journal of Physics, 72(june 2004), 786-789
  101. J.Shen, X.Jin, C.Zhou: A color image encryption algorithm based on magic cube transformation and modular arithmetic operation,
    Lecture Notes in Computer Science, 2005, 270-280
  102. M.Trenkler: An algorithms for making magic cubes,
    The Pi Mu Epsilon Journal (USA), Vol.12, No. 2, pp.105-106, Spring 2005 [PDF-file, 63kB]

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Last revision - 04.04.2006