Skip to main content

Abstract

The expression magic square is commonly used for any arrangement of squares, like the squares on a board for playing chess or draughts, in which the individual square cells contain numbers such that the sum of the numbers along any row, column or diagonal produces the same result. A magic square containing n rows and n columns is called an order n magic square. Mathematically, it is often useful to consider the square as a particular square matrix possessing the magic constant S n which is equal to the common sum of its rows, columns and diagonals.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

  1. Andrews, W. S., Magic Squares and Cubes, La Salle, Illinois: Open Court Publishing Co., 1908, repr. New York: Dover, 1960.

    MATH  Google Scholar 

  2. Arnauld, A., ‘Solution d’un des plus célèbres et des plus difficiles problèmes d’arithmétique appelé communément les quarrés magiques’, in Nouveaux Eléments de Géométrie, Paris: Charles Savreux, 1667.

    Google Scholar 

  3. Bachet de Méz iriac, C.-G., Problèmes plaisants et délectables qui se font par les nombres, Lyon, 1612; 2nd (augmented) ed., Lyon, 1624; 3rd ed. with additions by Labosne A., Paris: Gauthier-Villars, 1879; reprnt, with forward by J. Itard, Paris: Blanchard, 1959.

    Google Scholar 

  4. Benson, W. H. & Jacoby, O., New recreations with magic squares, New York: Dover, 1976.

    Google Scholar 

  5. Cazalas, E., Carrés magiques au degré n, séries numérales de G. Tarry, avec un aperçu historique et une bibliographie des figures magiques, Paris: Hermann, 1934.

    MATH  Google Scholar 

  6. Ch’e n Te-K’un, ‘Some Chinese Islamic magic squares porcelain’, Journal of Asian Art, Nanyang University, Singapore, Inaugural Issue (1972), 146–159.

    Google Scholar 

  7. Gardner, M., More mathematical puzzles and diversions, 1961; repub. Harmondsworth: Penguin, 1971.

    Google Scholar 

  8. Hermelink, H., ‘Zur Frühgeschichte der magischen Quadrate in Westeuropa’ [On the early history of magic squares in Western Europe], Sudhoffs Archiv, vol. 4(1981), 313–338.

    Google Scholar 

  9. Kraitchik, M., Mathematical Recreations, New York: Norton, 1942; 2nd edition Dover, 1953.

    Google Scholar 

  10. Lehmer, D. N., ‘On the congruences connected with certain magic squares’, Transactions of the American Mathematical Society, vol. 31; 2 (1929), 529–551.

    Article  MathSciNet  MATH  Google Scholar 

  11. Martzloff, J.-C, A History of Chinese Mathematics, New York: Springer Verlag, 1997, tr. Stephen S. Wilson from Histoire des mathématiques chinoises, Paris: Masson, 1987.

    Google Scholar 

  12. Molk, J., (ed.), Encyclopédie des Sciences mathématiques pures et appliquées, t. 1, vol. 3, fasc. 1, ‘figures magiques’, pp. 62-75, Paris: Gauthiers-Villars, 1906; repub. Paris: Gabay, 1992.

    Google Scholar 

  13. Morgan, G, ‘Les “neuf palais” dans les manuscrits de Touen-Houang’, in Nouvelles contributions aux études de Touen-Houang sous la direction de Michel Soymié, Geneva: Droz, 1981, pp. 251–260.

    Google Scholar 

  14. Ollerenshaw, K. & Bondi, H., ‘Magic Squares of Order Four’, Philosophical Transactions of the Royal Society of London, series A, 306 (1982), 443–532.

    MathSciNet  MATH  Google Scholar 

  15. Rosu, A., ‘Les carrés magiques indiens et l’histoire des idées en Asie’, Zeitschrift der Deutschen Morgenländischen Gesellschaft, vol 139, 1 (1989), 119–158.

    Google Scholar 

  16. Sesiano, J., ‘An Arabic Treatise on the Construction of Bordered Magic Squares’, Historia Scientiarum, 42 (1991), 13–31.

    MathSciNet  Google Scholar 

  17. Sesiano, J., ‘Herstellungsverfahren magischer Quadrate aus islamischer Zeit (I & II’)’ [The methods of constructing magic squares in Islamic times], Sudhoffs Archiv, vol. 64, 2 (1980), 187–196 and vol. 71, 1(1987), 78-89.

    MathSciNet  MATH  Google Scholar 

  18. Tannery, P., ‘Le Traité de Manuel Moschopoulos sur les carrés magiques, texte grec et traduction’, Annuaire de l’Association pour l’encouragement des études grecques en France, 1886, pp. 88-118. Mémoires Scientifiques de Paul Tannery, vol. IV, Paris: Gauthier-Villars, 1920, 27–60.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 1999 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Chabert, JL. (1999). Magic Squares. In: Chabert, JL. (eds) A History of Algorithms. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-18192-4_3

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-18192-4_3

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-63369-3

  • Online ISBN: 978-3-642-18192-4

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics