Skip to main content
SpringerLink
Log in
Book cover

Combinatorial Matrix Theory and Generalized Inverses of Matrices pp 227–252Cite as

An Illustrated Introduction to Some Old Magic Squares from India

  • Chapter

Abstract

In this article we consider old magic squares from India associated with

  1. 1.

    Daivajna Varāhamihira (505–587 AD) and his Bṛhat Saṁhitā [39]: magic perfume;

  2. 2.

    Khajuraho 945 AD: Sir Alexander Cunningham (1814–1893) [14];

  3. 3.

    Dudhai (Jhansi district) early 11th century: Harold Hargreaves (b. 1876) [27];

  4. 4.

    Ṭhakkura Pherū (fl. 1291–1323): Gaṇitasārakaumudī: The Moonlight of the Essence of Mathematics [1];

  5. 5.

    Simon de la Loubère (1642–1729): Monsieur Vincent, Surat [3,15];

  6. 6.

    Major-General Robert Shortrede (1800–1868) [16], Gwalior 1483 [11, (1842)]; Andrew Hollingworth Frost (1819–1907) [23], Nasik [17, (1877)];

  7. 7.

    Nārāyaṇa Paṇḍita (fl. 1340–1400): Gaṇita Kaumudī [2, (1356)];

  8. 8.

    Srinivasa Aiyangar Ramanujan (1887–1920) [34,35,40,43]; Prasantha Chandra Mahalanobis (1893–1972).

Magic squares were once part of occult philosophy, but more recently, however, they form part of recreational mathematics. For the past 50 years or so, they have been studied in a matrix-theoretic setting. Our main interest is in the history and philosophy of magic squares and the related magic matrices and in the related bibliography and biographies. We try to illustrate our findings as much as possible and, whenever feasible, with images of postage stamps and other philatelic items.

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

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   54.99
Price excludes VAT (USA)
  • Durable hardcover 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

Learn about institutional subscriptions

References

  1. [c. 1315/2009-Pherū] Gaṇitasārakaumudī: The Moonlight of the Essence of Mathematics by Ṭhakkura Pherū [Ṭhakkura Pherū (fl. 1291–1323)]. In: English and edited with Introduction, Translation & Mathematical Commentary, by SaKHYa [Sreeramula Rajeswara Sarma, Takanori Kusuba, Takao Hayashi and Michio Yano]. pub. Manohar, New Delhi (2009). First published in Sanskrit (early 14th century)

    Google Scholar 

  2. [1356/1942-Nārāyaṇa] Gaṇita Kaumudī by Nārāyaṇa Paṇḍita (fl. 1340–1400), edited by Padmakara Dvivedi, Princess of Wales Saraswvati Bhavana Texts, no. 57, part II, Banares, 1942: reprinted as The Ganitha Koumudi Part-2, by Narayana Pandita, Sanskrit Edition (with introduction in English by Padmakara Dvivedi), Nabu Press (2011). First published (1356)

    Google Scholar 

  3. [1693/1986-La Loubère] The Kingdom of Siam by Simon de La Loubère, with an Introduction by David K. Wyatt, pub. Oxford University Press (1986): online at Southeast Asia Visions. Reprint of A New Historical Relation of the Kingdom of Siam, Tome II: Illustrated with Sculptures, by Simon de La Loubère, pub. Tho. Horne at the Royal Exchange, Francis Saunders at the New Exchange & Tho. Bennet at the Half-Moon in St. Pauls Church-yard, London (1693)

  4. [1769-Franklin/4th-Letter27] “Letter XXVII from Benj. Franklin, Esq; of Philadelphia [Benjamin Franklin (1706–1790)]. To Peter Collinson, Esq. at London [Peter Collinson (1694–1768)]: “According to your request I now send you the Arithmetical Curiosity [A Magic Square of Squares], of which this is the history … ,” In Experiments and Observations on Electricity [5, 4th ed., pp. 350–354 (1769)]. See also [7]

    Google Scholar 

  5. [1769/2010-Franklin/4th] Experiments and Observations on Electricity, made at Philadelphia in America: To which are added, letters and papers on philosophical subjects. The whole corrected, methodized, improved, and now first collected into one volume, and illustrated with copper plates, by Benjamin Franklin, [Fourth Edition] printed for David Henry and sold by Francis Newbery, at the Corner of St. Paul’s Church-Yard, London, 496 pp (1769). Reprinted as Gale ECCO Print Edition (27 May 2010). For “A Magic Square of Squares”, see also [4], and for the 5th Edition, see [8]

    Google Scholar 

  6. [1773/2012-Dubourg/Franklin] Œuvres de M. Franklin [Benjamin Franklin (1706–1790)], by Jacques Barbeu-Dubourg [Jacques Barbeu-Dubourg (1709–1779)], two volumes, Collection Léo Pariseau, pub. L’Auteur, Paris (1773). Reprinted by Nabu Press (2012)

    Google Scholar 

  7. [1774-Franklin/5th-Letter27] “Letter XXVII from Benj. Franklin, Esq; of Philadelphia [Benjamin Franklin (1706–1790)]. To Peter Collinson, Esq. at London [Peter Collinson (1694–1768)]: According to your request I now send you the Arithmetical Curiosity [A Magic Square of Squares], of which this is the history … ,” In Experiments and Observations on Electricity [8, 5th ed., pp. 358–362 (1774)]. See also [4]

    Google Scholar 

  8. [1774-Franklin/5th] Experiments and Observations on Electricity, made at Philadelphia in America: To which are added, letters and papers on philosophical subjects. The whole corrected, methodized, improved, and now collected into one volume, and illustrated with copper plates, by Benjamin Franklin, The Fifth Edition. Printed for F. Newbery, at the Corner of St. Paul’s Church-Yard, London, 514 pp. (1774). Reprinted as Gale ECCO Print Edition (16 June 2010). For “A Magic Square of Squares”, see also [7], and for the 4th Edition, see [5]

    Google Scholar 

  9. [1815-Dalby/Hutton-Magic] “Magic square” by Charles Hutton [Charles Hutton (1737–1823)], In A Philosophical and Mathematical Dictionary [10, vol. 2, pp. 5–9, Plate XIX, Fig. 1]

    Google Scholar 

  10. [1815-Hutton] A Philosophical and Mathematical Dictionary, containing an explanation of the terms, and an account of the several subjects, comprised under the heads mathematics [sic], astronomy, and philosophy both natural and experimental, with an historical account of the rise, progress and present state of these sciences; also memoirs of the lives and writings of the most eminent authors, both ancient and modern, who by their discoveries or improvements have contributed to the advancement of them, by Charles Hutton, New Edition, Volume 2, printed for the author, London (1815): online Google eBook. See also [9]

  11. [1841/1842-Shortreede/Shortrede] “On an ancient magic square, cut in a temple at Gwalior” by “Captain Shortreede” [Robert Shortrede (1800–1868)], Journal of the Asiatic Society of Bengal, New Series, vol. 11, no. 124 (vol. 71, no. 37), pp. 292–293 (1842)

    Google Scholar 

  12. [1844/1845-Newton/magic] “Magic squares” by I. Newton [Israel Newton (1763–1856)], “Sept. 28, 1844, in the 82d year of his age”, In A New Method of Ascertaining Interest and Discount [13, pp. 15–16 (1845)]

    Google Scholar 

  13. [1845-Newton/Interest] A New Method of Ascertaining Interest and Discount, at various rates, both simple and compound, and of interest on indorsements [sic]: also a few magic squares of a singular quality by Israel Newton of Norwich, Vt. [Israel Newton (1763–1856) of Norwich, Vermont], E.P. Walton and Sons, printers, Montpelier [Vermont]. For “Magic squares”, see [12]

    Google Scholar 

  14. [1864/1871-Cunningham] Four Reports made during the years 1862–63–64–65, Volume II, by Alexander Cunningham [Sir Alexander Cunningham (1814–1893)], pub. Archæological Survey of India, Simla (1871): online Google eBook. “Khajurâho/Khajurâya”, pp. 412–439, with magic square on p. 434

  15. [1865-Frost] “Invention of magic cubes, and construction of magic squares possessing additional properties” by Rev. A.H. Frost [Andrew Hollingworth Frost (1819–1907)] (communicated by Rev. Percival Frost (1817–1898)], The Quarterly Journal of Pure and Applied Mathematics, vol. 7, no. 25, pp. 92–96 (February 1865) & vol. 7, no. 26, pp. 97–102 (June 1865): online Google eBook

  16. [1868-Shortrede/obit] Unsigned untitled obituary of Major-General Robert Shortrede (1800–1866), Monthly Notices of the Royal Astronomical Society, vol. 29, no. 1, pp. 120–121 (13 November 1868)

    Google Scholar 

  17. [1877-Frost] “On the general properties of Nasik squares” by A.H. Frost [Andrew Hollingworth Frost (1819–1907)], The Quarterly Journal of Pure and Applied Mathematics, vol. 15, no. 57, pp. 34–49 (June 1877): online Google eBook

  18. [1883-Frost/EB-9th] “Magic square”, unsigned [by Andrew Hollingworth Frost (1819–1907)], In The Encyclopædia Britannica: A Dictionary of Arts, Sciences, and General Literature, Ninth Edition, pub. Charles Scribner’s Sons, New York, vol. 15, pp. 213–216 (1883)

    Google Scholar 

  19. [1885/1987-Iyer/Varāhamihira] The Brihat Samhita of Varaha Mihira, Part II, translated into English by N. Chidambaram Iyer, Aryan Miscellany, Samhita Series, printed at the Foster Press, Madras (1885). Reprinted as The Bṛhat Sa \(\dot{m}\) hitā of Varāh Mihira, translated into English with notes by N.C. Iyer, Sri Garib Das Oriental Series No. 43, pub. Sri Satguru, Delhi (1987). See also Bhat [39]

    Google Scholar 

  20. [1888-Barnard] “Theory of magic squares and of magic cubes” by F.A.P. Barnard [Frederick Augustus Porter Barnard (1809–1889)], Memoirs of the National Academy of Sciences USA, vol. 4, Sixth Memoir, pp. 207–270 (1888)

    Google Scholar 

  21. [1904-Schilling/Kielhorn] “Mathematische Gesellschaft in Göttingen: Mitteilungen und Nachrichten”, Jahresbericht der Deutschen Mathematiker-Vereinigung, vol. 13, pp. 383–384 (1904): online at DigiZeitschriften: Das Deutsche Digitale Zeitschriftenarchiv

  22. [1905/1906-Cashmore] “XI. Chess magic squares” by M. Cashmore, M.P.S., Addresses and Papers read at the Joint Meeting of the British and South African Associations for the Advancement of Science, held in South Africa, 1905, vol. 1, pp. 83–90 (1906). See also [38]

    Google Scholar 

  23. [1907/2009-Frost/Boyer] “Andrew Hollingworth Frost [Hull 1819–Cambridge 1907], Biographical sketch” by Christian Boyer: online at multimagie.com. Includes 1870 portrait from the Church Mission Society archives

  24. [1911/2010-Frost/EB-11th] “Magic square” by A.H. Frost [Andrew Hollingworth Frost (1819–1907)], In The Encyclopædia Britannica: A Dictionary of Arts, Sciences, Literature and General Information, 11th edition, vol. 17: Lord Chamberlain to Mecklenburg, pub. Cambridge University Press, pp. 310–313 (1911). Article reset as “1911 Encyclopædia Britannica/Magic Square” online at Wikisource

    Google Scholar 

  25. [1912/1921-Strindberg] “Schackspelets Hemligheter” [Chess Game Secrets], by August Strindberg [August Strindberg (1849–1912)], En Blå Bok, by August Strindberg, Del 1–4, pp. 307–308 (c. 1912). Reprinted in Samlade Skrifter av August Strindberg, 46–48, pub. A. Bonnier, Stockholm (1918–1921): online at Project Runeberg

  26. [1915-Projective] Projective Ornament, pub. The Manas Press, Rochester NY (1915): online at Hathi Trust

  27. [1915/1916-Progress] Annual Progress Report of the Superintendent, Hindu and Buddhist monuments, Northern Circle, for the year ending 31st March 1916, by H. Hargreaves [Harold Hargreaves (b. 1876)], Superintendent, Hindu and Buddhist monuments, Northern Circle, Simla, pub. Archæological Survey of India, Lahore (18 September 1916): selected pages online at McGill

  28. [1917-Andrews] Magic Squares and Cubes by W.S. Andrews [William Symes Andrews (1847–1929)], with chapters by other writers [Charles Albert Browne, Jr., Paul Carus, Lorraine Screven Frierson, H.M. Kingery, Charles Planck, D.F. Savage, Harry A. Sayles, Frederic A. Woodruff & John Worthington], 2nd edition, revised and enlarged, pub. Open Court, Chicago (1917): online at David Moews’s home page

  29. [1917-Carus/Franklin] “An analysis of the Franklin squares” by Paul Carus, In Magic Squares and Cubes [28, pp. 96–112 (1917)]

    Google Scholar 

  30. [1917/2012-KohtzBrunner] Das Magische Quadrat und die Erfindung des Schachspiels by Johannes Kohtz [Johannes Kohtz (1843–1918)] & Erich Brunner [Erich Anselm Brunner (1885–1938)], handwritten manuscript, ca. 100 pp., section by Kohtz (dated April 1917), Sächsische Landesbibliothek – Staats- und Universitätsbibliothek Dresden (SLUB), pdf file Mscr.Dresd.App.157 (2012): online in Dresden

  31. [1918-Anderson/supermagic1] “The 34 supermagic square” by Brigadier-General F.J. Anderson, C.B. [Brigadier-General Sir Francis James Anderson, CB KBE (1860–1920)], Science Progress in the Twentieth Century: A Quarterly Review of Scientific Work, Thought & Affairs, vol. 13, no. 49, pp. 86–96 (“Popular Science” section, July 1918): online at Hathi Trust. Article reset and republished as [32, 33]

  32. [1918-Anderson/supermagic2] “Survival of the mystical mathematician: riddle of a series of numbers found in ancient India”, Current Opinion, vol. 65, no. 6, pp. 376–378 (“Science and Discovery” section, December 1918): online Google eBook. Article originally published as [31] , but this version has some introductory words though omits the last paragraphs on graphs with pretty patterns and Dürer’s Melencolia I. Article reset and republished as [33]

  33. [1919-Anderson/supermagic3] “The 34 supermagic square: mathematical recreations of ancient origin”, Scientific American Supplement, vol. 87, no. 2246, pp. 44–45 (18 January 1919): online Google eBook. Article originally published as [31, 32]

  34. [1957-Tata] Notebooks of Srinivasa Ramamujan, Volume I, Notebook 1, and Volume II, Notebook 2, pub. Tata Institute of Fundamental Research, Bombay (1957)

    Google Scholar 

  35. [1967-Ranganathan] Ramanujan: The Man and the Mathematician, by S.R. Ranganathan, pub. Asia Publishing House, Bombay (1967)

    Google Scholar 

  36. [1969-Cammann1] “Islamic and Indian magic squares, Part I” by Schuyler Cammann [Schuyler Van Rensselaer Cammann (1912–1991)], History of Religions, vol. 8, no. 3, pp. 181–209 (February 1969): online at JSTOR. For Part II, see [37]

  37. [1969-Cammann2] “Islamic and Indian magic squares, Part II” by Schuyler Cammann, History of Religions, vol. 8, no. 4, pp. 271–299 (May 1969): online at JSTOR. For Part I, see [36]

  38. [1981-Bidev/Kritik] “Bedenken gegen Anzahlbestimmung von Schach-Nasiks 8×8: Kritik der Arbeit von M. Cashmore, Capetown 1907: Chess magic squares [22]” by Pavle Bidev [Pavle Bidev (1912–1988)], typewritten manuscript [16 pp.], pub. Selbstverlag, YU-Igalo (c. 1981)

    Google Scholar 

  39. [1982/1987-Bhat] Varāhamihira’s Bṛhat Saṁhitā, Part Two by M. Ramakrishna Bhat, with English translation, exhaustive notes and literary comments, pub. Motilal Banarsidass, Delhi, 1982; 2nd revised edition (1987), reprinted (1993). See also Iyer [19]

    Google Scholar 

  40. [1985-Berndt] Ramanujan’s Notebooks, Part 1 by Bruce C. Berndt, pub. Springer, New York (1985)

    Google Scholar 

  41. [1986-Bidev/Stammt] Stammt Schach aus Alt Indien oder China? Schach-seine Urgeschichte, Schachmagische Mathematik, Schachmystische Philosophie, Part 1: German, Part 2: English “Did chess originate in China or India? Chess and magic squares” by Pavle Bidev, pub. Selbstverlag, YU-Igalo (1986)

    Google Scholar 

  42. [1987-Hayashi/HistMath] “Varāhamihira’s pandiagonal magic square of order four” by Takao Hayashi, Historia Mathematica, vol. 14, no. 2, pp. 159–166 (May 1987): online at ScienceDirect

  43. [1991-Kanigel] The Man Who Knew Infinity: A Life of the Genius Ramanujan by Robert Kanigel, pub. Charles Scribner’s Sons (1991)

    Google Scholar 

  44. [2006-BrewBooks/Franklin] Franklin’s “most magically magical” 16×16 “magic square of squares” displayed [7, p. 360 (1774)] as exhibit i061706 346 at the British Library, London, photo by Fabrizio Pivari (1 June 2006): online at BrewBooks

  45. [2007-Hayashi] “Magic squares in Indian mathematics” by Takao Hayashi, In Encyclopaedia of Classical Indian Sciences [46, pp. 222–233 (2007)]

    Google Scholar 

  46. [2007-SelinNarasimha] Encyclopaedia of Classical Indian Sciences: Natural Science, Technology and Medicine, edited by Helaine Selin & Roddam Narasimha, pub. Universities Press, Hyderabad (2007)

    Google Scholar 

  47. [2008-Hayashi/Selin] “Magic squares in Indian mathematics” by Takao Hayashi, In Encyclopedia of the History of Science, Technology, and Medicine in Non-Western Cultures [48, Part 13, pp. 1252–1259 (2008)]

    Google Scholar 

  48. [2008-Selin] Encyclopedia of the History of Science, Technology, and Medicine in Non-Western Cultures, 2nd edition, edited by Helaine Selin, pub. Springer, Berlin (2008): online at SpringerLink. For “Magic squares in Indian mathematics”, see [47]

  49. [2008-Pasles] Benjamin Franklin’s Numbers: An Unsung Mathematical Odyssey by Paul C. Pasles, pub. Princeton University Press (2008)

    Google Scholar 

  50. [2010-Styan/Pohle] “Some comments on old magic squares illustrated with postage stamps” by George P.H. Styan, invited talk presented in The Frederick V. Pohle Colloquium in the History of Mathematics, hosted by the Department of Mathematics & Computer Science at Adelphi University, Garden City NY (13 October 2010): beamer file online at McGill

  51. [2011-The Hindu] “Year-long celebration: A statement by the Ramanujan Mathematical Society on the series of activities planned in 2012 to mark the anniversary”, The Hindu (26 December 2011): online at The Hindu

  52. [2012-Manipal1] “An illustrated introduction to some magic squares from India” by George P.H. Styan, Invited talk presented at the International Workshop and Conference on Combinatorial Matrix Theory and Generalized Inverses of Matrices, Manipal University, Manipal (Karnataka), India (2–7 & 10–11 January 2012). Video (updated 17 February 2012): online at Vimeo. This talk is based, in part, on the talk [50] and then revised as the talk [56]

  53. [2012-Manipal2] “An introduction to Ramanujan’s magic squares” by George P.H. Styan, invited talk prepared for the International Conference on Combinatorial Matrix Theory and Generalized Inverses of Matrices, Manipal University, Manipal (Karnataka), India (2–7 & 10–11 January 2012): beamer file online at McGill. This follows up on the talk [52] and then combined with [52] as the talk [56]

  54. [2012-Manipal3] “An illustrated introduction to Yantra magic squares and Agrippa-type magic matrices” by George P.H. Styan, Götz Trenkler & Ka Lok Chu, In Lectures on Matrix and Graph Methods [55, pp. 159–220 (2012)]

    Google Scholar 

  55. [2012-Manipal/Lectures] Lectures on Matrix and Graph Methods, edited by Ravindra B. Bapat, Steve Kirkland, K. Manjunatha Prasad, and & Simo Puntanen, pub. Manipal University Press (2012). Lecture notes from the International Workshop and Conference on Combinatorial Matrix Theory and Generalized Inverses of Matrices, Manipal University, Manipal (Karnataka), India (2–7 & 10–11 January 2012). See also [54]

    Google Scholar 

  56. [2012-Waterloo] “An illustrated introduction to some magic squares from India” by George P.H. Styan, talk presented at the Annual Meeting of the Canadian Society for the History and Philosophy of Mathematics (CSHPM-2012), University of Waterloo, Waterloo, Ontario (27–29 May 2012): beamer pdf file (updated 19 June 2012): 44 pp. This talk is based, in part, on the talks [50, 52, 53]

    Google Scholar 

  57. [2012-G75] “An illustrated introduction to Euler and Fitting factorizations and Anderson graphs for classic magic matrices” by George P.H. Styan, Special invited talk presented (on 19 July 2012) in the Special Session to celebrate George P.H. Styan’s 75th Birthday at the International Conference on Trends and Perspectives in Linear Statistical Inference (LINSTAT-2012) & 21st International Workshop on Matrices and Statistics (IWMS-2012), Mathematical Research and Conference Center of the Institute of Mathematics of the Polish Academy of Sciences, Bedlewo (near Poznań), Poland (16–20 July 2012): handout [58]; see also [59]

    Google Scholar 

  58. [2012-G75bib] “An illustrated and annotated bibliography for Anderson graphs, Bragdon’s “magic lines” diagrams and Moran’s “sequence designs”, Report 2012-06, Dept. of Mathematics and Statistics, McGill University, 10 pp. (17 July 2012). Handout (75 references) for the talk [57], expanded as [59]

    Google Scholar 

  59. [2012-MagicBib] “An illustrated, annotated and indexed bibliography for magic squares, with special emphasis on Anderson graphs, Knight’s Tour diagrams, tessellated magic squares, and magic squares from India” by George P.H. Styan, Report 2012-07, Dept. of Mathematics and Statistics, McGill University, updated December 3, 2012. Expansion of Report 2012-06 [58]

    Google Scholar 

  60. [2012-Ramanujan/Wkp] “Srinivasa Ramanujan”: online at Wikipedia, the free encyclopedia, last modified on 3 September 2012 at 00:11

Download references

Acknowledgements

This paper is based, in part, on the invited talks [52, 53, 56] presented at the International Workshop and Conference on Combinatorial Matrix Theory and Generalized Inverses of Matrices, Manipal University, Manipal (Karnataka), India, 2–7 & 10–11 January 2012; I am very grateful to Professor K. Manjunatha Prasad and Professor T.E.S. Raghavan who reminded me of Ramanujan’s work on magic squares and to Dr. B. Chaluvaraju for drawing my attention to “Bangalore University’s old collections in the library which deal with Yantras and magic squares.” In addition, many thanks go to Miguel Angel Amela, Nicolas C. Ammerlaan, Thomas W. Ammerlaan, Pavel Chebotarev, Daniel J.H. Rosenthal, Evelyn Matheson Styan, and Götz Trenkler for their help. This article is based also, in part, on talks [50, 56] given at the Annual Meeting of the Canadian Society for the History and Philosophy of Mathematics (CSHPM), University of Waterloo, 27–29 May 2012, and in the Frederick V. Pohle Colloquium in the History of Mathematics, hosted by the Department of Mathematics & Computer Science at Adelphi University, Garden City, NY, 13 October 2010. This research was supported, in part, by the Natural Sciences and Engineering Research Council of Canada.

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, McGill University, 805 ouest rue Sherbrooke Street West, Montreal, Quebec, Canada, H3A 0B9

    George P. H. Styan

  2. Dawson College, 3040 Sherbrooke Street West, Westmount, Montreal, Quebec, Canada, H3Z 1A4

    Ka Lok Chu

Authors
  1. George P. H. Styan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ka Lok Chu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to George P. H. Styan .

Editor information

Editors and Affiliations

  1. Indian Statistical Institute, S.J.S. Sansanwal Marg 7, New Delhi, 110016, India

    Ravindra B. Bapat

  2. Hamilton Institute, National University of Ireland, Kilkenny, Maynooth, Co. Kildar, Kilkenny, Ireland

    Steve J. Kirkland

  3. Department of Statistics, Manipal University, manipal.edu, Manipal, 576104, Karnataka, India

    K. Manjunatha Prasad

  4. , School of Information Sciences, University of Tampere, Kalevantie 4,, Tampere, 33014, Finland

    Simo Puntanen

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer India

About this chapter

Cite this chapter

Styan, G.P.H., Chu, K.L. (2013). An Illustrated Introduction to Some Old Magic Squares from India. In: Bapat, R., Kirkland, S., Prasad, K., Puntanen, S. (eds) Combinatorial Matrix Theory and Generalized Inverses of Matrices. Springer, India. https://doi.org/10.1007/978-81-322-1053-5_18

Download citation

Publish with us

Policies and ethics

Search

Navigation