An introduction to Ramanujan's magic squares

1 An introduction toRamanujan s magic squaresGeorge P. H. Styan2 January 18, 20122 Thisbeamer file is for an invited talk presented as a video on Tuesday, 10 January 2012, at theInternational Workshop and Conference on Combinatorial Matrix Theory and Generalized Inverses of Matrices,Manipal University, Manipal (Karnataka), India, 2 11 January 2012. This research was supported, in part, by theNatural Sciences and Engineering Research Council of P. H. Styan3 Ramanujan smagic squaresAcknowledgements: January 18, 2012B3-01aThis beamer file is for an invited talk presented on Tuesday, 10 January 2012, atthe International Workshop and Conference on Combinatorial Matrix Theory andGeneralized Inverses of Matrices, Manipal University, Manipal (Karnataka), India,2 11 January am very grateful to Professor Prasad and the Workshop participants whoreminded me of Ramanujan s work on magic squares and to Dr.

2 B. Chaluvarajufor drawing our attention to Bangalore University s old collections in the librarywhich deal with Yantras and magic squares . In addition, many thanks go to Pavel Chebotarev and Ka Lok Chu for their research was supported, in part, by the Natural Sciences and EngineeringResearch Council of P. H. Styan4 Ramanujan s magic squaresSrinivasa Aiyangar Ramanujan (1887 1920)B3-01bSrinivasa Aiyangar Ramanujan (1887 1920)was born in Erode and lived in Kumbakonam(both then in Madras Presidency, both now in Tamil Nadu),and died in Chetput (Madras, now Chennai).George P. H. Styan5 Ramanujan s magic squaresErode, Kumbakonam, ChennaiB3-02aGeorge P. H. Styan6 Ramanujan s magic squaresKumbakonam (near Thanjavur)B3-02bRamanujan lived most of his life in Kumbakonam, an ancient capital of theChola Empire. The dozen or so major temples dating from this period madeKumbakonam a magnet to pilgrims from throughout South Raja Chola I, popularly known as Raja Raja the Great,ruled the Chola Empire between 985 and 1014 P.

3 H. Styan7 Ramanujan smagic squares22 December = National Mathematics DayB3-03 Srinivasa Aiyangar Ramanujan was born on 22 December 1887,and on 22 December 1962 and on 22 December 2011,India Post issued a postage stamp in his 22 December 2011, Prime Minister Dr. Manmohan Singh in Chennaideclared 22 December as National Mathematics Day, and declared2012 as National Mathematical Year. [The Hindu,27 December 2011.]George P. H. Styan8 Ramanujan s magic squaresGauss, Euler, Cauchy, Newton, and ArchimedesB3-04 Ramanujan s talent was said9by the English mathematicianGodfrey Harold Hardy (1877 1947)to be in the same league as that ofGauss, Euler, Cauchy, Newton, and SrinivasaRamanujan ,Wikipedia, 7 January 2012, p. P. H. Styan10 Ramanujan smagic squaresTrinity College, Cambridge, and G. H. HardyB3-05 From 1914 1919 Ramanujan workedwith G. H. Hardy at Trinity College, 7:51 PMG H HardyPage 1 of 111 Photographs(left panel: Ramanujan, centre) from Srinivasa Ramanujan ,Wikipedia, 7 January 2012, p.

4 P. H. Styan12 Ramanujan smagic squaresBerndt 1985B3-06 Ramanujan s work on magic squaresis presented, in some detail, inChapter 1 (pp. 16 24) ofRamanujan s Notebooks, Part I,byBruce C. Berndt (Springer 1985) The origin of Chapter 1probably is found inRamanujan s early school daysand is therefore much earlier thanthe remainder of the notebooks. George P. H. Styan13 Ramanujan s magic squaresTata Institute 1957B3-07 Ramanujan s work on magic squares was also presented,photographed from its original form, inNotebooks of Srinivasa Ramanujan,Volume I, Notebook 1, and Volume II, Notebook 2,pub. Tata Institute of Fundamental Research, Bombay, P. H. Styan14 Ramanujan s magic squaresRamanujan s 3 3 magic matrixB3-08In Berndt 1985, Corollary 1, p. 17, we find:In a 3 3 magic square,the elements in themiddle row, middle column,and each [main] diagonalare in arithmetic P.

5 H. Styan15 Ramanujan s magic squaresRamanujan s 3 3 magic matrixR3B3-09 And so we have the general form for a 3 3 magic matrixR3= h+uh u+vh vh u vhh+u+vh+vh+u vh u =hE3+uU3+vV3=h 1 1 11 1 11 1 1 +u 1 1 0 1 0 10 1 1 +v 0 1 1 1 0 11 1 0 ..George P. H. Styan16 Ramanujan s magic squaresRamanujan s 4 4 magic matrixR4B3-10 Berndt (op. cit., p. 21) presentsR4= a+p d+s c+q b+rc+r b+q a+s d+pb+s c+p d+r a+qd+q a+r b+p c+s = a d c bc b a db c d ad a b c + p s q rr q s ps p r qq r p s ,the sum of two orthogonal Latin squares (Graeco-Latin square).George P. H. Styan17 Ramanujan s magic squaresRamanujan s 5 5 magic squareB3-11 Ramanujan (Tata Institute 1957, Volume II, Notebook 2, p. 12 = original p. 8)gives this 5 5 magic square, which is also thesum of two orthogonal Latin squares (Graeco-Latin square).

6 George P. H. Styan18 Ramanujan s magic squaresRamanujan s 7 7 and 8 8 magic squaresB3-12 Berndt (op. cit.) reports two 7 7 (p. 24) and two 8 8 (p. 22) magic squares (but apparently no 6 6) by Ramanujan, includingR7= 1 49 41 33 25 17 918 10 2 43 42 34 2635 27 19 11 3 44 3645 37 29 28 20 12 413 5 46 38 30 22 2123 15 14 6 47 39 3140 32 24 16 8 7 48 ,R8= 1 62 59 8 9 54 51 1660 7 2 61 52 15 10 536 57 64 3 14 49 56 1163 4 5 58 55 12 13 5017 46 43 24 25 38 35 3244 23 18 45 36 31 26 3722 41 48 19 30 33 40 2747 20 21 42 39 28 29 34 ,and says thatR8is constructed from four 4 4 magic squares .We find thatR8may be constructed from two4 4 magic P. H. Styan19 Ramanujan s magic squaresRamanujan s 8 8 magic matrixR8B3-13 The classic fully- magic Nasik (pandiagonal) matrix with magic summ(R8) =260R8= 1 62 59 8 9 54 51 1660 7 2 61 52 15 10 536 57 64 3 14 49 56 1163 4 5 58 55 12 13 5017 46 43 24 25 38 35 3244 23 18 45 36 31 26 3722 41 48 19 30 33 40 2747 20 21 42 39 28 29 34 = R(11)8040404 + 04R(12)80404 + 0404R(21)804 + 040404R(22)8 ,whereR(11)8,R(12)8,R(21)8,R(22)8are 4 4 fully- magic Nasik (pandiagonal) matriceseach with magic sum 130=12m(R8).

7 Moreover, the magic matricesR(11)8,R(12)8,R(21)8,R(22)8are interchangeable and sothere are 4! =24 fully- magic Nasik (pandiagonal) 8 8 matrices P. H. Styan20 Ramanujan s magic squaresRamanujan s 4 4 magic submatricesR(11)8,R(12)8,R(21)8,R(22)8B3 -14aFurthermore,R(12)8=R(11)8+8X,R(21)8= R(11)8+16X,R(22)8=R(11)8+24X,where the fully- magic Nasik (pandiagonal) 4 4 matricesR(11)8= 1 62 59 860 7 2 616 57 64 363 4 5 58 ,X= 1 1 11 111 11 1 11 111 1 ,with magic sumsm(R(11)8) =12m(R8) =130 andm(X) = so we may writeRamanujan s 8 8 magic matrix as the sum of two Kronecker products:R8= 1 62 59 8 9 54 51 1660 7 2 61 52 15 10 536 57 64 3 14 49 56 1163 4 5 58 55 12 13 5017 46 43 24 25 38 35 3244 23 18 45 36 31 26 3722 41 48 19 30 33 40 2747 20 21 42 39 28 29 34 =(1 11 1) R(11)8+8(0 12 3) P.

8 H. Styan21 Ramanujan s magic squaresRamanujan s magic matrixR8and the Agrippa magic matrixA4B3-14bThe 4 4 magic -basis matrixB4= 311 31 1 11 111 13 1 13 = DX,whereD= 3 0 0 00 1 0 00 0 1 00 0 0 3 ,whileX= 1 1 11 111 11 1 11 111 1 is the doubly-balanced 4 4 magic matrix used in our two-Kronecker-productsconstruction of Ramanujan s 8 8 magic matrixR8. The 4 4 Agrippa Cardanomagic matrixA4=12(4B4 B 4+ (42+1)E4)= 4 14 15 19 7 6 125 11 10 816 2 3 13 .George P. H. Styan22 Ramanujan s magic squaresThe Man Who Knew InfinityB3-15 For a model of the biographer s art we recommendThe Man who Knew infinity :A Life of the Genius Ramanujan,by Robert Kanigel,pub. Charles Scribner s Sons 1991;Washington Square Press 1992 George P. H. Styan23 Ramanujan s magic squaresThe Man Who Knew InfinityB3-1612-01-07 4:17 PMTHE MAN WHO KNEW infinity : A Life of the Genius Ramanujan - Robert KanigelPage 2 of 4 Club selectionQuality Paperback Book Club selectionNational Book Critics Circle Award finalist, 1992 Los Angeles Times Book Prize finalist, 1991 Library Journal, Best Sci-Tech Books of 1991 New York Public Library, Book to Remember, 1991 New York Times Book Review, Notable Books of the Year,1991 Film option held by Matt Brown/ Edward R.

9 PressmanSelected reviews: Kirkus (starred), Publisher's Weekly, Booklist(starred), New York Times, New York Times Book Review, LosAngeles Times, Christian Science Monitor, San FranciscoChronicle, Boston Phoenix, Washington Post, New York Review ofBooks, Byte, Science, Indian Express (India), New Scientist,Virginia Quarterly Review, American Mathematical Monthly, TheIndependent ( ), Times Literary Supplement ( ), FarEastern Economic Review, American Scientist, Isis, SewaneeReviewOriginal Scribners hardcover edition, 1991"A fascinating account of Ramanujan's life which reads like asad romantic novel." -- Julius Axelrod, Nobel laureate"This is the best biography of a mathematician, in fact of anyscientist, that I have ever read." -- Bruce Berndt, Universityof Illinois" of the best scientific biographies I've everseen." -- John Gribbin, author of In Search of Schrodinger's Cat-- From the dustjacket of the Scribner's hardcover edition,1991challenging philosophical ideasthat is actually a compelling read.

10 "-- Los Angeles Times"One of the finest, bestdocumented biographies everpublished about a modernmathematician." -- MartinGardner, Raleigh News andObserver"The most luminous expressionever of two three-dimensionallives along both personal andprofessional axes. As apresentation of genius interactingwith genius, I've seen nothing tocompare with it." -- Hugh Kenner,Byte"A superbly [Kanigel's] exceptionaltriumph is in the telling of thiswonderful human story." -- Science"Even a complete innumerate canenjoy Mr. Kanigel's richly desmontratesconsiderable psychologicalacumen in his portait of the twocentral thoroughlycaptivating book." -- New YorkTimes Book Review"An extraordinary compellingbiography, richly textured withsocial, psychological, personal,and mathematical , romantic tale that wouldbe beyond the scope ofimagination were it not, after all,true. [Other] reviewers justlypraise this book as one of the bestscientific biographies everwritten.