an introduction to ramanujan's magic squares · an introduction to ramanujan’s magic squares...

An introduction toRamanujan’s magic squares

George P. H. Styan2

January 18, 2012

2This beamer 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 Canada.

George P. H. Styan3 Ramanujan’s magic squares

Acknowledgements: January 18, 2012 B3-01a

This 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 2012.

I am very grateful to Professor Prasad and the Workshop participants whoreminded me of Ramanujan’s work on magic squares and to Dr. 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 help.This research was supported, in part, by the Natural Sciences and EngineeringResearch Council of Canada.


Srinivasa Aiyangar Ramanujan (1887–1920) B3-01b

Srinivasa 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).


Erode, Kumbakonam, Chennai B3-02a


Kumbakonam (near Thanjavur) B3-02b

Ramanujan lived most of his life in Kumbakonam, an ancient capital of theChola Empire. The dozen or so major temples dating from this period made

Kumbakonam a magnet to pilgrims from throughout South India.

Raja Raja Chola I, popularly known as Raja Raja the Great,ruled the Chola Empire between 985 and 1014 CE.


22 December = National Mathematics Day B3-03

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 honour.

On 22 December 2011, Prime Minister Dr. Manmohan Singh in Chennaideclared 22 December as National Mathematics Day, and declared

2012 as National Mathematical Year. [The Hindu, 27 December 2011.]


Gauss, Euler, Cauchy, Newton, and Archimedes B3-04

Ramanujan’s talent was said9 by the English mathematicianGodfrey Harold “G.H.” Hardy (1877–1947)

to be in the same league as that ofGauss, Euler, Cauchy, Newton, and Archimedes.

9“Srinivasa Ramanujan”, Wikipedia, 7 January 2012, p. 1.


Trinity College, Cambridge, and G. H. Hardy B3-05

From 1914–1919 Ramanujan workedwith G. H. Hardy at Trinity College, Cambridge11.

12-01-07 7:51 PMG H Hardy

Page 1 of 1

11Photographs (left panel: Ramanujan, centre) from “Srinivasa Ramanujan”, Wikipedia, 7 January 2012, p. 8.


Berndt 1985 B3-06

Ramanujan’s work on magic squaresis presented, in some detail, in

Chapter 1 (pp. 16–24) ofRamanujan’s Notebooks, Part I, byBruce C. Berndt (Springer 1985)

“The origin of Chapter 1probably is found in

Ramanujan’s early school daysand is therefore much earlier thanthe remainder of the notebooks.”


Tata Institute 1957 B3-07

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, 1957.


Ramanujan’s 3× 3 magic matrix B3-08

In Berndt 1985, Corollary 1, p. 17, we find:

In a 3× 3 magic square,the elements in the

middle row, middle column,and each [main] diagonal

are in arithmetic progression.


Ramanujan’s 3× 3 magic matrix R3 B3-09

And so we have the general form for a 3× 3 magic matrix

R3 =

h + u h − u + v h − v

h − u − v h h + u + v

h + v h + u − v h − u

= hE3 + uU3 + vV3

= h

1 1 1

1 1 1

1 1 1

+ u

1 −1 0

−1 0 1

0 1 −1

+ v

0 1 −1

−1 0 1

1 −1 0

.

.



Berndt (op. cit., p. 21) presents

R4 =

a+ p d + s c + q b + r

c + r b + q a+ s d + p

b + s c + p d + r a+ q

d + q a+ r b + p c + s

=

a d c b

c b a d

b c d a

d a b c

+

p s q r

r q s p

s p r q

q r p s

,

the sum of two orthogonal Latin squares (Graeco-Latin square).


Ramanujan’s 5× 5 magic square B3-11

Ramanujan (Tata Institute 1957, Volume II, Notebook 2, p. 12 = original p. 8)gives this 5× 5 magic square, which is also the

sum of two orthogonal Latin squares (Graeco-Latin square).


Ramanujan’s 7× 7 and 8× 8 magic squares B3-12

Berndt (op. cit.) reports two 7× 7 (p. 24) and two 8× 8 (p. 22)magic squares (but apparently no 6× 6) by Ramanujan, including

R7 =

1 49 41 33 25 17 9

18 10 2 43 42 34 26

35 27 19 11 3 44 36

45 37 29 28 20 12 4

13 5 46 38 30 22 21

23 15 14 6 47 39 31

40 32 24 16 8 7 48

, R8 =

1 62 59 8 9 54 51 16

60 7 2 61 52 15 10 53

6 57 64 3 14 49 56 11

63 4 5 58 55 12 13 50

17 46 43 24 25 38 35 32

44 23 18 45 36 31 26 37

22 41 48 19 30 33 40 27

47 20 21 42 39 28 29 34

,

and says that R8 is “constructed from four 4× 4 magic squares”.

We find that R8 may be constructed from two 4× 4 magic squares.



The classic fully-magic Nasik (pandiagonal) matrix with magic sum m(R8) = 260

R8 =

1 62 59 8 9 54 51 16

60 7 2 61 52 15 10 53

6 57 64 3 14 49 56 11

63 4 5 58 55 12 13 50

17 46 43 24 25 38 35 32

44 23 18 45 36 31 26 37

22 41 48 19 30 33 40 27

47 20 21 42 39 28 29 34

=

R(11)8 04

04 04

+

04 R(12)8

04 04

+

04 04

R(21)8 04

+

04 04

04 R(22)8

,

where R(11)8 ,R(12)

8 ,R(21)8 ,R(22)

8 are 4× 4 fully-magic Nasik (pandiagonal) matriceseach with magic sum 130 = 1

2m(R8).

Moreover, the magic matrices R(11)8 ,R(12)

8 ,R(21)8 ,R(22)

8 are interchangeable and sothere are 4! = 24 fully-magic Nasik (pandiagonal) 8× 8 matrices like R8.


Ramanujan’s 4× 4 magic submatrices R(11)8 ,R(12)

8 ,R(21)8 ,R(22)

8 B3-14a

Furthermore,

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 matrices

R(11)8 =

1 62 59 8

60 7 2 61

6 57 64 3

63 4 5 58

, X =

1 −1 −1 1

−1 1 1 −1

1 −1 −1 1

−1 1 1 −1

,

with magic sums m(R(11)8 ) = 1

2m(R8) = 130 and m(X) = 0. And so we may writeRamanujan’s 8× 8 magic matrix as the sum of two Kronecker products:

R8 =

1 62 59 8 9 54 51 16

60 7 2 61 52 15 10 53

6 57 64 3 14 49 56 11

63 4 5 58 55 12 13 50

17 46 43 24 25 38 35 32

44 23 18 45 36 31 26 37

22 41 48 19 30 33 40 27

47 20 21 42 39 28 29 34

=(1 1

1 1

)⊗ R(11)

8 + 8(0 1

2 3

)⊗ X.


Ramanujan’s magic matrix R8 and the Agrippa magic matrix A4 B3-14b

The 4× 4 magic-basis matrix

B4 =

−3 1 1 −3

1 −1 −1 1

−1 1 1 −1

3 −1 −1 3

= −DX,

where

D =

3 0 0 0

0 1 0 0

0 0 1 0

0 0 0 3

, while X =

1 −1 −1 1

−1 1 1 −1

1 −1 −1 1

−1 1 1 −1

is the doubly-balanced 4× 4 magic matrix used in our “two-Kronecker-productsconstruction” of Ramanujan’s 8× 8 magic matrix R8. The 4× 4 Agrippa–Cardanomagic matrix

A4 = 12

(4B4 − B′

4 + (42 + 1)E4)=

4 14 15 1

9 7 6 12

5 11 10 8

16 2 3 13

.


The Man Who Knew Infinity B3-15

For a“model of the biographer’s art”

we recommend

The Man who Knew Infinity:A Life of the Genius Ramanujan,

by Robert Kanigel,pub. Charles Scribner’s Sons 1991;Washington Square Press 1992



12-01-07 4:17 PMTHE MAN WHO KNEW INFINITY: A Life of the Genius Ramanujan - Robert Kanigel

Page 2 of 4http://www.robertkanigel.com/_i__b_the_man_who_knew_infinity__b___a_life_of_the_genius_ramanujan__i__58016.htm

Book-of-the-Month Club selectionQuality Paperback Book Club selectionNational Book Critics Circle Award finalist, 1992Los Angeles Times Book Prize finalist, 1991Library Journal, Best Sci-Tech Books of 1991New York Public Library, Book to Remember, 1991New York Times Book Review, Notable Books of the Year,1991Film option held by Matt Brown/Edward R. Pressman

Selected 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 (U.K.), Times Literary Supplement (U.K.), FarEastern Economic Review, American Scientist, Isis, SewaneeReview

Original 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

"Enthralling...One 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,1991

challenging philosophical ideasthat is actually a compelling read."-- 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 craftedbiography...[Kanigel's] exceptionaltriumph is in the telling of thiswonderful human story." -- Science

"Even a complete innumerate canenjoy Mr. Kanigel's richly detailedbook...Kanigel desmontratesconsiderable psychologicalacumen in his portait of the twocentral figures...A thoroughlycaptivating book." -- New YorkTimes Book Review

"An extraordinary compellingbiography, richly textured withsocial, psychological, personal,and mathematical details...Awarm, 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." -- American MathematicalMonthly

"Perspicacious, informed,imaginative, [The Man Who KnewInfinity] is to my mind the bestmathematical biography I haveever read...[It]is not just a brilliantbiography of Ramanujan, thegenius born in a hovel. Nor is it a'life and works,' part one Life, parttwo Works. It is a sensitive andintimate portrait of Ramanujan,the human being, who lived formathematics....You do not need tohave any sympathy at all withmathematics to read The Man Who

12-0

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Tim

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Scien

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onito

r, S

an F

ranc

isco

Chro

nicle

, Bos

ton

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ton

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of

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"A fa

scin

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's lif

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read

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e a

sad

rom

antic

nov

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Nob

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bio

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hy o

f a m

athe

mat

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n, in

fact

of a

nysc

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that

I ha

ve e

ver r

ead.

" -- B

ruce

Ber

ndt,

Uni

vers

ityof

Illin

ois

"Ent

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..One

of t

he b

est s

cien

tific

bio

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I've

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-- Jo

hn G

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In S

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's C

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the

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chal

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philo

soph

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idea

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a c

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read

."--

Los

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"One

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tdo

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." --

Hug

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New

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extr

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soci

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this

boo

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he b

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scie

ntifi

c bi

ogra

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s ev

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ritte

n." -

- Am

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athe

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"Per

spic

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us, i

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,im

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an W

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new

Infin

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]is n

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Times Literary Supplement

"This is the first comprehensivebiography [of Ramanujan] and islikely to remain the canonicalreference book for all aspects ofhis brief life for many years tocome...[I] can recommend itwholeheartedly to bothmathematicians and to generalreaders." -- The MathematicalGazette

"Poignant and absorbing." --

German edition, Verlag Vieweg,1993

Original Washington SquarePress paperback edition, 1992

12-0

1-07

4:1

7 PM

THE

MAN

WH

O K

NEW

INFI

NIT

Y: A

Life

of t

he G

eniu

s Ra

man

ujan

- R

ober

t Kan

igel

Page

4 o

f 4ht

tp:/

/ww

w.r

ober

tkan

igel

.com

/_i_

_b_t

he_m

an_w

ho_k

new

_inf

inity

__b_

__a_

life_

of_t

he_g

eniu

s_ra

man

ujan

__i_

_580

16.h

tm

Tim

es L

itera

ry S

uppl

emen

t

"Thi

s is

the

first

com

preh

ensi

vebi

ogra

phy

[of R

aman

ujan

] and

islik

ely

to re

mai

n th

e ca

noni

cal

refe

renc

e bo

ok fo

r all

aspe

cts

ofhi

s br

ief l

ife fo

r man

y ye

ars

toco

me.

..[I]

can

reco

mm

end

itw

hole

hear

tedl

y to

bot

hm

athe

mat

icia

ns a

nd to

gen

eral

read

ers."

-- T

he M

athe

mat

ical

Gaz

ette

"Poi

gnan

t and

abs

orbi

ng."

--

Ger

man

edi

tion,

Ver

lag

Vie

weg

,19

93 Orig

inal

Was

hing

ton

Squa

rePr

ess

pape

rbac

k ed

ition

, 199

2



1 Scribner’s hardcover, 19912 U.K. hardcover, Scribner’s, 19913 Washington Square Press paperback, 19924 U.K. paperback, Abacus, 19925 Indian edition, Rupa, 19926 German edition, Vieweg Verlag, 19937 Cassette book, National Library for the Blind, 19938 Japanese edition, Kousakusha, 19949 Korean edition, Science Books, 200010 Chinese editions, Shanghai Scientific, 2002, 200811 Italian edition, Rizzoli, 200312 Thai edition, Matichon, 200713 Audio edition, Blackstone Audio, 200714 Greek edition, Travlos, 2008


Der das Unendliche kannte; L’uomo che vide l’infinito B3-18



Times Literary Supplement

"This is the first comprehensivebiography [of Ramanujan] and islikely to remain the canonicalreference book for all aspects ofhis brief life for many years tocome...[I] can recommend itwholeheartedly to bothmathematicians and to generalreaders." -- The MathematicalGazette

"Poignant and absorbing." --

German edition, Verlag Vieweg,1993

Original Washington SquarePress paperback edition, 1992

12-0

1-07

4:1

7 PM

THE

MAN

WH

O K

NEW

INFI

NIT

Y: A

Life

of t

he G

eniu

s Ra

man

ujan

- R

ober

t Kan

igel

Page

4 o

f 4ht

tp:/

/ww

w.r

ober

tkan

igel

.com

/_i_

_b_t

he_m

an_w

ho_k

new

_inf

inity

__b_

__a_

life_

of_t

he_g

eniu

s_ra

man

ujan

__i_

_580

16.h

tm

Tim

es L

itera

ry S

uppl

emen

t

"Thi

s is

the

first

com

preh

ensi

vebi

ogra

phy

[of R

aman

ujan

] and

islik

ely

to re

mai

n th

e ca

noni

cal

refe

renc

e bo

ok fo

r all

aspe

cts

ofhi

s br

ief l

ife fo

r man

y ye

ars

toco

me.

..[I]

can

reco

mm

end

itw

hole

hear

tedl

y to

bot

hm

athe

mat

icia

ns a

nd to

gen

eral

read

ers."

-- T

he M

athe

mat

ical

Gaz

ette

"Poi

gnan

t and

abs

orbi

ng."

--

Ger

man

edi

tion,

Ver

lag

Vie

weg

,19

93 Orig

inal

Was

hing

ton

Squa

rePr

ess

pape

rbac

k ed

ition

, 199

2



Italian edition, L'uomo che vide l'infinito, Rizzoli,2003

Japanese hardcover edition, Kousakusha, 1994

mathematics to read The Man WhoKnew Infinity with pleasure; but bythe time you have finished it,some of Ramanujan's love of hissubject will probably have rubbedoff on you, and you will havebegun to appreciate the hypnoticfascination that it exerts uponthose who make it their life'swork." -- New York Review of Books

Washington Square Presspaperback, 1992

Abacus paperback edition [U.K.],1992

Review Excerpts, U.K.Edition

"This is a fine example of a workof popularising mathematics, anddeserves a wide readership." --New Scientist

A "fascinating biography...Kanigelhas written an exciting andthoughtful book." -- TheIndependent

"This is a grand read, enthrallingin every paragraph...the sort ofbook that makes you realise whata marvellous thing it is to behuman." -- Huddershield DailyExaminer

"A noteworthy performance." --Times Literary Supplement

12-0

1-07

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THE

MAN

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Prasantha Chandra Mahalanobis (1893–1972) B3-19

In The Man who Knew Infinity we learn that (Prologue, pp. 1–2):

One day in the summer of 1913, a twenty-year-old Bengalifrom an old and prosperous Calcutta family stood

in the chapel of King’s College, Cambridge, England.

Prasantha Chandra Mahalanobis (1893–1972) was smitten.


Mahalanobis (1893–1972) & Ramanujan (1887–1920) B3-20

Scarcely off the boat from India and planning to study in London,Mahalanobis had come up to Cambridge

on the train for the day to sightsee.

The next day he met with the provost, and soon,to his astonishment and delight, he was a student at

King’s College, Cambridge.

He had been at Cambridge for about six monthswhen his mathematics tutor asked him,

“Have you met your wonderful countryman Ramanujan?”

He had not yet met him, but he had heard of him.


Are you warm at night? asked Mahalanobis B3-21

Soon Mahalanobis did meet Ramanujan, and the two became friends;on Sunday mornings, after breakfast, they’d go for long walks,

talk about life, philosophy, mathematics.

Later, looking back, Mahalanobis would date the flowering of theirfriendship to one day in the fall following Ramanujan’s arrival.

He’d gone to see him at his place in Whewell’s Court.Cambridge was deserted. And cold.

Are you warm at night? asked Mahalanobis,seeing Ramanujan beside the fire.

No, replied the mathematician from always-warm Madras,he slept with his overcoat on, wrapped in a shawl.


Sarangapani Street, Kumbakonam & Whewell’s Court, Cambridge B3-22

Until early 1914 Ramanujan lived in a traditional home onSarangapani Street in Kumbakonam. The family home is now a museum.

From 1914–1919 Ramanujan lived in Whewell’s Court, a 5-minute walk fromHardy’s rooms. Whewell’s Court was a 3-story stone warren of rooms laced witharched Gothic windows and pierced at intervals by staircases leading to rooms.


Ramanujan had enough blankets B3-23

Figuring his friend hadn’t enough blankets,Mahalanobis stepped back into the little sleeping alcove

on the other side of the fireplace.

The bedspread was loose, as if Ramanujan had just gotten up.Yet the blankets lay perfectly undisturbed,

tucked neatly under the mattress.

Yes, Ramanujan had enough blankets;he just didn’t know what to do with them.

Gently, patiently, Mahalanobis showed him how you peeled them back,made a little hollow for yourself, slipped inside ...


Ramanujan: the one superlatively great mathematician B3-24

For five years, walled off from India by World War One (1914–1918),Ramanujan would remain in strange, cold, distant England, fashioning,

through 21 major papers, an enduring mathematical legacy.

Then, he would go home to India to a hero’s welcome.

“Srinivasa Ramanujan”, an Englishman would later say of him,“was a mathematician so great that his name transcends jealousies,

the one superlatively great mathematician whomIndia has produced in the last thousand years.”


an introduction to ramanujan's magic squares · an introduction to ramanujan’s magic squares...

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