Asia Pacific Mathematics Newsletter

Shrikhande, “Euler’s Spoiler”,
Turns 100
Nithyanand Rao

Portrait by Mohan R (Courtesy of The Shrikhande Family)Relatives, well-wishers and dignitaries kept
arriving to greet Professor Shrikhande. Seatedon the lawns, he would adjust his hearing aid—
trying to hear over the firecrackers in the background—thank them and smile, and now and then burst into ahearty chuckle, trying not to look in the direction of
the intense light drenching the table.
Sharadchandra Shankar Shrikhande, celebrating his
100th birthday on 19 October 2017, wasn’t too keen toremain in the spotlight. The bright light on the pole
was turned away, but visitors kept coming to greet him
and seek his blessings, some aware of his great
mathematical achievements—in particular, the one that

ensured his name would be associated with Euler, oneof the greatest mathematicians in history. It was 58 years
ago that Shrikhande, along with his mentor R.C. Boseand their collaborator E.T. Parker, proved Euler wrong
and made the headlines.
Late in his life, the legendary Swiss mathematician
Leonhard Euler (1707–1783) began a long paper [1]pondering a puzzle he couldn’t find an answer to.
Although he was almost completely blind by then, hisalready prodigious productivity had increased,
distractions having been reduced. He had always madethe most of his phenomenal memory and ability to

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calculate in his head and, after his loss of vision, he useda scribe to record his discoveries. The puzzle he wasconsidering was this: Imagine that there are 36 officers
belonging to six different military regiments, eachregiment having six officers of different ranks. How
does one arrange them in the form of a square suchthat each row and column has six officers, and no rankor regiment appears more than once in a row orcolumn?

Euler (Emanuel Handmann via Wikimedia Commons)Mathematicians, professional or otherwise, like toponder such puzzles for recreation, and not a few
mathematical gems have emerged from such playing
around. In many such puzzles, a simplification helps
clarify the picture. Consider nine officers instead of 36,
belonging to three different regiments, each regiment
having one officer of each of the three ranks. The
problem, as before, is to arrange them in a square suchthat no rank or regiment is repeated in a row or column.
This is actually two problems in one. One can arrange
the nine officers such that no regiment—labelled 1, 2
or 3 below—is repeated in a row or column. Such anarrangement is called a Latin square (since Euler
originally used Latin letters instead of numbers).

One can also arrange the officers such that norank—red, green or blue below—is repeated in a row
or column to get another Latin square.

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How does one satisfy both the requirements—
neither rank nor regiment repeated in a row or column?
If we combine, or superpose, the two squares, we getanother square as below, where the numbers are now
coloured.

In the square above, both requirements are satisfied.
And there is only one red 1 and only one green 2; only
once does each such combination of rank and regimentappear in the square. Two such superposed Latin
squares, therefore, are called orthogonal to each other.
Orthogonal Latin squares were, in fact, aroundmuch longer before Euler. Amulets bearing Latin
squares were known in medieval Islam, circa 1200 [2].
Choi Seok-Jeong (1646–1715), a Korean government
official and mathematician, used orthogonal Latin
squares of order 9 to construct magic squares—one inwhich the entries in any of the rows or columns or
diagonals add up to the same number. (Euler himself
was interested in magic squares.)
The pair of Latin squares in the above example is oforder n = 3. Euler observed that a similar exercise canbe done for orders 4 and 5, and also whenever n is an
odd number or is divisible by four. Now, the 36 officers
problem, in the language of Latin squares, is to find apair of orthogonal Latin squares of order 6. But Euler
found himself unable to do it for n = 6. He couldn’t findan arrangement to solve the puzzle, and concluded that
one didn’t exist at all (although his proof wasn’t fully
correct). He went on to conjecture that no solutions
existed for a number that, like six, left a remainder of
two when divided by four. These are numbers such as
10,14,18, 22, and so on, which are called “oddly even”
numbers. (There are no orthogonal Latin squares oforder 2 as well.)Asia Pacific Mathematics Newsletter

This conjecture by Euler was made in 1782. In 1901,
a French mathematician named Gaston Tarry (1843–
1913) proved that n = 6 was indeed impossible bylaboriously checking all possible cases. But Euler’s
conjecture that orthogonality was impossible for all
oddly even numbers remained to be resolved. Until
1959, when R.C. Bose, Shrikhande and E.T. Parker
disproved the conjecture.

(Courtesy of The Shrikhande Family)“The suddenness with which complete successcame,” Bose recalled later [3], “in a problem which had
baffled mathematicians for over one and three-quarters
centuries startled the authors as much as anyone else.
What makes this even more surprising is that theconcepts employed were not even close to the frontiersof deep modern mathematics.”
The result was announced in the annual meeting of
the American Mathematical Society held in New Yorkin April 1959. “Bose, Ernie (Parker) and I,” Shrikhande
said later [4], “had the rare privilege of seeing our works
reported on the front page of the Sunday Edition of the
New York Times of April 26, 1959.”
The three of them, noted the NYT report [5], “arenow known among their colleagues as ‘Euler’s spoilers’”.

Shrikhande and Bose were, at the time, at the
University of North Carolina, Chapel Hill, in the
Department of Mathematical Statistics, which had beenfounded in 1946 by the statistician Harold Hotelling(1895–1973). Bose had been a faculty there since 1949,
and Shrikhande, who had joined for a PhD in 1947,
had been Bose’s first PhD student. But his path hadn’t
been straightforward. Born on 19 October 1917, thefifth of ten siblings, in Sagar, now in Madhya Pradesh,
his family faced severe financial difficulties. Despite

that, Shrikhande’s father, who worked at a flour mill,
was determined to educate his children. Shrikhande
did well academically and won scholarships that helped
him complete his BSc Honours at the Government
College of Science (now known as the Institute of
Science) in Nagpur with a first rank and a gold medal.
“At this point of time,” he wrote in an autobiographicalessay [6], “I was badly in need of a gainful employment,
which however was not available.” Unable to find a job,
he instead made his way, in January 1940, to Kolkata
where financial assistance was available for him to join
the Indian Statistical Institute (ISI). Founded by P.C.
Mahalanobis (1893–1972), ISI at that time “was locatedin four of five rooms in the Presidency College” where
Mahalanobis was a professor of physics. But thoserooms housed a hotbed of young talent in statistics
many of whom would go on to make major contributions.
One of them was Raj Chandra Bose (1901–1987),
whom Shrikhande describes as “the major architectof my research involvement in combinatorics”, the fieldof mathematics that studies how to combine different
things, whether certain combinations are possible atall, and what combination is most appropriate given
certain constraints.
After a year at ISI, Shrikhande had a short-lived
appointment at the college in Jabalpur where he haddone his intermediate studies (the equivalent of two
years after Class 10 today) and then joined as a lecturer
in mathematics at the Government College of Sciencein Nagpur. But he kept making trips to Calcutta in theearly 1940s to work with Bose, who introduced him to
the theory of statistical design of experiments, a field
where Latin squares proved to be more than amusement.
In the 1920s, R.A. Fisher (1890–1962), the pioneering
British statistician and biologist, had applied them tothe design of field experiments in agriculture. His ideas
were later developed by Bose.
Suppose one wishes to test the effects of threedifferent agricultural fertilizers on the growth of a crop.
Now, it is likely that the fertility of soil in different
patches of a large plot of land varies naturally, perhaps
because the amount of moisture in the soil varies. How
does one test how good the fertilizers are whileaccounting for this natural variation in fertility of thesoil? Fisher’s solution to this problem was to divide thefield into cells of a three-by-three square and apply thethree fertilizers in the pattern of a Latin square of order
3 (such as the one shown previously). Because of thispattern, one can use statistical analysis to eliminate thebias due to variation in natural soil fertility along a row
or column of the field.

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Now, if instead of one variety of crop, what if we
had three? The solution to accommodate this variable
is to use a Latin square of order 3 orthogonal to theearlier one. The first Latin square tells us where thethree different fertilizers are to be applied and thesecond Latin square tells us where the three different
varieties of the crop are to be planted. The orthogonal
Latin square obtained by combining them would show
the combination of crop and fertilizer to be applied ineach cell.

Bose showed that a complete
set of mutually orthogonal
Latin squares is essentiallythe same as a finite projectiveplane
In fact, one can accommodate more variables bysuperposing additional orthogonal Latin squares. Thisis useful in the design of experiments wherever there
are many variables involved, such as in medical researchwhere one may need to test a medical treatment for its
efficacy on patients of different age, weight, stages of adisease and so on, while needing to eliminate any effects
due to such unavoidable variations. For a given ordern, however, there can be a maximum of n − 1 mutuallyorthogonal Latin squares—“mutual” because any pairof them will be orthogonal. But it’s not necessary thatthe entire set of n −1 mutually orthogonal Latin squares
exist. If they do exist, they are said to form a complete
set of mutually orthogonal Latin squares. Such acomplete set can be constructed if n is a prime
number—divisible only by one and itself, such as 3, 5,
7, 11 and so on—or a power of a prime number. In
1938, Bose showed [7] how to construct a complete set

R.C. Bose(Mac Tutor History of Mathematics Archive,
Uni­versity of St Andrews)

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of mutually orthogonal Latin squares in such a case, aresult that, unknown to him, had been discovered
earlier in 1896 by E.H. Moore (1862–1932), andindependently by W.L. Stevens in 1939 [2]. In this work,
he was influenced by F.W. Levi, a German Jewish
mathematician who, having fled the Nazis, became thehead of the Department of Mathematics at the
University of Calcutta. (Levi had shown that an
attempted proof of Euler’s conjecture by H.F. MacNeish,an American mathematician, was wrong. Others had
made unsuccessful attempts earlier as well.) Bose also
showed that a complete set of mutually orthogonal
Latin squares is essentially the same as a finite projective
plane, a concept from geometry.

B o se’s ar r ival at No r th
Carolina “proved a turning
point in my academic career,”
wrote Shrikhande
For mathematicians, there are few things more
beautiful than such surprising connections between
disparate areas of mathematics. What’s more, projective
geometry too, like Euler’s conjecture, had what would
seem an unlikely provenance—Renaissance artists.
They were interested in perspective, in representing
three-dimensional objects and scenes on the(Courtesy of The Shrikhande Family)Asia Pacific Mathematics Newsletter

two-dimensional surfaces of their paintings. To draw,
for example, square-shaped tiles on a floor, they would
not paint them all to have the same size with their sides
parallel to each other—this would look like a view fromabove. Instead, two of the opposing sides of a tile would
be drawn as if they were approaching each other, withthe tiles further away drawn smaller.
This effect can be readily seen from an image ofrailway tracks—the parallel tracks, as they recede,
appear to meet at the horizon. Projective geometry
takes this idea seriously and imagines a point at“infinity” where a family of parallel lines “meet”. Or to
put it differently, a point at infinity can be thought ofas a point which lies on each member of a family ofparallel lines. This trick ensures that any two lines ona plane, parallel or not, will meet. Different families ofparallel lines meet at different points at infinity, andthese points would make up a line at infinity. Such aline at infinity, along with an ordinary (finite) plane,makes up a finite projective plane. (Mathematically, a
finite projective plane has the property that lines andpoints in it are dual to each other—one can exchange
their roles, and the plane and its properties would still
remain the same. Any two lines, for example, intersect
at only one point, and any two points lie on only one
line.)
If there exists a finite projective plane of order n,which is a prime or a prime power, Bose showed, thata complete set of mutually orthogonal Latin squares
exists for order n. Whether this condition is necessary—that is, is there an n which is not a prime or a prime
power for which one can find a complete set of mutuallyorthogonal Latin squares—is unknown. (This isequivalent to the question of whether a projective planeof non-prime power order exists.) We do know [2] that
there exists no complete set of mutually orthogonal
Latin squares for order 10, but since 10 is not a prime
or a prime power, the problem remains open.
Bose had become famous for his work on block
designs by the time he left Calcutta for North Carolina.
Shrikhande was already there, having been selected, in
1947, for a government scholarship programme thatsent Indians abroad for higher studies. He was laterjoined by his wife, Shakuntala, whose heart he had won
over a game of badminton in Nagpur, and theirchildren. Bose too arrived, but then left, only to return
as permanent faculty in 1949. “His arrival there proved
a turning point in my academic career,” wrote
Shrikhande [6]. “Under his inspiring guidance I wasfortunate to complete my [PhD] thesis in about a year.”
Shrikhande found time to interact with and help his

fellow students too. Another student of Bose at the time,
K.A. Bush, recalls that a particular affinity developed
between him and Shrikhande. “I turned to him forsupport,” wrote Bush in an essay commemorating
Shrikhande’s work [8], “because I instinctively
recognized his compassionate nature almost from our
first meeting.” (There were other Indians too in the
Department who would go on to make a name forthemselves. R.R. Bahadur (1924–1997) and Gopinath
Kallianpur (1925–2015), who would later become
Director of ISI, were also students. S.N. Roy (1906–
1964) joined later as faculty.)
Shrikhande returned to India in 1953, to his
position at Nagpur, till 1958 when he resigned andaccepted a visiting position at Chapel Hill. “My two
years stay there till 1960 was the most productive periodof my research career,” he would write later [6].

M

athematicians, when unable to make progress
on a particular problem, sometimes find it useful
to consider related problems. For Euler’s conjecture,
the first chink appeared via a related conjecture. H F
MacNeish gave a lower bound [9] to the number ofmutually orthogonal Latin squares of any order n: it
would be one less than the smallest prime number thatappears when n is written out as a product of its prime
number factors (called its prime power decomposition).
Any number can be written as the product of prime
numbers. The number 22, for example, is 2 × 11, bothprime numbers. For 22, MacNeish guaranteed, theminimum number of mutually orthogonal Latin
squares was one, meaning that there’s at least one Latin
square of this order. He then conjectured that this lower
bound is the same as the upper bound. Thus, if
MacNeish’s conjecture (later worked on by H.B. Mann)
was true, then the maximum number of mutuallyorthogonal Latin squares of order 22 (which leaves
remainder two when divided by four), would have been
one—there would be only one Latin square of thatorder, implying that Euler’s conjecture would be true.
Proving MacNeish’s conjecture would mean proving
Euler’s conjecture, but disproving MacNeish’s conjecture
does not necessarily disprove Euler’s conjecture.
Parker, nevertheless, saw an opening. He was, at thetime, working at Remington Rand Univac, an earlycomputer company. His work on this problem, his sister
Edythe Parker Woodruff later wrote, “was done entirely
outside of company work; it was his recreation. He did
it out of his pure love for the subject.” [10]. Parker found[11] four mutually orthogonal Latin squares of order

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Asia Pacific Mathematics Newsletter

Shrikhande and Bose at work (Courtesy of The Shrikhande Family)

21 (we now know there are five [2]), whereas there
should have been only two according to MacNeish’s
conjecture.
He had used a method used to design statisticalexperiments called balanced incomplete block designs,
which Bose had contributed greatly to. Suppose, in theearlier example of the testing of different fertilizers ona crop, we divide a plot into three columns because wehave reason to believe that the soil fertility is constantin a given column but varies between columns. How
does one then test the effect of three different fertilizers
on a given crop, eliminating the natural fertility
variation between columns? One could divide eachcolumn into two and use two fertilizers in each column.
For example, one cou ld have t he fol low ing
arrangement—the first column has fertilizer 2 and 3
applied, the second column has fertilizer 1 and 3
applied, and the third column has fertilizer 1 and 2
applied. This is a balanced incomplete block design—it’s “balanced” because each pair of fertilizers has thesame probability of occurring in this scheme, and“incomplete” because in no column do we have all threefertilizers used. Today, Latin squares and balanced
incomplete block designs have found applications as
varied as in communication systems, cryptography,
error-control coding, and clinical trials of medicines.

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Parker believed his disproof of MacNeish’s conjecture
to be “an unimportant near miss” [5]. But “Parker’s
result”, as a later paper he wrote with Bose andShrikhande noted, “though it did not disprove Euler’s
conjecture threw serious doubts on its correctness.”[12]. Building on this, Bose and Shrikhande constructedorthogonal Latin squares of order 22 [13], [14]. They
used a generalisation of balanced incomplete block
designs called pairwise balanced designs.
Parker became aware of this work, and he came up
with two orthogonal Latin squares of order 10 througha different method [15]. There ensued “feverish
correspondence” [16] between the three mathematicians,
culminating in their proof that there did indeed exist
orthogonal Latin squares of all orders other than six(and two). Euler was wrong.
Their proof had bridged geometry, combinatorics
and statistics. “It is a striking example of the unity ofscience,” said Bose of this famous result [3], “that theinitial impulse which lead to a solution of the problem
propounded by Euler, came from the practical needsof agricultural experimentation, thus reversing themore usual passage from basic to applied knowledge.”
Later in his life, in an autobiographical essay [17],Bose recalled what had happened the day after the result
was announced at the AMS meeting. “The scienceAsia Pacific Mathematics Newsletter

editor of the New York Times came to interview us,” hewrote, “and next morning our picture appeared on thefirst page of the Sunday edition of the New York Times
with a description of our work. I was staying at a small
hotel. Next morning when I went to pay the bill, thecashier looked at me and asked, ‘Is it your picture in
the New York Times?’ ‘Yes, it is my picture,’ I said. He
replied, ‘You must have done something. The front pageof the New York Times cannot be bought for a million
dollars.’”
Shrikhande’s second son, Mohan, was in school
when his teacher stopped him on the stairs and asked
him if he had seen that day’s New York Times. In thenews report, recalled the late Dutch combinatorist J J
Seidel (1919–2001) in an address on Shrikhande’s 70thbirthday in 1987 [4], “The three scientists stood in frontof, and partly made invisible a huge 10 × 10 tableau
showing the Latin square that spoiled Euler’s conjecture.
The next day, dozens of school-boys had computed themissing pieces.”As his children—Mohan, Asha and Anil,grandchildren, and great-grandchildren took
to the stage to speak lovingly of how deeply he had
influenced them, and how grateful they were for his
presence and openness, Shrikhande’s life flashed by on
the screen behind. Shrikhande the young man travellingto distant shores, the loving husband and father, thefamily man who remained close to his siblings, themathematician and mentor, the yoga practitioner, andthe doting grandfather and great-grandfather. From
Nagpur there also arrived a message from his academic
descendants, expressing their gratitude to him forhaving started and sustained research in statistics andcombinatorics in that city by founding the Departmentof Statistics at the Institute of Science in Nagpur in 1955.
All these facets of “Gramps”, as his grandchildren callhim, were not orthogonal to each other but blended
harmoniously.
After his famous work, Shrikhande returned to
India in 1960, joining Banaras Hindu University, which
he left in 1963 to join the University of Bombay where
he was the first head of the Department of Mathematics
and the Director of the UGC Centre for Advanced
Study in Mathematics. Navin Singhi, who did his PhD
with Shrikhande at the University of Bombay in
1972–73 and went on to collaborate with him and Bose,
remembers Shrikhande’s simplicity and unassuming
style, and also his focussed approach to his work. As
head of the Department, he had to do a lot of

administrative work, his office often a continuous
stream of visitors. But Shrikhande would always
nonchalantly return to his work as soon as he had
cleared off such frequent interruptions.

“All his work had the hallmark
of being simple and elegant,yet deep”
His students remember his endearing personality
and habits. “He used empty envelopes left over fromincoming mails for writing research papers. He almost
never used normal paper or notebook to write down,”
Singhi recalls. “As a chairman, he used to get a lot of
mail. He had developed his own techniques on how to
open an envelope of incoming mail carefully so that its
inside could be used to write mathematics. These covers
which most people throw away were the ones which
carried most of Shrikhande’s research work, with me
and with others.”
The late Vasanti Bhat-Nayak, who did her PhD with
Shrikhande at the University of Bombay, had joined
the Department as a lecturer after an unhappy spell at
the Tata Institute of Fundamental Research. She felt
welcomed by the cordial environment Shrikhande had
nurtured and his gentle encouragement that restored
her shattered confidence. “Shrikhande would always
introduce me to others as his colleague and not as his
student,” she wrote in a commemorative essay. “I wasnever ridiculed even if I made a mathematically wrong
statement.” She later became head of the Department
herself.
Shrikhande made many other notable contributions
to combinatorics and design theory. In particular, while
investigating Latin squares, he published anotherlandmark paper [18] in the same year, 1959, on what
has come to be known as the Shrikhande graph, which
has connections to algebra, group theory and topology.
“All his work,” said Sharad Sane, a combinatorist at IIT
Bombay, “had the hallmark of being simple and elegant,yet deep.” This echoes the words of the late statistician
J.N. Srivastava (1933–2010) who, like Shrikhande, did
his PhD with Bose, and wrote that [8], “In any given
combinatorial setting, Professor Shrikhande has the
knack of discovering questions which are deep, elegant
and significant. This gives a fundamental character toall his work...”Mohan Shrikhande, a renowned combinatorist
himself, and who retired from the Central Michigan
University in 2015, did not begin his career as a
combinatorist, having done his PhD in a different area.

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Asia Pacific Mathematics Newsletter“I wasn’t particularly interested in combinatorics
until after my PhD,” he said, “which was when I read apaper my father sent me.” He went on to co-author
papers with some of his father’s colleagues and students,
including Bose, Singhi and Sane.
Sane, and S.B. Rao—former director of both theIndian Statistical Institute, Kolkata, and the C.R. Rao
Advanced Institute of Mathematics, Statistics andComputer Science, Hyderabad—had both worked with
Shrikhande at the University of Bombay, and, at thebirthday party, spoke of the deep influence his work
has had on combinatorics and design theory, and themany mathematicians he had mentored, in India andabroad.

Shrikhande lived with his children and grandchildren
in the US. He returned to India in 2009, and now lives
in Vijayawada, in the care of the in-laws of Nita
Kulkarni, one of his grandchildren, in the ashram theyrun for underprivileged girls. On his birthday, which
also happened to be Deepavali, the festival of lights, thegirls sang songs, drew a rangoli in the form of a
Shrikhande graph, and prepared a special cake for him.
Shrikhande’s fellow Euler’s spoilers, Bose and Parker,
have passed away in 1987 and 1991 respectively, but
their work has had an enduring impact. “The techniques
they developed,” said Sane, “have now become an
important part of the study of design theory andcombinatorics.”
Shrikhande hasn’t stopped learning, however. Whenliving with his family in the US for example, heenthusiastically learnt how to use email, said his granddaughter Nita. These days, he still enjoys trying to fillin numbers in a square—in the form of Sudoku puzzles,
which is just a special kind of Latin square.
“Yeah, I’m trying. But it’s getting very difficult,” hesaid, laughing.
References(Courtesy of The Shrikhande Family)

And they spoke of his humility. The late Canadianmathematician Ralph G. Stanton (1923–2010) foundthis out when he first met Shrikhande at the American
Mathematical Society meeting where the disproof of
Euler’s conjecture was announced. Stanton foundShrikhande “much more interested in discussing othermathematical topics in design theory than in dwellingon his achievement.” [8]. As Srivastava wrote, “Manypeople, in the midst of their achievements, get puffed
up. Not so with Professor Shrikhande. There is a certain
humility about him which is worth emulating.”
Yashodhan Hatwalne, a physicist at the Raman
Research Institute in Bengaluru, whose mother is thesister of Shrikhande’s late wife Shakuntala, used to takelong walks with Shrikhande when they were in the US.
He knew about Shrikhande’s fame and achievement.
“But he never talked about all this. Not once,” says
Hatwalne.
Shrikhande retired in 1978, but accepted the
directorship of the Mehta Research Institute (now theHarish-Chandra Research Institute) in Allahabad,
1983–86. After his wife Shakuntala passed away in 1988,

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[1] L. Euler, Investigations on a New Type of Magic
Square, 1782; http://eulerarchive.maa.org/docs/
translations/E530.pdf[2] A. Dey, Orthogonal Latin Squares and the Falsityof Euler’s Conjecture, in: R. Bhatia, C.S. Rajan,
A.I. Singh (eds), Connected at Infinity II. Texts andReadings in Mathematics, Vol. 67 (Hindustan
Book Agency, Gurgaon, 2013), pp. 1–17.
[3] H.H. Frisinger, The Solution of a Famous
Two-Centuries Old Problem: The Leonhard
Euler-Latin Square Conjecture, Historia
Mathematica 8(1) (1981) 56–60.
[4] Souvenir of the Instructional Workshop on Discrete
Mathematics – 20 December 1993 to 8 January 1994,
Department of Mathematics, University of
Bombay.
[5] J.A. Osmundsen, Major Mathematical Conjecture
Propounded 177 Years Ago is Disproved; 3
Mathematicians Solve Old Puzzle. New YorkTimes, 26 April 1959.
[6] S.S. Shrikhande, Combinatorics and I, in: J.K.
Ghosh, S.K. Mitra and K.R. Parthasarathy (eds).
Glimpses of India’s Statistical Heritage (Wiley
Eastern, New Delhi, 1993).
[7] R.C. Bose, On the Application of the Properties
of Galois Fields to the Problem of Construction

Asia Pacific Mathematics Newsletterof Hyper-Graeco-Latin Squares, Sankhyā 3(4)(1938) 323–338.
[8] D.S. Meek and R.G. Stanton (eds), Selected Papersof S.S. Shrikhande, Vol. I, II, (Charles Babbage
Research Centre, Winnipeg, 1985).
[9] H.F. MacNeish, Euler Squares, Ann. Math. 23(3)(1922) 221–227.
[10] E.P. Woodruff, Greco-Latin Squares Problem
Solved by E.T. Parker, Ph.D. and Colleagues.
O n l i n e ; h t t p s : / / w e b. a r c h i v e . o r g / w e b /20050426040727/http://www2.msstate.edu/~eaddy/html/etparker.htm[11] E.T. Parker, Construction of Some Sets of
Mutually Orthogonal Latin Squares, Proc. Amer.
Math. Soc. 10(6) (1959) 946–949.
[12] R.C. Bose, S.S. Shrikhande and E.T. Parker,
Further Results on the Construction of Mutually
Orthogonal Latin Squares and the Falsity ofEuler’s Conjecture, Canad. J. Math. 12 (1960)
189–203.

[13] R.C. Bose and S.S. Shrikhande, On the Falsity ofEuler’s Conjecture about the Non-Existence of
Two Orthogonal Latin Squares of Order 4t+2,
Proc. Natl. Acad. Sci. USA 45(5) (1959) 734–737.
[14] R .C. B os e and S.S. Shrik hande, On theConstruction of Sets of Mutually Orthogonal
Latin Squares and the Falsity of a Conjecture ofEuler, Trans. Amer. Math. Soc. 95(2) (1960)
191–209.
[15] E.T. Parker, Orthogonal Latin Squares, Proc. Natl.
Acad. Sci. USA 45(6) (1959) 859–862.
[16] M. Gardner. Mathematical Games, Scientific
American 201(5) (1959) 181–192.
[17] R.C. Bose, Autobiography of a Mathematical
Statistician, in: J. Gani (ed), The Making of
Statisticians (Springer, New York, 1982).
[18] S.S. Shrikhande, The Uniqueness of the L 2
Association Scheme, Ann. Math. Stat. 30(3)(1959) 781–798.

Nithyanand Rao
A lapsed physics student, Nithyanand Rao is now a science writer based in
Bangalore, and one of the co-founders and contributing editors of Bhãvanã.

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