Shrikhande, “Euler's Spoiler”

Asia Pacific Mathematics Newsletter Shrikhande, “Euler’s Spoiler”, Turns 100 Nithyanand Rao Portrait by Mohan R (Courtesy of The Shrikhande Family) elatives, well-wishers and dignitaries kept ensured his name would be associated with Euler, one arriving to greet Professor Shrikhande. Seated of the greatest mathematicians in history. It was 58 years on the lawns, he would adjust his hearing aid— ago that Shrikhande, along with his mentor R.C. Bose Rtrying to hear over the firecrackers in the background— and their collaborator E.T. Parker, proved Euler wrong thank them and smile, and now and then burst into a and made the headlines. hearty chuckle, trying not to look in the direction of the intense light drenching the table. Late in his life, the legendary Swiss mathematician Sharadchandra Shankar Shrikhande, celebrating his Leonhard Euler (1707–1783) began a long paper [1] 100th birthday on 19 October 2017, wasn’t too keen to pondering a puzzle he couldn’t find an answer to. remain in the spotlight. The bright light on the pole Although he was almost completely blind by then, his was turned away, but visitors kept coming to greet him already prodigious productivity had increased, and seek his blessings, some aware of his great distractions having been reduced. He had always made mathematical achievements—in particular, the one that the most of his phenomenal memory and ability to December 2018, Volume 8 No 1 15 Asia Pacific Mathematics Newsletter calculate in his head and, after his loss of vision, he used a scribe to record his discoveries. The puzzle he was considering was this: Imagine that there are 36 officers belonging to six different military regiments, each regiment having six officers of different ranks. How does one arrange them in the form of a square such that each row and column has six officers, and no rank or regiment appears more than once in a row or column? 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 get another square as below, where the numbers are now coloured. In the square above, both requirements are satisfied. Euler (Emanuel Handmann via Wikimedia Commons) And there is only one red 1 and only one green 2; only once does each such combination of rank and regiment Mathematicians, professional or otherwise, like to appear in the square. Two such superposed Latin ponder such puzzles for recreation, and not a few squares, therefore, are called orthogonal to each other. mathematical gems have emerged from such playing Orthogonal Latin squares were, in fact, around around. In many such puzzles, a simplification helps much longer before Euler. Amulets bearing Latin clarify the picture. Consider nine officers instead of 36, squares were known in medieval Islam, circa 1200 [2]. belonging to three different regiments, each regiment Choi Seok-Jeong (1646–1715), a Korean government having one officer of each of the three ranks. The official and mathematician, used orthogonal Latin problem, as before, is to arrange them in a square such squares of order 9 to construct magic squares—one in that no rank or regiment is repeated in a row or column. which the entries in any of the rows or columns or This is actually two problems in one. One can arrange diagonals add up to the same number. (Euler himself the nine officers such that no regiment—labelled 1, 2 was interested in magic squares.) or 3 below—is repeated in a row or column. Such an The pair of Latin squares in the above example is of arrangement is called a Latin square (since Euler order n = 3. Euler observed that a similar exercise can originally used Latin letters instead of numbers). be 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 a pair of orthogonal Latin squares of order 6. But Euler found himself unable to do it for n = 6. He couldn’t find an 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 One can also arrange the officers such that no 10,14,18, 22, and so on, which are called “oddly even” rank—red, green or blue below—is repeated in a row numbers. (There are no orthogonal Latin squares of or column to get another Latin square. order 2 as well.) 16 December 2018, Volume 8 No 1 Asia Pacific Mathematics Newsletter This conjecture by Euler was made in 1782. In 1901, that, Shrikhande’s father, who worked at a flour mill, a French mathematician named Gaston Tarry (1843– was determined to educate his children. Shrikhande 1913) proved that n = 6 was indeed impossible by did well academically and won scholarships that helped laboriously checking all possible cases. But Euler’s him complete his BSc Honours at the Government conjecture that orthogonality was impossible for all College of Science (now known as the Institute of oddly even numbers remained to be resolved. Until Science) in Nagpur with a first rank and a gold medal. 1959, when R.C. Bose, Shrikhande and E.T. Parker “At this point of time,” he wrote in an autobiographical disproved the conjecture. essay [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 located in four of five rooms in the Presidency College” where Mahalanobis was a professor of physics. But those rooms 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 architect of my research involvement in combinatorics”, the field of mathematics that studies how to combine different things, whether certain combinations are possible at (Courtesy of The Shrikhande Family) all, and what combination is most appropriate given certain constraints. “The suddenness with which complete success After a year at ISI, Shrikhande had a short-lived came,” Bose recalled later [3], “in a problem which had appointment at the college in Jabalpur where he had baffled mathematicians for over one and three-quarters done his intermediate studies (the equivalent of two centuries startled the authors as much as anyone else. years after Class 10 today) and then joined as a lecturer What makes this even more surprising is that the in mathematics at the Government College of Science concepts employed were not even close to the frontiers in Nagpur. But he kept making trips to Calcutta in the of deep modern mathematics.” early 1940s to work with Bose, who introduced him to The result was announced in the annual meeting of the theory of statistical design of experiments, a field the American Mathematical Society held in New York where Latin squares proved to be more than amusement. in April 1959. “Bose, Ernie (Parker) and I,” Shrikhande In the 1920s, R.A. Fisher (1890–1962), the pioneering said later [4], “had the rare privilege of seeing our works British statistician and biologist, had applied them to reported on the front page of the Sunday Edition of the the design of field experiments in agriculture. His ideas New York Times of April 26, 1959.” were later developed by Bose. The three of them, noted theNYT report [5], “are Suppose one wishes to test the effects of three now known among their colleagues as ‘Euler’s spoilers’”. different 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 hrikhande and Bose were, at the time, at the because the amount of moisture in the soil varies. How SUniversity of North Carolina, Chapel Hill, in the does one test how good the fertilizers are while Department of Mathematical Statistics, which had been accounting for this natural variation in fertility of the founded in 1946 by the statistician Harold Hotelling soil? Fisher’s solution to this problem was to divide the (1895–1973). Bose had been a faculty there since 1949, field into cells of a three-by-three square and apply the and Shrikhande, who had joined for a PhD in 1947, three fertilizers in the pattern of a Latin square of order had been Bose’s first PhD student. But his path hadn’t 3 (such as the one shown previously). Because of this been straightforward. Born on 19 October 1917, the pattern, one can use statistical analysis to eliminate the fifth of ten siblings, in Sagar, now in Madhya Pradesh, bias due to variation in natural soil fertility along a row his family faced severe financial difficulties. Despite or column of the field. December 2018, Volume 8 No 1 17 Asia Pacific Mathematics Newsletter Now, if instead of one variety of crop, what if we of mutually orthogonal Latin squares in such a case, a had three? The solution to accommodate this variable result that, unknown to him, had been discovered is to use a Latin square of order 3 orthogonal to the earlier in 1896 by E.H.

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