Solved by
 Edythe Parker Woodruff, Ph. D.

Ernest Tilden Parker (1926-1991), Ph. D. and Professor of Mathematics was a great grandson of Rev. Oliver Eady and a brother of the author of this article.  As a mathematician, he and two colleagues succeeded in constructing a counter example to a conjecture stated by Leonard Euler 177 years earlier.  A photo of them with an article describing their work appeared on the front page of the Sunday New York Times on April 26, 1959.  The example was the cover illustration and story for the November 1959 Scientific American magazine.  The work was published in the Proceedings of the National Academy of Sciences in June of 1959 and later in the Proceedings of the American Mathematical Society.

Here is the problem:  On a 10 x 10 matrix or grid (that is, a "checkerboard" without colors), place each of the hundred numbers: 00 to 99.  Arrange them so that:

1.  In each row there appears exactly one number beginning with each of the ten digits.
2.  In each column there appears exactly one number beginning with each of the ten digits.
3.  In each row there appears exactly one number ending in each of the ten digits.
4.  In each column there appears exactly one number ending in each of the ten digits.

Each of the numbers 00 through 99 must be used exactly once in the arrangement.  If such an arrangement is found, for historic reasons it has the name of a Graeco-Latin square.

Euler had said that this could not be done for a similar 6 X 6 arrangement using six different first and second digits.  He further said it could not be done for 10 x 10, 14 x 14, 18 x 18, etc.--that is, those sizes that are an odd  number times two.  In 1900 he was proven correct about the impossibility of 6 x 6.  Parker, R.C. Bose, and S. S. Shrikhande together showed that Euler was wrong for all those other sizes.

Let me quote some from the Scientific American magazine article by Martin Gardner, pp 181-188.  Parts of it includes quotes from the three.  "The history of mathematics is filled with shrewd conjectures--intuitive guesses by men of great mathematical insight--that often wait for centuries before they are proved or disproved.  When this finally happens it is a mathematical event of first magnitude...."

"The story of how Parker, Bose, and Shrikhande managed to find Graeco-Latin squares of orders 10, 14, 18, 22 (and so on) begins in 1958, when Parker made a discovery that cast grave doubt on the correctness of Euler's conjecture.  Following Parker's lead, Bose developed some strong general rules for the construction of large-order Graeco-Latin squares.  Then Bose and Shrikhande, applying these rules, were able to construct a Graeco-Latin square of order 22...."

"When Parker saw the results obtained by Bose and Shrikhande, he was able to develop a new method that led to his construction of an order-10 Graeco- Latin square...."

 "At this stage, the three mathematicians conclude their report, there ensued a feverish correspondence between Bose and Shrikhande on the one hand and Parker on the other.  Methods were refined more and more; it was ultimately established that Euler's conjecture is wrong for all values of n = 4k + 2, where n is greater than 6.  The suddenness with which complete success came in a problem that had baffled mathematicians for almost two centuries startled the authors as much as anyone else."

As Dr. Parker's sister, I would like to add a couple of things.  The Euler conjecture had occupied much of E. T.'s thought for many years before he presented the 1958 paper that told Bose and Shrikhande of his work to date.  (I do not know how much effort they had put into the problem, but likely also a great deal of prior thought had occurred.)   At the time of the breakthrough, E. T. was employed by Remington Rand Univac, an early computer company, but the construction of this example was done entirely by abstract thought and no computer use was involved.  Even if a computer had found an example for one size, that would not have proven that all of these others exist.  E. T.'s work on this problem was done entirely outside of company work; it was his recreation.  He did it out of his pure love for the subject.  Later he joined the faculty of the University of Illinois Mathematics Department, where he became a professor.

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