Notes for Ernest Tilden PARKER

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Notes for Ernest Tilden PARKER

GRECO-LATIN SQUARES PROBLEM
Solved by
E. T. PARKER, Ph. D. and COLLEAGUES
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.

Note from Edythe Parker Woodruff: Ernest Tilden Parker legally changed
his name to E. T. Parker about 1982. His father changed his name from
Toney MacCager Parker to Thomas Mack Parker. Thomas M. Parker appears on
E. T. Parker's birth certificate.

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