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Euler's Graeco-Roman Squares Conjecture


EulersGraecoRoman

Euler conjectured that there do not exist Euler squares of order n=4k+2 for k=1, 2, .... In fact, MacNeish (1921-1922) published a purported proof of this conjecture (Bruck and Ryser 1949). While it is true that no such square of order six exists, such squares were found to exist for all other orders of the form 4k+2 by Bose, Shrikhande, and Parker in 1959 (Wells 1986, p. 77), refuting the conjecture (and establishing unequivocally the invalidity of MacNeish's "proof").


See also

36 Officer Problem, Euler Square, Latin Square

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References

Bose, R. C. "On the Application of the Properties of Galois Fields to the Problem of Construction of Hyper-Graeco-Latin Squares." Indian J. Statistics 3, 323-338, 1938.Bose, R. C.; Shrikhande, S. S.; and Parker, E. T. "Further Results on the Construction of Mutually Orthogonal Latin Squares and the Falsity of Euler's Conjecture." Canad. J. Math. 12, 189, 1960.Bruck, R. H. and Ryser, H. J. "The Nonexistence of Certain Finite Projective Planes." Canad. J. Math. 1, 88-93, 1949.Levi, F. W. Second lecture in Finite Geometrical Systems: Six Public Lectures Delivered in February, 1940, at the University of Calcutta. Calcutta, India: University of Calcutta, 1942.MacNeish, H. F. "Euler Squares." Ann. Math. 23, 221-227, 1921-1922.Mann, H. B. "On Orthogonal Latin Squares." Bull. Amer. Math. Soc. 51, 185-197, 1945.Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 77, 1986.

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Euler's Graeco-Roman Squares Conjecture

Cite this as:

Weisstein, Eric W. "Euler's Graeco-Roman Squares Conjecture." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/EulersGraeco-RomanSquaresConjecture.html

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