Euler conjectured that there do not exist Euler squares of order for , 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 by Bose, Shrikhande, and Parker in 1959 (Wells 1986, p. 77), refuting the conjecture (and establishing unequivocally the invalidity of MacNeish's "proof").
Euler's Graeco-Roman Squares Conjecture
See also
36 Officer Problem, Euler Square, Latin SquareExplore with Wolfram|Alpha
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.Referenced on Wolfram|Alpha
Euler's Graeco-Roman Squares ConjectureCite this as:
Weisstein, Eric W. "Euler's Graeco-Roman Squares Conjecture." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/EulersGraeco-RomanSquaresConjecture.html