How can a delegation of six regiments, each of which sends a colonel, a lieutenant-colonel, a major, a captain, a lieutenant, and a sub-lieutenant be arranged in a regular 
array such that no row or column
duplicates a rank or a regiment? The answer is that no such arrangement is
possible.
36 Officer Problem
See also
Euler's Graeco-Roman Squares Conjecture, Latin Square, Trigonometry Angles--Pi/3, Trigonometry Angles--Pi/6Explore with Wolfram|Alpha
References
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.Parker, E. T. "Orthogonal Latin Squares." Not. Amer. Math. Soc. 6, 276, 1959.Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, p. 31, 1999.Tarry, G. "Le problème de 36 officiers." Compte Rendu de l'Assoc. Français Avanc. Sci. Naturel 1, 122-123, 1900.Tarry, G. "Le problème de 36 officiers." Compte Rendu de l'Assoc. Français Avanc. Sci. Naturel 2, 170-203, 1901.Referenced on Wolfram|Alpha
36 Officer ProblemCite this as:
Weisstein, Eric W. "36 Officer Problem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/36OfficerProblem.html