|Order 3||Order 4||Order 5|
The arrangement of the Latin characters alone and of the Greek characters alone each forms a Latin square. A Graeco-Latin square can therefore be decomposed into two "orthogonal" Latin squares. Orthogonality here means that every pair (s, t) from the Cartesian product S×T occurs exactly once.
In the 1780s Euler demonstrated methods for constructing Graeco-Latin squares where n is odd or a multiple of 4. Observing that no order-2 square exists and unable to construct an order-6 square (see thirty-six officers problem), he conjectured that none exist for any oddly even number Indeed, the non-existence of order-6 squares was definitely confirmed in 1901 by Gaston Tarry through exhaustive enumeration of all possible arrangements of symbols. However, Euler's conjecture resisted solution for a very long time. In 1959, R.C. Bose and S. S. Shrikhande found some counterexamples; then Parker found a counterexample of order 10. In 1960, Parker, Bose, and Shrikhande showed Euler's conjecture to be false for all Thus, Graeco-Latin squares exist for all orders n ≥ 3 except
The Talmud Yerushalmi and Graeco-Roman Culture, Vol. 1. and the Talmud Yerushalmi and Graeco-Roman Culture, Vol. 2. (Reviews of Books)
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