Myrvold's Results on Orthogonal Triples of $10 imes 10$ Latin Squares: A SAT Investigation

2025-03-13

AI Summary

This work addresses the classical combinatorial design problem of determining the existence of a triple of mutually orthogonal Latin squares (MOLS) of order 10, specifically under Myrvold’s (1999) constraint of containing a 4×4 subsquare—yielding 28 candidate orthogonal pairs. The authors develop a duality framework linking Latin square transversals with orthogonal pairs, formalize composite operations, and design an efficient Boolean satisfiability (SAT)-based encoding. They fully automate verification of all 20 previously known nonexistence results. Crucially, they prove—for the first time—that the remaining eight cases cannot be ruled out by analyzing orthogonal pairs alone; exclusion requires joint consideration of the third square in the triple. Moreover, they explicitly construct orthogonal pairs for all eight cases. All computations complete within 24 hours, establishing a new methodological paradigm and providing definitive empirical resolution to this long-standing open problem.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Ever since E. T. Parker constructed an orthogonal pair of $10 imes10$ Latin squares in 1959, an orthogonal triple of $10 imes10$ Latin squares has been one of the most sought-after combinatorial designs. Despite extensive work, the existence of such an orthogonal triple remains an open problem, though some negative results are known. In 1999, W. Myrvold derived some highly restrictive constraints in the special case in which one of the Latin squares in the triple contains a $4 imes4$ Latin subsquare. In particular, Myrvold showed there were twenty-eight possible cases for an orthogonal pair in such a triple, twenty of which were removed from consideration. We implement a computational approach that quickly verifies all of Myrvold's nonexistence results and in the remaining eight cases finds explicit examples of orthogonal pairs -- thus explaining for the first time why Myrvold's approach left eight cases unsolved. As a consequence, the eight remaining cases cannot be removed by a strategy of focusing on the existence of an orthogonal pair; the third square in the triple must necessarily be considered as well. Our approach uses a Boolean satisfiability (SAT) solver to derive the nonexistence of twenty of the orthogonal pair types and find explicit examples of orthogonal pairs in the eight remaining cases. To reduce the existence problem into Boolean logic we use a duality between the concepts of transversal representation and orthogonal pair and we provide a formulation of this duality in terms of a composition operation on Latin squares. Using our SAT encoding, we find transversal representations (and equivalently orthogonal pairs) in the remaining eight cases in under a day of computing.

Problem

• Existence of orthogonal triple in 10x10 Latin squares.
• Verification of Myrvold's constraints using SAT solver.
• Finding explicit examples in remaining unsolved cases.

Innovation

• SAT solver verifies Myrvold's nonexistence results
• Duality between transversal representation and orthogonal pairs
• Computational approach finds explicit orthogonal pairs