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

TL;DR

The paper tackles the longstanding question of the existence of a $3\operatorname{MOLS}(10)$ by examining Myrvold's division of $10\times10$ Latin squares containing a $4\times4$ subsquare into 28 transversal-representation pair (TRP) types. It introduces a duality between orthogonality and transversal representations via a composition operation on column-Latin squares and encodes the TRP problem as SAT, using a Latin-square/transpose-based formulation with color constraints and symmetry breaking. The authors show that 20 of Myrvold's cases are unsatisfiable in under short time on a computing cluster, while the remaining eight cases are satisfiable, providing explicit TRPs and demonstrating that elimination requires considering the third square in the triple rather than just two squares. This work highlights the necessity of a triple-wide perspective and demonstrates a powerful SAT-based approach with a duality-driven encoding, offering concrete TRPs for the eight open cases and suggesting future work on ruling out or confirming triple existence with more structure in the base square $L$.

Abstract

Ever since E. T. Parker constructed an orthogonal pair of $10\times10$ Latin squares in 1959, an orthogonal triple of $10\times10$ 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 $4\times4$ 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 two hours of computing on a large computing cluster.

Figures (13)

• Figure 1: A transversal representation pair of Latin squares of order four. Each transversal of $D$ is highlighted in a different colour, and the row representations of the transversals are given in $D'$.
• Figure 2: A transversal representation pair of $4\times4$ column-Latin squares. Note that the highlighted entries of $D_1$ are not transversals, but their row representations when placed in a $4\times4$ array do form a column-Latin square.
• Figure 3: The Latin square $L$ (left) and its possible transversal types (right). White cells represent symbols in $\{0,1,2,3\}$, light cells represent symbols in the rectangles $\Delta$ and $\Gamma$, and dark cells represent the symbols $\{4,5,\dotsc,9\}$ in $\Sigma$. The cells of $\Sigma$ are not shown in absolute positions; in actuality, each row and column of $\Sigma$ has exactly two dark cells. Similarly, the transversal types are shown up to a permutation of the first six entries and the last four entries.
• Figure 4: An example $4\times4$ Costas Latin square.
• Figure 5: A scatterplot of the solver's running time for each pair type. The median running time is shown as a solid black line. Timeouts are not plotted but are used in determining the median.

Theorems & Definitions (20)

• definition thmcounterdefinition
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• lemma thmcounterlemma: myrvold1999negative
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• proof
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