Searching for Orthogonal Latin Squares via Cells Mapping and BOINC-Based Cube-and-Conquer

This study focuses on searching for pairs of orthogonal diagonal Latin squares of order 10. Consider a cells mapping in accordance to which one diagonal Latin square is mapped to another one. Given a certain cells mapping schema, the problem is to find a pair of orthogonal diagonal Latin squares of order 10 such that they match the schema (or to prove that such a pair does not exist). The problem is reduced to the Boolean satisfiability problem (SAT). Three mapping schemes are considered, and for each of them a SAT instance is constructed. If a satisfying assignment is found for an instance, the corresponding pair of orthogonal Latin squares can be easily extracted from it. The Cube-and-Conquer approach is used to solve the instances. The cubing phase is performed on a sequential look-ahead SAT solver, while on the conquer phase an experiment in a BOINC-based volunteer computing project is launched. In the experiment, for two out of three schemes orthogonal pairs are found.