this paper we consider only exhaustive searching techniques rather than more radical ones such as genetic algorithms, simulated annealing or the like. Among search algorithms, we consider only backtracking methods to which the cardinality of the constraints is irrelevant. This narrowing of our focus is in no way intended to slight any of the alternative methods. Merely, our research is what it is and not another thing. It is extremely easy and natural to represent existence problems such as our QG1--QG7 in terms of consistent labelling. To generate a quasigroup of order v is to fill in each of the v

The method proposed by Davis, Putnam, Logemann, and Loveland for propositional reasoning, often referred to as the Davis-Putnam method, is one of the major practical methods for the satisfiability (SAT) problem of propositional logic. We show how to implement the Davis-Putnam method efficiently using the trie data structure for propositional clauses. A new technique of indexing only the first and last literals of clauses yields a unit propagation procedure whose complexity is sublinear to the number of occurrences of the variable in the input. We also show that the Davis-Putnam method can work better when unit subsumption is not used. We illustrate the performance of our programs on some quasigroup problems. The efficiency of our programs has enabled us to solve some open quasigroup problems.

The Davis-Putnam method is one of the major practical methods for the satisfiability (SAT) problem of propositional logic. We show how to implement the DavisPutnam method efficiently using the trie data structure for propositional clauses by presenting seven implementations of the method. We propose a new technique for implementing unit propagation whose complexity is sublinear to the number of occurrences of the variable in the input. We present the performance of our programs on some quasigroup problems. The efficiency of our programs allowed us to solve some open quasigroup problems.