A Latin square is an n X n array filled with n different symbols, each occurring exactly once in each row and exactly once in each column. A Costas array can be regarded geometrically as a set of n points, each at the center of a square in an n X n squares and each row or column contains only one point, and all of the vectors between each pair of dots are distinct. A Costas Latin square of order n is a set of n disjoint Costas arrays of the same order.

Costas Latin squares are important combinatorial structures in combinatorial design theory, and are well studied in decades. Some Costas Latin squares are found in recent years. But there are still some open problems about the existence of Costas Latin squares with some properties including idempotency, orthogonality, and some quasigroup properties. In this paper, we describe a method for solving some open cases using state-of-the-art SAT solvers.