Avoiding partial Latin squares and intricacy

AbstractIn this paper we consider the following problem: Given a partial n × n latin square P on symbols 1, 2,…, n, is it possible to find an n × n latin square L on the same symbols which differs from P in every cell? In other words, is P avoidable? We show that all 2k × 2k partial latin squares for k ⩾ 2 are avoidable and give some results on odd partial latin squares. We also use these results to show that the intricacy of avoiding partial latin squares is two and of avoiding more general arrays is at most three