Constraint programming is a methodology for solving difficult combinatorial problems. In the methodology, one makes three design decisions: the constraint model, the search algorithm for solving the model, and the heuristic for guiding the search. Previous work has shown that the three design decisions can greatly influence the efficiency of a constraint programming approach. However, what has not been explicitly addressed in previous work is to what level, if any, the three design decisions can be made independently. In this paper we use crossword puzzle generation as a case study to examine this question. We draw the following general lessons from our study. First, that the three design decisions—model, algorithm, and heuristic—are mutually dependent. As a consequence, in order to solve a problem using constraint programming most efficiently, one must exhaustively explore the space of possible models, algorithms, and heuristics. Second, that if we do assume some form of independence when making our decisions, the resulting decisions can be sub-optimal by orders of magnitude.