There exist implemented algorithms which can provide all possible solutions to any crossword puzzle grid for any given set of words. A formula is derived, within a Bayesian framework, and it provides an estimate of the number of solutions which can be constructed from a given dictionary for any given crossword puzzle without direct recourse to computation of all solutions. The formula is constructed to account for any puzzle geometry and any given dictionary, natural language or otherwise. The number of solutions estimated by the formula, for a variety of puzzles and dictionaries, is compared with the actual number in each solution set. The formula is shown to be only partially effective but to be capable of further development. The formula is useful as an indicator of the time required for computing all solutions, if any exist.