Summary: Hashing techniques for accessing a table without searching it are usually designed to perform efficiently on the average over all possible contents of the table. If the table contents are known in advance, we might be able to choose a hashing function with guaranteed efficient (worst-case) performance. Such a technique has been called ''perfect hashing'' by Sprugnoli and others. In this paper, we address the question of whether perfect hashing is feasible in principle as a general technique, or whether it must rely on special qualities of the table contents. We approach the question by counting the number of functions which must be searched to be sure of finding a perfect hashing function. We present upper and lower bounds on the size of this search space, with attention to the tradeoff between the size of the search space and the size of the hash table.