Filling the Gaps

Problem text... At the largest conference on coding and cryptography the following theorem needed a proof or a counterexample: Suppose you are given a set of words of equal length; each word consisting of 0's, 1's and/or *'s. Furthermore suppose the pattern of *'s is different for all words in the set. By this we mean: if you replace all 0's and 1's by say $ you obtain different words. The claim