As a part of the international human genome project, large-scale genomic maps of human and other model organisms are being generated. More recently, mapping using various anchoring (as opposed to the traditional "fingerprinting") strategies have been proposed based largely on mathematical models. In all of the theoretical work dealing with anchoring, an anchor has been idealized as a point on a continuous, infinite-length genome. In general, it is not desirable to make these assumptions, since in practice they may be violated under a variety of actual biological situations. Here we analyze a discrete model that can be used to predict the expected progress made when mapping by random anchoring. By virtue of keeping all three length scales (genome length, clone length, and probe length) finite, our results for the random anchoring strategy are derived in full generality, which contain previous results as special cases and hence can have broad application for planning mapping experiments or assessing the accuracy of the continuum models. Finally, we pose a challenging nonrandom anchoring model corresponding to a more efficient mapping scheme.