Some partial Latin cubes and their completions

Abstract

It is well known that all n×n partial Latin squares with at most n−1 entries are completable. Our intent is to extend this well known statement to partial Latin cubes. We show that if an n×n×n partial Latin cube contains at most n−1 entries, no two of which occupy the same row, then the partial Latin cube is completable. Also included in this paper is the problem of completing 2×n×n partial Latin boxes with at most n−1 entries. Given certain sufficient conditions, we show when such partial Latin boxes are completable and then extendable to a deeper Latin box.