Some partial Latin cubes and their completions

It iswell known that all n×n partial Latin squareswith atmost 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 showwhen such partial Latin boxes are completable and then extendable to a deeper Latin box. © 2011 Elsevier Ltd. All rights reserved.