Abstract. We show that for all integers m 4 there exists a 2m × 2m × m latin cuboid that cannot be completed to a 2m× 2m× 2m latin cube. We also show that for all even m > 2 there exists a (2m−1) × (2m−1) × (m−1) latin cuboid that cannot be extended to any (2m−1) × (2m−1) × m latin cuboid.Key words. latin cube, latin cuboid, transversal, orthogonal latin squares AMS subject classification. 05B15 DOI. 10.1137/110825534 1. Introduction. There is a celebrated result due to Marshall Hall [6] that every latin rectangle is completable to a latin square. However, the equivalent statement in higher dimensions is not true. The purpose of this paper is to investigate the extent to which it fails.We think of a 3-dimensional array as having layers stacked on top of each other. It also has lines of cells in three directions, obtained from fixing two coordinates and allowing the third to vary. The lines obtained by varying the first, second, and third coordinates will be known respectively as columns, rows, and stacks. The first, second, and third coordinates themselves will be referred to as the indices of the rows, columns, and layers.An n × n × k latin cuboid is a 3-dimensional array containing n different symbols positioned so that every symbol occurs exactly once in each row and column and at most once in each stack. An n × n × n latin cuboid is a latin cube of order n. Every layer of a latin cuboid is a latin square, and we will present our cuboids by displaying the latin squares corresponding to each layer. Each individual layer is composed of a set of n 2 entries, each of which is a triple (r, c, s), where s is the symbol in row r and column c. The layer in which a given entry resides will always be made clear by the context.We say that an n× n× k latin cuboid has order n. It is extendible if it is contained in some n× n× (k + 1) latin cuboid and it is completable if it is contained in some latin cube of order n. Our aim is to investigate how "thin" (that is, how small k can be, relative to n) nonextendible and noncompletable latin cuboids can be. We will refer to an n × n × k latin cuboid as being less than half-full, half-full, or more than half-full