We prove a general result on completing objects similar to Latin rectangles in which the number of occurrences of each symbol is prescribed, each cell contains multiple symbols, and no cell contains repeated symbols. This generalizes several results in the literature, and leads to confirming a conjecture of Cavenagh, Hämäläinen, Lefevre, and Stones. An $$r\times s$$ $$\lambda $$ -Latin rectangle L is an $$r\times s$$ array in which each cell contains a multiset of $$\lambda $$ elements from the set $$\{1,\dots ,n\}$$ of symbols such that each symbol occurs at most $$\lambda $$ times in each row and column. If $$r=s=n$$ , then L is a $$\lambda $$ -Latin square. A $$\lambda $$ -Latin rectangle is simple if no symbol is repeated in any cell. Cavenagh et al. asked for conditions that ensure a simple $$\lambda $$ -Latin rectangle can be extended to a simple $$\lambda $$ -Latin square. We solve this problem in a more general setting by allowing the number of occurrences of each symbol to be prescribed. Cavenagh et al. conjectured that for each $$r, \lambda $$ there exists some $$n(r, \lambda )$$ such that for any $$n \geqslant n(r, \lambda )$$ , every simple partial $$\lambda $$ -Latin square of order r (each cell contains at most $$\lambda $$ symbols) embeds in a simple $$\lambda $$ -Latin square of order n. We confirm this conjecture.

Abstract. Can we color the [Formula: see text] cells of an [Formula: see text] cube [Formula: see text] with [Formula: see text] colors in such a way that each layer parallel to each face contains each color exactly once and that the coloring is symmetric so that [Formula: see text] for distinct [Formula: see text], and [Formula: see text] for [Formula: see text]? Using transportation networks, we show that such a coloring is possible if and only if [Formula: see text] (mod 3) (with two exceptions: [Formula: see text] can be 1, but [Formula: see text] cannot be 3). Motivated by the designs of experiments, the study of these objects (without symmetry) was initiated by Kishen and Fisher in the 1940s. These objects are also closely related to orthogonal arrays whose existence has been extensively investigated, and they are natural three-dimensional analogues of symmetric latin squares.