Every latin hypercube of order 5 has transversals

Abstract

We prove that for all $n>1$ every latin $n$-dimensional cube of order 5 has transversals. We find all 123 paratopy classes of layer-latin cubes of order 5 with no transversals. For each $n\ge 3$ and $q\ge 3$ we construct a $\left(2q-2\right)\times q\times \cdots \times q$ latin $n$-dimensional cuboid of order $q$ with no transversals. Moreover, we find all paratopy classes of nonextendible and noncompletable latin cuboids of order 5.