  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 $n\geq 3$ and $q\geq 3$ we construct a
(2q-2)-layer latin n-dimensional cuboid with no transversals. Moreover, we find
all paratopy classes of nonextendible and noncompletable latin cuboids of order
5.
