Every latin hypercube of order 5 has transversals.

We prove that for all n>1 $n\gt 1$ every latin n $n$‐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≥3 $n\ge 3$ and q≥3 $q\ge 3$ we construct a (2q−2)×q×⋯×q $(2q-2)\times q\times {\rm{\cdots }}\times q$ latin n $n$‐dimensional cuboid of order q $q$ with no transversals. Moreover, we find all paratopy classes of nonextendible and noncompletable latin cuboids of order 5. [ABSTRACT FROM AUTHOR]