Every latin hypercube of order 5 has transversals

We prove that for all every latin ‐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 and we construct a latin ‐dimensional cuboid of order with no transversals. Moreover, we find all paratopy classes of nonextendible and noncompletable latin cuboids of order 5.