McKay, BrendanWanless, Ian2015-12-100895-4801http://hdl.handle.net/1885/39594We count all latin cubes of order n < 6 and latin hypercubes of order n < 5 and dimension d < 5. We classify these (hyper)cubes into isotopy classes and paratopy classes (main classes). For the same values of n and d we classify all d-ary quasigroups of order n into isomorphism classes and also count them according to the number of identity elements they possess (meaning we have counted the d-ary loops). We also give an exact formula for the number of (isomorphism classes of) d-ary quasigroups of order 3 for every d. Then we give a number of constructions for d-ary quasigroups with a specific number of identity elements. In the process, we prove that no 3-ary loop of order n can have exactly n- 1 identity elements (but no such result holds in dimensions other than 3). Finally, we give some new examples of latin cuboids which cannot be extended to latin cubes.Keywords: Identity elements; Jacobsthal sequence; Latin cube; Latin cuboid n-ary quasigroup; Latin hypercube; MDS code; n-ary loop; Total loop; Set theory; Geometry Identity elements; Jacobsthal sequence; Latin cube; Latin cuboid n-ary quasigroup; Latin hypercube; MDS code; n-ary loop; Total loop200810.1137/0706938742016-02-24