Latin Hypercubes Realizing Integer Partitions

For an integer partition h1+…+hn=N, a 2-realization of this partition is a latin square of order N with disjoint subsquares of orders h1,…,hn. The existence of 2-realizations is a partially solved problem posed by Fuchs. In this paper, we extend Fuchs' problem to m-ary quasigroups, or, equivalently, latin hypercubes. We construct latin cubes for some partitions with at most two distinct parts and highlight how the new problem is related to the original.