Partitions and balanced matchings of an n-dimensional cube

A non-empty set A⊆X=X1×⋯×Xn is a box if A=A1×⋯×An and Ai⊆Xi for each i∈[n]. Two boxes A,B⊂X are dichotomous if Ai=Xi∖Bi for some i∈[n]. Using a cube tiling code of Rn designed by Lagarias and Shor, a certain class of partitions of an n-dimensional cube into 2n pairwise dichotomous boxes is constructed. Additionally, for every prime number n≥3 perfect matchings of the graph of the unit cube [0,1]n with faulty vertices (0,…,0) and (1,…,1) in which the number of edges in every direction is the same are presented.