Partitions and balanced matchings of an -dimensional cube

A non-empty set A ⊆ X = X 1 × ⋯ × X n is a box if A = A 1 × ⋯ × A n and A i ⊆ X i for each i ∈ [ n ] . Two boxes A , B ⊂ X are dichotomous if A i = X i ∖ B i for some i ∈ [ n ] . Using a cube tiling code of R n designed by Lagarias and Shor, a certain class of partitions of an n -dimensional cube into 2 n 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.